Backport: docs: update docs to use gateway pattern (#9684)

This is an automated backport of #9591 to the release-v5.0 branch.

Co-authored-by: josh <144584931+dancer@users.noreply.github.com>

Author: vercel-ai-sdk[bot]

content/docs/02-foundations/05-streaming.mdx

@@ -46,11 +46,10 @@ While streaming interfaces can greatly enhance user experiences, especially with
 However, regardless of the speed of your model, the AI SDK is designed to make implementing streaming UIs as simple as possible. In the example below, we stream text generation from OpenAI's `gpt-4.1` in under 10 lines of code using the SDK's [`streamText`](/docs/reference/ai-sdk-core/stream-text) function:
 
 ```ts
-import { openai } from '@ai-sdk/openai';
 import { streamText } from 'ai';
 
 const { textStream } = streamText({
-  model: openai('gpt-4.1'),
+  model: 'openai/gpt-4.1',
   prompt: 'Write a poem about embedding models.',
 });
 

content/docs/03-ai-sdk-core/10-generating-structured-data.mdx

@@ -124,12 +124,11 @@ When you use the `array` output strategy, the schema specifies the shape of an a
 With `streamObject`, you can also stream the generated array elements using `elementStream`.
 
 ```ts highlight="7,18"
-import { openai } from '@ai-sdk/openai';
 import { streamObject } from 'ai';
 import { z } from 'zod';
 
 const { elementStream } = streamObject({
-  model: openai('gpt-4.1'),
+  model: 'openai/gpt-4.1',
   output: 'array',
   schema: z.object({
     name: z.string(),
@@ -176,11 +175,10 @@ You can use the `output` setting to set the output format to `no-schema` in thos
 and omit the schema parameter.
 
 ```ts highlight="6"
-import { openai } from '@ai-sdk/openai';
 import { generateObject } from 'ai';
 
 const { object } = await generateObject({
-  model: openai('gpt-4.1'),
+  model: 'openai/gpt-4.1',
   output: 'no-schema',
   prompt: 'Generate a lasagna recipe.',
 });
@@ -212,12 +210,12 @@ const { object } = await generateObject({
 You can access the reasoning used by the language model to generate the object via the `reasoning` property on the result. This property contains a string with the model's thought process, if available.
 
 ```ts
-import { openai, OpenAIResponsesProviderOptions } from '@ai-sdk/openai';
+import { OpenAIResponsesProviderOptions } from '@ai-sdk/openai';
 import { generateObject } from 'ai';
 import { z } from 'zod';
 
 const result = await generateObject({
-  model: openai('gpt-5'),
+  model: 'openai/gpt-5',
   schema: z.object({
     recipe: z.object({
       name: z.string(),