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Mirrors > Home > MPE Home > Th. List > Mathboxes > isriscg | Structured version Visualization version GIF version |
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
isriscg | ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2900 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps)) | |
2 | 1 | anbi1d 631 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps))) |
3 | oveq1 7157 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 RngIso 𝑠) = (𝑅 RngIso 𝑠)) | |
4 | 3 | eleq2d 2898 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑠))) |
5 | 4 | exbidv 1918 | . . 3 ⊢ (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠))) |
6 | 2, 5 | anbi12d 632 | . 2 ⊢ (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)))) |
7 | eleq1 2900 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps)) | |
8 | 7 | anbi2d 630 | . . 3 ⊢ (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps))) |
9 | oveq2 7158 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝑅 RngIso 𝑠) = (𝑅 RngIso 𝑆)) | |
10 | 9 | eleq2d 2898 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑆))) |
11 | 10 | exbidv 1918 | . . 3 ⊢ (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) |
12 | 8, 11 | anbi12d 632 | . 2 ⊢ (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) |
13 | df-risc 35255 | . 2 ⊢ ≃𝑟 = {〈𝑟, 𝑠〉 ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))} | |
14 | 6, 12, 13 | brabg 5419 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 RingOpscrngo 35166 RngIso crngiso 35233 ≃𝑟 crisc 35234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-iota 6309 df-fv 6358 df-ov 7153 df-risc 35255 |
This theorem is referenced by: isrisc 35257 risc 35258 |
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