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Mirrors > Home > MPE Home > Th. List > Mathboxes > risci | Structured version Visualization version GIF version |
Description: Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.) |
Ref | Expression |
---|---|
risci | ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅 ≃𝑟 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex2 3517 | . . 3 ⊢ (𝐹 ∈ (𝑅 RngIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)) | |
2 | risc 35258 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) | |
3 | 1, 2 | syl5ibr 248 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝑅 ≃𝑟 𝑆)) |
4 | 3 | 3impia 1113 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅 ≃𝑟 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∃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: riscer 35260 |
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