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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 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex2 2818 | . . 3 ⊢ (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)) | |
| 2 | risc 37993 | . . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))) | |
| 3 | 1, 2 | imbitrrid 246 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RingOpsIso 𝑆) → 𝑅 ≃𝑟 𝑆)) |
| 4 | 3 | 3impia 1118 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsIso 𝑆)) → 𝑅 ≃𝑟 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∃wex 1779 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 RingOpscrngo 37901 RingOpsIso crngoiso 37968 ≃𝑟 crisc 37969 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 df-ov 7434 df-risc 37990 |
| This theorem is referenced by: riscer 37995 |
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