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| Mirrors > Home > MPE Home > Th. List > Mathboxes > isrisc | Structured version Visualization version GIF version | ||
| Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| Ref | Expression |
|---|---|
| isrisc.1 | ⊢ 𝑅 ∈ V |
| isrisc.2 | ⊢ 𝑆 ∈ V |
| Ref | Expression |
|---|---|
| isrisc | ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isrisc.1 | . 2 ⊢ 𝑅 ∈ V | |
| 2 | isrisc.2 | . 2 ⊢ 𝑆 ∈ V | |
| 3 | isriscg 37991 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 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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