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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 RngIso 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrisc.1 | . 2 ⊢ 𝑅 ∈ V | |
2 | isrisc.2 | . 2 ⊢ 𝑆 ∈ V | |
3 | isriscg 36557 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∃wex 1781 ∈ wcel 2106 Vcvv 3466 class class class wbr 5132 (class class class)co 7384 RingOpscrngo 36467 RngIso crngiso 36534 ≃𝑟 crisc 36535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pr 5411 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-opab 5195 df-iota 6475 df-fv 6531 df-ov 7387 df-risc 36556 |
This theorem is referenced by: riscer 36561 |
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