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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 33783 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 (class class class)co 6650 RingOpscrngo 33693 RngIso crngiso 33760 ≃𝑟 crisc 33761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-iota 5851 df-fv 5896 df-ov 6653 df-risc 33782 |
This theorem is referenced by: riscer 33787 |
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