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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 36142 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))) | |
4 | 1, 2, 3 | mp2an 689 | 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 1782 ∈ wcel 2106 Vcvv 3432 class class class wbr 5074 (class class class)co 7275 RingOpscrngo 36052 RngIso crngiso 36119 ≃𝑟 crisc 36120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-iota 6391 df-fv 6441 df-ov 7278 df-risc 36141 |
This theorem is referenced by: riscer 36146 |
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