Theorem isrisc 37971

Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
isrisc.1	⊢ 𝑅 ∈ V
isrisc.2	⊢ 𝑆 ∈ V

Assertion

Ref	Expression
isrisc	⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

Distinct variable groups:   𝑅,𝑓   𝑆,𝑓

Proof of Theorem isrisc

Step	Hyp	Ref	Expression
1		isrisc.1	. 2 ⊢ 𝑅 ∈ V
2		isrisc.2	. 2 ⊢ 𝑆 ∈ V
3		isriscg 37970	. 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)))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1775   ∈ wcel 2105  Vcvv 3477   class class class wbr 5147  (class class class)co 7430  RingOpscrngo 37880   RingOpsIso crngoiso 37947   ≃_𝑟 crisc 37948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-iota 6515  df-fv 6570  df-ov 7433  df-risc 37969

This theorem is referenced by:  riscer  37974