Theorem isrisc 38887

Description: Obsolete theorem, use isbrric2 20733 instead. The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

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 38886	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
4	1, 2, 3	mp2an 705	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   class class class wbr 5103  (class class class)co 7412  RingOpscrngo 38796   RingOpsIso crngoiso 38863   ≃_𝑟 crisc 38864

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6487  df-fv 6539  df-ov 7415  df-risc 38885

This theorem is used by:  riscer  38890