Theorem isrisc 37157

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 37156	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
4	1, 2, 3	mp2an 689	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∃wex 1780   ∈ wcel 2105  Vcvv 3473   class class class wbr 5148  (class class class)co 7412  RingOpscrngo 37066   RingOpsIso crngoiso 37133   ≃_𝑟 crisc 37134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7415  df-risc 37155

This theorem is referenced by:  riscer  37160