Theorem isrisc 36558

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 RngIso 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 36557	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
4	1, 2, 3	mp2an 690	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  Vcvv 3466   class class class wbr 5132  (class class class)co 7384  RingOpscrngo 36467   RngIso crngiso 36534   ≃_𝑟 crisc 36535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5283  ax-nul 5290  ax-pr 5411

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3426  df-v 3468  df-dif 3938  df-un 3940  df-in 3942  df-ss 3952  df-nul 4310  df-if 4514  df-sn 4614  df-pr 4616  df-op 4620  df-uni 4893  df-br 5133  df-opab 5195  df-iota 6475  df-fv 6531  df-ov 7387  df-risc 36556

This theorem is referenced by:  riscer  36561