Theorem isrisc 32954

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 32953	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
4	1, 2, 3	mp2an 704	1 ⊢ (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  (class class class)co 6549  RingOpscrngo 32863   RngIso crngiso 32930   ≃_𝑟 crisc 32931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-risc 32952

This theorem is referenced by:  riscer  32957