Theorem risc 38046

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

Assertion

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

Distinct variable groups:   𝑅,𝑓   𝑆,𝑓

Proof of Theorem risc

Step	Hyp	Ref	Expression
1		isriscg 38044	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
2	1	bianabs 541	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 ≃𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2113   class class class wbr 5093  (class class class)co 7352  RingOpscrngo 37954   RingOpsIso crngoiso 38021   ≃_𝑟 crisc 38022

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-iota 6442  df-fv 6494  df-ov 7355  df-risc 38043

This theorem is referenced by:  risci  38047