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Theorem isrisc 37992
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 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓

Proof of Theorem isrisc
StepHypRef Expression
1 isrisc.1 . 2 𝑅 ∈ V
2 isrisc.2 . 2 𝑆 ∈ V
3 isriscg 37991 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
41, 2, 3mp2an 692 1 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  (class class class)co 7431  RingOpscrngo 37901   RingOpsIso crngoiso 37968  𝑟 crisc 37969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569  df-ov 7434  df-risc 37990
This theorem is referenced by:  riscer  37995
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