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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 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 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 36557 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
41, 2, 3mp2an 690 1 (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
Colors of variables: wff setvar class
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
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