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Theorem isriscg 36446
Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
isriscg ((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
Distinct variable groups:   𝑅,𝑓   𝑆,𝑓
Allowed substitution hints:   𝐴(𝑓)   𝐵(𝑓)

Proof of Theorem isriscg
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2826 . . . 4 (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
21anbi1d 631 . . 3 (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3 oveq1 7365 . . . . 5 (𝑟 = 𝑅 → (𝑟 RngIso 𝑠) = (𝑅 RngIso 𝑠))
43eleq2d 2824 . . . 4 (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑠)))
54exbidv 1925 . . 3 (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)))
62, 5anbi12d 632 . 2 (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠))))
7 eleq1 2826 . . . 4 (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
87anbi2d 630 . . 3 (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9 oveq2 7366 . . . . 5 (𝑠 = 𝑆 → (𝑅 RngIso 𝑠) = (𝑅 RngIso 𝑆))
109eleq2d 2824 . . . 4 (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑆)))
1110exbidv 1925 . . 3 (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
128, 11anbi12d 632 . 2 (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
13 df-risc 36445 . 2 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
146, 12, 13brabg 5497 1 ((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107   class class class wbr 5106  (class class class)co 7358  RingOpscrngo 36356   RngIso crngiso 36423  𝑟 crisc 36424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-iota 6449  df-fv 6505  df-ov 7361  df-risc 36445
This theorem is referenced by:  isrisc  36447  risc  36448
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