Theorem isriscg 39011

Description: Obsolete theorem, use isbrric2 20758 instead. The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
isriscg	⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 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.

Step	Hyp	Ref	Expression
1		eleq1 2848	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
2	1	anbi1d 643	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3		oveq1 7427	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑠))
4	3	eleq2d 2846	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑠)))
5	4	exbidv 1953	. . 3 ⊢ (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠)))
6	2, 5	anbi12d 644	. 2 ⊢ (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠))))
7		eleq1 2848	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
8	7	anbi2d 642	. . 3 ⊢ (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9		oveq2 7428	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑅 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑆))
10	9	eleq2d 2846	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
11	10	exbidv 1953	. . 3 ⊢ (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
12	8, 11	anbi12d 644	. 2 ⊢ (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))
13		df-risc 39010	. 2 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
14	6, 12, 13	brabg 5513	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingOpsIso 𝑆))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   class class class wbr 5102  (class class class)co 7420  RingOpscrngo 38921   RingOpsIso crngoiso 38988   ≃_𝑟 crisc 38989

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423  df-risc 39010

This theorem is used by:  isrisc  39012  risc  39013