Theorem isriscg 38676

Description: Obsolete theorem, use isbrric2 20638 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 2854	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
2	1	anbi1d 643	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3		oveq1 7430	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑠))
4	3	eleq2d 2852	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑠)))
5	4	exbidv 1954	. . 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 2854	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
8	7	anbi2d 642	. . 3 ⊢ (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9		oveq2 7431	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑅 RingOpsIso 𝑠) = (𝑅 RingOpsIso 𝑆))
10	9	eleq2d 2852	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RingOpsIso 𝑠) ↔ 𝑓 ∈ (𝑅 RingOpsIso 𝑆)))
11	10	exbidv 1954	. . 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 38675	. 2 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RingOpsIso 𝑠))}
14	6, 12, 13	brabg 5529	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 1570  ∃wex 1812   ∈ wcel 2146   class class class wbr 5114  (class class class)co 7423  RingOpscrngo 38586   RingOpsIso crngoiso 38653   ≃_𝑟 crisc 38654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-iota 6499  df-fv 6551  df-ov 7426  df-risc 38675

This theorem is used by:  isrisc  38677  risc  38678