Theorem isriscg 36852

Description: The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)

Assertion

Ref	Expression
isriscg	⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 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.

Step	Hyp	Ref	Expression
1		eleq1 2822	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
2	1	anbi1d 631	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3		oveq1 7416	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 RngIso 𝑠) = (𝑅 RngIso 𝑠))
4	3	eleq2d 2820	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑠)))
5	4	exbidv 1925	. . 3 ⊢ (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)))
6	2, 5	anbi12d 632	. 2 ⊢ (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠))))
7		eleq1 2822	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
8	7	anbi2d 630	. . 3 ⊢ (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9		oveq2 7417	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑅 RngIso 𝑠) = (𝑅 RngIso 𝑆))
10	9	eleq2d 2820	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑆)))
11	10	exbidv 1925	. . 3 ⊢ (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
12	8, 11	anbi12d 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 36851	. 2 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
14	6, 12, 13	brabg 5540	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107   class class class wbr 5149  (class class class)co 7409  RingOpscrngo 36762   RngIso crngiso 36829   ≃_𝑟 crisc 36830

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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

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 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6496  df-fv 6552  df-ov 7412  df-risc 36851

This theorem is referenced by:  isrisc  36853  risc  36854