Theorem isriscg 32953

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 2676	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∈ RingOps ↔ 𝑅 ∈ RingOps))
2	1	anbi1d 737	. . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps)))
3		oveq1 6556	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 RngIso 𝑠) = (𝑅 RngIso 𝑠))
4	3	eleq2d 2673	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑓 ∈ (𝑟 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑠)))
5	4	exbidv 1837	. . 3 ⊢ (𝑟 = 𝑅 → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)))
6	2, 5	anbi12d 743	. 2 ⊢ (𝑟 = 𝑅 → (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠))))
7		eleq1 2676	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ RingOps ↔ 𝑆 ∈ RingOps))
8	7	anbi2d 736	. . 3 ⊢ (𝑠 = 𝑆 → ((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ↔ (𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps)))
9		oveq2 6557	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑅 RngIso 𝑠) = (𝑅 RngIso 𝑆))
10	9	eleq2d 2673	. . . 4 ⊢ (𝑠 = 𝑆 → (𝑓 ∈ (𝑅 RngIso 𝑠) ↔ 𝑓 ∈ (𝑅 RngIso 𝑆)))
11	10	exbidv 1837	. . 3 ⊢ (𝑠 = 𝑆 → (∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠) ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))
12	8, 11	anbi12d 743	. 2 ⊢ (𝑠 = 𝑆 → (((𝑅 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑠)) ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))
13		df-risc 32952	. 2 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
14	6, 12, 13	brabg 4919	1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅 ≃𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  (class class class)co 6549  RingOpscrngo 32863   RngIso crngiso 32930   ≃_𝑟 crisc 32931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-iota 5768  df-fv 5812  df-ov 6552  df-risc 32952

This theorem is referenced by:  isrisc  32954  risc  32955