Theorem riscer 33787

Description: Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
riscer	⊢ ≃𝑟 Er dom ≃𝑟

Proof of Theorem riscer

Dummy variables 𝑓 𝑔 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-risc 33782	. . 3 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
2	1	relopabi 5245	. 2 ⊢ Rel ≃𝑟
3		eqid 2622	. 2 ⊢ dom ≃𝑟 = dom ≃𝑟
4		vex 3203	. . . . . . 7 ⊢ 𝑟 ∈ V
5		vex 3203	. . . . . . 7 ⊢ 𝑠 ∈ V
6	4, 5	isrisc 33784	. . . . . 6 ⊢ (𝑟 ≃𝑟 𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)))
7		rngoisocnv 33780	. . . . . . . . . 10 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RngIso 𝑠)) → ◡𝑓 ∈ (𝑠 RngIso 𝑟))
8	7	3expia 1267	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → ◡𝑓 ∈ (𝑠 RngIso 𝑟)))
9		risci 33786	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ ◡𝑓 ∈ (𝑠 RngIso 𝑟)) → 𝑠 ≃𝑟 𝑟)
10	9	3expia 1267	. . . . . . . . . 10 ⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠 ≃𝑟 𝑟))
11	10	ancoms 469	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠 ≃𝑟 𝑟))
12	8, 11	syld 47	. . . . . . . 8 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠 ≃𝑟 𝑟))
13	12	exlimdv 1861	. . . . . . 7 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠 ≃𝑟 𝑟))
14	13	imp 445	. . . . . 6 ⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑠 ≃𝑟 𝑟)
15	6, 14	sylbi 207	. . . . 5 ⊢ (𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟)
16		vex 3203	. . . . . . 7 ⊢ 𝑡 ∈ V
17	5, 16	isrisc 33784	. . . . . 6 ⊢ (𝑠 ≃𝑟 𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
18		eeanv 2182	. . . . . . . . . . 11 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
19		rngoisoco 33781	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡))) → (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡))
20	19	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡)))
21		risci 33786	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡)
22	21	3expia 1267	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟 ≃𝑟 𝑡))
23	22	3adant2 1080	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟 ≃𝑟 𝑡))
24	20, 23	syld 47	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
25	24	exlimdvv 1862	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → (∃𝑓∃𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
26	18, 25	syl5bir 233	. . . . . . . . . 10 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
27	26	3expb 1266	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
28	27	adantlr 751	. . . . . . . 8 ⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
29	28	imp 445	. . . . . . 7 ⊢ ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) ∧ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟 ≃𝑟 𝑡)
30	29	an4s 869	. . . . . 6 ⊢ ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ∧ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟 ≃𝑟 𝑡)
31	6, 17, 30	syl2anb 496	. . . . 5 ⊢ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)
32	15, 31	pm3.2i 471	. . . 4 ⊢ ((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
33	32	ax-gen 1722	. . 3 ⊢ ∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
34	33	gen2 1723	. 2 ⊢ ∀𝑟∀𝑠∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
35		dfer2 7743	. 2 ⊢ ( ≃𝑟 Er dom ≃𝑟 ↔ (Rel ≃𝑟 ∧ dom ≃𝑟 = dom ≃𝑟 ∧ ∀𝑟∀𝑠∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))))
36	2, 3, 34, 35	mpbir3an 1244	1 ⊢ ≃𝑟 Er dom ≃𝑟

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   class class class wbr 4653  ^◡ccnv 5113  dom cdm 5114   ∘ ccom 5118  Rel wrel 5119  (class class class)co 6650   Er wer 7739  RingOpscrngo 33693   RngIso crngiso 33760   ≃_𝑟 crisc 33761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-grpo 27347  df-gid 27348  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-rngohom 33762  df-rngoiso 33775  df-risc 33782

This theorem is referenced by: (None)