Theorem riscer 32957

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 32952	. . 3 ⊢ ≃𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
2	1	relopabi 5167	. 2 ⊢ Rel ≃𝑟
3		eqid 2610	. 2 ⊢ dom ≃𝑟 = dom ≃𝑟
4		vex 3176	. . . . . . 7 ⊢ 𝑟 ∈ V
5		vex 3176	. . . . . . 7 ⊢ 𝑠 ∈ V
6	4, 5	isrisc 32954	. . . . . 6 ⊢ (𝑟 ≃𝑟 𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)))
7		rngoisocnv 32950	. . . . . . . . . 10 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RngIso 𝑠)) → ◡𝑓 ∈ (𝑠 RngIso 𝑟))
8	7	3expia 1259	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → ◡𝑓 ∈ (𝑠 RngIso 𝑟)))
9		risci 32956	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ ◡𝑓 ∈ (𝑠 RngIso 𝑟)) → 𝑠 ≃𝑟 𝑟)
10	9	3expia 1259	. . . . . . . . . 10 ⊢ ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠 ≃𝑟 𝑟))
11	10	ancoms 468	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (◡𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠 ≃𝑟 𝑟))
12	8, 11	syld 46	. . . . . . . 8 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠 ≃𝑟 𝑟))
13	12	exlimdv 1848	. . . . . . 7 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠 ≃𝑟 𝑟))
14	13	imp 444	. . . . . 6 ⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑠 ≃𝑟 𝑟)
15	6, 14	sylbi 206	. . . . 5 ⊢ (𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟)
16		vex 3176	. . . . . . 7 ⊢ 𝑡 ∈ V
17	5, 16	isrisc 32954	. . . . . 6 ⊢ (𝑠 ≃𝑟 𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
18		eeanv 2170	. . . . . . . . . . 11 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
19		rngoisoco 32951	. . . . . . . . . . . . . 14 ⊢ (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡))) → (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡))
20	19	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡)))
21		risci 32956	. . . . . . . . . . . . . . 15 ⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡)
22	21	3expia 1259	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟 ≃𝑟 𝑡))
23	22	3adant2 1073	. . . . . . . . . . . . 13 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔 ∘ 𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟 ≃𝑟 𝑡))
24	20, 23	syld 46	. . . . . . . . . . . 12 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
25	24	exlimdvv 1849	. . . . . . . . . . 11 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → (∃𝑓∃𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
26	18, 25	syl5bir 232	. . . . . . . . . 10 ⊢ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
27	26	3expb 1258	. . . . . . . . 9 ⊢ ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟 ≃𝑟 𝑡))
28	27	adantlr 747	. . . . . . . 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 444	. . . . . . 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 865	. . . . . 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 495	. . . . 5 ⊢ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡)
32	15, 31	pm3.2i 470	. . . 4 ⊢ ((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
33	32	ax-gen 1713	. . 3 ⊢ ∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
34	33	gen2 1714	. 2 ⊢ ∀𝑟∀𝑠∀𝑡((𝑟 ≃𝑟 𝑠 → 𝑠 ≃𝑟 𝑟) ∧ ((𝑟 ≃𝑟 𝑠 ∧ 𝑠 ≃𝑟 𝑡) → 𝑟 ≃𝑟 𝑡))
35		dfer2 7630	. 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 1237	1 ⊢ ≃𝑟 Er dom ≃𝑟

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Rel wrel 5043  (class class class)co 6549   Er wer 7626  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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847

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-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-map 7746  df-grpo 26731  df-gid 26732  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-rngohom 32932  df-rngoiso 32945  df-risc 32952

This theorem is referenced by: (None)