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Theorem riscer 34069
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.
StepHypRef Expression
1 df-risc 34064 . . 3 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
21relopabi 5389 . 2 Rel ≃𝑟
3 eqid 2748 . 2 dom ≃𝑟 = dom ≃𝑟
4 vex 3331 . . . . . . 7 𝑟 ∈ V
5 vex 3331 . . . . . . 7 𝑠 ∈ V
64, 5isrisc 34066 . . . . . 6 (𝑟𝑟 𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)))
7 rngoisocnv 34062 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑓 ∈ (𝑠 RngIso 𝑟))
873expia 1114 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑓 ∈ (𝑠 RngIso 𝑟)))
9 risci 34068 . . . . . . . . . . 11 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ 𝑓 ∈ (𝑠 RngIso 𝑟)) → 𝑠𝑟 𝑟)
1093expia 1114 . . . . . . . . . 10 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠𝑟 𝑟))
1110ancoms 468 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠𝑟 𝑟))
128, 11syld 47 . . . . . . . 8 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠𝑟 𝑟))
1312exlimdv 1998 . . . . . . 7 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠𝑟 𝑟))
1413imp 444 . . . . . 6 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑠𝑟 𝑟)
156, 14sylbi 207 . . . . 5 (𝑟𝑟 𝑠𝑠𝑟 𝑟)
16 vex 3331 . . . . . . 7 𝑡 ∈ V
175, 16isrisc 34066 . . . . . 6 (𝑠𝑟 𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
18 eeanv 2315 . . . . . . . . . . 11 (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
19 rngoisoco 34063 . . . . . . . . . . . . . 14 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡))) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡))
2019ex 449 . . . . . . . . . . . . 13 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)))
21 risci 34068 . . . . . . . . . . . . . . 15 ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)) → 𝑟𝑟 𝑡)
22213expia 1114 . . . . . . . . . . . . . 14 ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟𝑟 𝑡))
23223adant2 1123 . . . . . . . . . . . . 13 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟𝑟 𝑡))
2420, 23syld 47 . . . . . . . . . . . 12 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2524exlimdvv 1999 . . . . . . . . . . 11 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2618, 25syl5bir 233 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
27263expb 1113 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2827adantlr 753 . . . . . . . 8 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2928imp 444 . . . . . . 7 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) ∧ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
3029an4s 904 . . . . . 6 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ∧ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
316, 17, 30syl2anb 497 . . . . 5 ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡)
3215, 31pm3.2i 470 . . . 4 ((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3332ax-gen 1859 . . 3 𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3433gen2 1860 . 2 𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
35 dfer2 7900 . 2 ( ≃𝑟 Er dom ≃𝑟 ↔ (Rel ≃𝑟 ∧ dom ≃𝑟 = dom ≃𝑟 ∧ ∀𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))))
362, 3, 34, 35mpbir3an 1405 1 𝑟 Er dom ≃𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wal 1618   = wceq 1620  wex 1841  wcel 2127   class class class wbr 4792  ccnv 5253  dom cdm 5254  ccom 5258  Rel wrel 5259  (class class class)co 6801   Er wer 7896  RingOpscrngo 33975   RngIso crngiso 34042  𝑟 crisc 34043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-er 7899  df-map 8013  df-grpo 27627  df-gid 27628  df-ablo 27679  df-ass 33924  df-exid 33926  df-mgmOLD 33930  df-sgrOLD 33942  df-mndo 33948  df-rngo 33976  df-rngohom 34044  df-rngoiso 34057  df-risc 34064
This theorem is referenced by: (None)
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