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Theorem riscer 35147
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 35142 . . 3 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
21relopabi 5687 . 2 Rel ≃𝑟
3 eqid 2818 . 2 dom ≃𝑟 = dom ≃𝑟
4 vex 3495 . . . . . . 7 𝑟 ∈ V
5 vex 3495 . . . . . . 7 𝑠 ∈ V
64, 5isrisc 35144 . . . . . 6 (𝑟𝑟 𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)))
7 rngoisocnv 35140 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑓 ∈ (𝑠 RngIso 𝑟))
873expia 1113 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑓 ∈ (𝑠 RngIso 𝑟)))
9 risci 35146 . . . . . . . . . . 11 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ 𝑓 ∈ (𝑠 RngIso 𝑟)) → 𝑠𝑟 𝑟)
1093expia 1113 . . . . . . . . . 10 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠𝑟 𝑟))
1110ancoms 459 . . . . . . . . 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 1925 . . . . . . 7 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠𝑟 𝑟))
1413imp 407 . . . . . 6 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑠𝑟 𝑟)
156, 14sylbi 218 . . . . 5 (𝑟𝑟 𝑠𝑠𝑟 𝑟)
16 vex 3495 . . . . . . 7 𝑡 ∈ V
175, 16isrisc 35144 . . . . . 6 (𝑠𝑟 𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
18 exdistrv 1947 . . . . . . . . . . 11 (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
19 rngoisoco 35141 . . . . . . . . . . . . . 14 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡))) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡))
2019ex 413 . . . . . . . . . . . . 13 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)))
21 risci 35146 . . . . . . . . . . . . . . 15 ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)) → 𝑟𝑟 𝑡)
22213expia 1113 . . . . . . . . . . . . . 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 1926 . . . . . . . . . . 11 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2618, 25syl5bir 244 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
27263expb 1112 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2827adantlr 711 . . . . . . . 8 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2928imp 407 . . . . . . 7 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) ∧ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
3029an4s 656 . . . . . 6 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ∧ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
316, 17, 30syl2anb 597 . . . . 5 ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡)
3215, 31pm3.2i 471 . . . 4 ((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3332ax-gen 1787 . . 3 𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3433gen2 1788 . 2 𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
35 dfer2 8279 . 2 ( ≃𝑟 Er dom ≃𝑟 ↔ (Rel ≃𝑟 ∧ dom ≃𝑟 = dom ≃𝑟 ∧ ∀𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))))
362, 3, 34, 35mpbir3an 1333 1 𝑟 Er dom ≃𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105   class class class wbr 5057  ccnv 5547  dom cdm 5548  ccom 5552  Rel wrel 5553  (class class class)co 7145   Er wer 8275  RingOpscrngo 35053   RngIso crngiso 35120  𝑟 crisc 35121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-er 8278  df-map 8397  df-grpo 28197  df-gid 28198  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054  df-rngohom 35122  df-rngoiso 35135  df-risc 35142
This theorem is referenced by: (None)
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