Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  riscer Structured version   Visualization version   GIF version

Theorem riscer 36904
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 36899 . . 3 𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}
21relopabiv 5821 . 2 Rel ≃𝑟
3 eqid 2733 . 2 dom ≃𝑟 = dom ≃𝑟
4 vex 3479 . . . . . . 7 𝑟 ∈ V
5 vex 3479 . . . . . . 7 𝑠 ∈ V
64, 5isrisc 36901 . . . . . 6 (𝑟𝑟 𝑠 ↔ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)))
7 rngoisocnv 36897 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑓 ∈ (𝑠 RngIso 𝑟))
873expia 1122 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑓 ∈ (𝑠 RngIso 𝑟)))
9 risci 36903 . . . . . . . . . . 11 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps ∧ 𝑓 ∈ (𝑠 RngIso 𝑟)) → 𝑠𝑟 𝑟)
1093expia 1122 . . . . . . . . . 10 ((𝑠 ∈ RingOps ∧ 𝑟 ∈ RingOps) → (𝑓 ∈ (𝑠 RngIso 𝑟) → 𝑠𝑟 𝑟))
1110ancoms 460 . . . . . . . . 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 1937 . . . . . . 7 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) → (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) → 𝑠𝑟 𝑟))
1413imp 408 . . . . . 6 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) → 𝑠𝑟 𝑟)
156, 14sylbi 216 . . . . 5 (𝑟𝑟 𝑠𝑠𝑟 𝑟)
16 vex 3479 . . . . . . 7 𝑡 ∈ V
175, 16isrisc 36901 . . . . . 6 (𝑠𝑟 𝑡 ↔ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
18 exdistrv 1960 . . . . . . . . . . 11 (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) ↔ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)))
19 rngoisoco 36898 . . . . . . . . . . . . . 14 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ (𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡))) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡))
2019ex 414 . . . . . . . . . . . . 13 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)))
21 risci 36903 . . . . . . . . . . . . . . 15 ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps ∧ (𝑔𝑓) ∈ (𝑟 RngIso 𝑡)) → 𝑟𝑟 𝑡)
22213expia 1122 . . . . . . . . . . . . . 14 ((𝑟 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((𝑔𝑓) ∈ (𝑟 RngIso 𝑡) → 𝑟𝑟 𝑡))
23223adant2 1132 . . . . . . . . . . . . 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 1938 . . . . . . . . . . 11 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → (∃𝑓𝑔(𝑓 ∈ (𝑟 RngIso 𝑠) ∧ 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2618, 25biimtrrid 242 . . . . . . . . . 10 ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
27263expb 1121 . . . . . . . . 9 ((𝑟 ∈ RingOps ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2827adantlr 714 . . . . . . . 8 (((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) → ((∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡)) → 𝑟𝑟 𝑡))
2928imp 408 . . . . . . 7 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ (𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps)) ∧ (∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
3029an4s 659 . . . . . 6 ((((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠)) ∧ ((𝑠 ∈ RingOps ∧ 𝑡 ∈ RingOps) ∧ ∃𝑔 𝑔 ∈ (𝑠 RngIso 𝑡))) → 𝑟𝑟 𝑡)
316, 17, 30syl2anb 599 . . . . 5 ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡)
3215, 31pm3.2i 472 . . . 4 ((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3332ax-gen 1798 . . 3 𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
3433gen2 1799 . 2 𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))
35 dfer2 8704 . 2 ( ≃𝑟 Er dom ≃𝑟 ↔ (Rel ≃𝑟 ∧ dom ≃𝑟 = dom ≃𝑟 ∧ ∀𝑟𝑠𝑡((𝑟𝑟 𝑠𝑠𝑟 𝑟) ∧ ((𝑟𝑟 𝑠𝑠𝑟 𝑡) → 𝑟𝑟 𝑡))))
362, 3, 34, 35mpbir3an 1342 1 𝑟 Er dom ≃𝑟
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107   class class class wbr 5149  ccnv 5676  dom cdm 5677  ccom 5681  Rel wrel 5682  (class class class)co 7409   Er wer 8700  RingOpscrngo 36810   RngIso crngiso 36877  𝑟 crisc 36878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-map 8822  df-grpo 29777  df-gid 29778  df-ablo 29829  df-ass 36759  df-exid 36761  df-mgmOLD 36765  df-sgrOLD 36777  df-mndo 36783  df-rngo 36811  df-rngohom 36879  df-rngoiso 36892  df-risc 36899
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator