Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmdprd Unicode version

Theorem dmdprd 15236
 Description: The domain of definition of the internal direct product, which states that is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dmdprd.z Cntz
dmdprd.0
dmdprd.k mrClsSubGrp
Assertion
Ref Expression
dmdprd DProd SubGrp
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem dmdprd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . . . . 5 SubGrp
21a1i 10 . . . 4 SubGrp
3 fex 5749 . . . . . . 7 SubGrp
43expcom 424 . . . . . 6 SubGrp
54adantr 451 . . . . 5 SubGrp
65adantrd 454 . . . 4 SubGrp
7 df-sbc 2992 . . . . . 6 SubGrp SubGrp
8 simpr 447 . . . . . . 7
9 simpr 447 . . . . . . . . . 10
109dmeqd 4881 . . . . . . . . . . 11
11 simplr 731 . . . . . . . . . . 11
1210, 11eqtrd 2315 . . . . . . . . . 10
139, 12feq12d 5381 . . . . . . . . 9 SubGrp SubGrp
1412difeq1d 3293 . . . . . . . . . . . 12
159fveq1d 5527 . . . . . . . . . . . . 13
169fveq1d 5527 . . . . . . . . . . . . . 14
1716fveq2d 5529 . . . . . . . . . . . . 13
1815, 17sseq12d 3207 . . . . . . . . . . . 12
1914, 18raleqbidv 2748 . . . . . . . . . . 11
209imaeq1d 5011 . . . . . . . . . . . . . . . 16
2114imaeq2d 5012 . . . . . . . . . . . . . . . 16
2220, 21eqtrd 2315 . . . . . . . . . . . . . . 15
2322unieqd 3838 . . . . . . . . . . . . . 14
2423fveq2d 5529 . . . . . . . . . . . . 13
2515, 24ineq12d 3371 . . . . . . . . . . . 12
2625eqeq1d 2291 . . . . . . . . . . 11
2719, 26anbi12d 691 . . . . . . . . . 10
2812, 27raleqbidv 2748 . . . . . . . . 9
2913, 28anbi12d 691 . . . . . . . 8 SubGrp SubGrp
3029adantlr 695 . . . . . . 7 SubGrp SubGrp
318, 30sbcied 3027 . . . . . 6 SubGrp SubGrp
327, 31syl5bbr 250 . . . . 5 SubGrp SubGrp
3332ex 423 . . . 4 SubGrp SubGrp
342, 6, 33pm5.21ndd 343 . . 3 SubGrp SubGrp
3534anbi2d 684 . 2 SubGrp SubGrp
36 df-br 4024 . . 3 DProd DProd
37 fvex 5539 . . . . . . . . . . . 12
3837rgenw 2610 . . . . . . . . . . 11
39 ixpexg 6840 . . . . . . . . . . 11
4038, 39ax-mp 8 . . . . . . . . . 10
4140rabex 4165 . . . . . . . . 9
4241mptex 5746 . . . . . . . 8 g
4342rnex 4942 . . . . . . 7 g
4443rgen2w 2611 . . . . . 6 SubGrp Cntz mrClsSubGrp g
45 df-dprd 15233 . . . . . . 7 DProd SubGrp Cntz mrClsSubGrp g
4645fmpt2x 6190 . . . . . 6 SubGrp Cntz mrClsSubGrp g DProd SubGrp Cntz mrClsSubGrp
4744, 46mpbi 199 . . . . 5 DProd SubGrp Cntz mrClsSubGrp
4847fdmi 5394 . . . 4 DProd SubGrp Cntz mrClsSubGrp
4948eleq2i 2347 . . 3 DProd SubGrp Cntz mrClsSubGrp
50 fveq2 5525 . . . . . . 7 SubGrp SubGrp
51 feq3 5377 . . . . . . 7 SubGrp SubGrp SubGrp SubGrp
5250, 51syl 15 . . . . . 6 SubGrp SubGrp
53 fveq2 5525 . . . . . . . . . . . 12 Cntz Cntz
54 dmdprd.z . . . . . . . . . . . 12 Cntz
5553, 54syl6eqr 2333 . . . . . . . . . . 11 Cntz
5655fveq1d 5527 . . . . . . . . . 10 Cntz
5756sseq2d 3206 . . . . . . . . 9 Cntz
5857ralbidv 2563 . . . . . . . 8 Cntz
5950fveq2d 5529 . . . . . . . . . . . 12 mrClsSubGrp mrClsSubGrp
60 dmdprd.k . . . . . . . . . . . 12 mrClsSubGrp
6159, 60syl6eqr 2333 . . . . . . . . . . 11 mrClsSubGrp
6261fveq1d 5527 . . . . . . . . . 10 mrClsSubGrp
6362ineq2d 3370 . . . . . . . . 9 mrClsSubGrp
64 fveq2 5525 . . . . . . . . . . 11
65 dmdprd.0 . . . . . . . . . . 11
6664, 65syl6eqr 2333 . . . . . . . . . 10
6766sneqd 3653 . . . . . . . . 9
6863, 67eqeq12d 2297 . . . . . . . 8 mrClsSubGrp
6958, 68anbi12d 691 . . . . . . 7 Cntz mrClsSubGrp
7069ralbidv 2563 . . . . . 6 Cntz mrClsSubGrp
7152, 70anbi12d 691 . . . . 5 SubGrp Cntz mrClsSubGrp SubGrp
7271abbidv 2397 . . . 4 SubGrp Cntz mrClsSubGrp SubGrp
7372opeliunxp2 4824 . . 3 SubGrp Cntz mrClsSubGrp SubGrp
7436, 49, 733bitri 262 . 2 DProd SubGrp
75 3anass 938 . 2 SubGrp SubGrp
7635, 74, 753bitr4g 279 1 DProd SubGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   w3a 934   wceq 1623   wcel 1684  cab 2269  wral 2543  crab 2547  cvv 2788  wsbc 2991   cdif 3149   cin 3151   wss 3152  csn 3640  cop 3643  cuni 3827  ciun 3905   class class class wbr 4023   cmpt 4077   cxp 4687  ccnv 4688   cdm 4689   crn 4690  cima 4692  wf 5251  cfv 5255  (class class class)co 5858  cixp 6817  cfn 6863  c0g 13400   g cgsu 13401  mrClscmrc 13485  cgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231 This theorem is referenced by:  dmdprdd  15237  dprdgrp  15240  dprdf  15241  dprdcntz  15243  dprddisj  15244  dprdres  15263  subgdmdprd  15269 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-dprd 15233
 Copyright terms: Public domain W3C validator