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Theorem dmdprd 15551
 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 2956 . . . . 5 SubGrp
21a1i 11 . . . 4 SubGrp
3 fex 5961 . . . . . . 7 SubGrp
43expcom 425 . . . . . 6 SubGrp
54adantr 452 . . . . 5 SubGrp
65adantrd 455 . . . 4 SubGrp
7 df-sbc 3154 . . . . . 6 SubGrp SubGrp
8 simpr 448 . . . . . . 7
9 simpr 448 . . . . . . . . . 10
109dmeqd 5064 . . . . . . . . . . 11
11 simplr 732 . . . . . . . . . . 11
1210, 11eqtrd 2467 . . . . . . . . . 10
139, 12feq12d 5574 . . . . . . . . 9 SubGrp SubGrp
1412difeq1d 3456 . . . . . . . . . . . 12
159fveq1d 5722 . . . . . . . . . . . . 13
169fveq1d 5722 . . . . . . . . . . . . . 14
1716fveq2d 5724 . . . . . . . . . . . . 13
1815, 17sseq12d 3369 . . . . . . . . . . . 12
1914, 18raleqbidv 2908 . . . . . . . . . . 11
209, 14imaeq12d 5196 . . . . . . . . . . . . . . 15
2120unieqd 4018 . . . . . . . . . . . . . 14
2221fveq2d 5724 . . . . . . . . . . . . 13
2315, 22ineq12d 3535 . . . . . . . . . . . 12
2423eqeq1d 2443 . . . . . . . . . . 11
2519, 24anbi12d 692 . . . . . . . . . 10
2612, 25raleqbidv 2908 . . . . . . . . 9
2713, 26anbi12d 692 . . . . . . . 8 SubGrp SubGrp
2827adantlr 696 . . . . . . 7 SubGrp SubGrp
298, 28sbcied 3189 . . . . . 6 SubGrp SubGrp
307, 29syl5bbr 251 . . . . 5 SubGrp SubGrp
3130ex 424 . . . 4 SubGrp SubGrp
322, 6, 31pm5.21ndd 344 . . 3 SubGrp SubGrp
3332anbi2d 685 . 2 SubGrp SubGrp
34 df-br 4205 . . 3 DProd DProd
35 fvex 5734 . . . . . . . . . . . 12
3635rgenw 2765 . . . . . . . . . . 11
37 ixpexg 7078 . . . . . . . . . . 11
3836, 37ax-mp 8 . . . . . . . . . 10
3938rabex 4346 . . . . . . . . 9
4039mptex 5958 . . . . . . . 8 g
4140rnex 5125 . . . . . . 7 g
4241rgen2w 2766 . . . . . 6 SubGrp Cntz mrClsSubGrp g
43 df-dprd 15548 . . . . . . 7 DProd SubGrp Cntz mrClsSubGrp g
4443fmpt2x 6409 . . . . . 6 SubGrp Cntz mrClsSubGrp g DProd SubGrp Cntz mrClsSubGrp
4542, 44mpbi 200 . . . . 5 DProd SubGrp Cntz mrClsSubGrp
4645fdmi 5588 . . . 4 DProd SubGrp Cntz mrClsSubGrp
4746eleq2i 2499 . . 3 DProd SubGrp Cntz mrClsSubGrp
48 fveq2 5720 . . . . . . 7 SubGrp SubGrp
49 feq3 5570 . . . . . . 7 SubGrp SubGrp SubGrp SubGrp
5048, 49syl 16 . . . . . 6 SubGrp SubGrp
51 fveq2 5720 . . . . . . . . . . . 12 Cntz Cntz
52 dmdprd.z . . . . . . . . . . . 12 Cntz
5351, 52syl6eqr 2485 . . . . . . . . . . 11 Cntz
5453fveq1d 5722 . . . . . . . . . 10 Cntz
5554sseq2d 3368 . . . . . . . . 9 Cntz
5655ralbidv 2717 . . . . . . . 8 Cntz
5748fveq2d 5724 . . . . . . . . . . . 12 mrClsSubGrp mrClsSubGrp
58 dmdprd.k . . . . . . . . . . . 12 mrClsSubGrp
5957, 58syl6eqr 2485 . . . . . . . . . . 11 mrClsSubGrp
6059fveq1d 5722 . . . . . . . . . 10 mrClsSubGrp
6160ineq2d 3534 . . . . . . . . 9 mrClsSubGrp
62 fveq2 5720 . . . . . . . . . . 11
63 dmdprd.0 . . . . . . . . . . 11
6462, 63syl6eqr 2485 . . . . . . . . . 10
6564sneqd 3819 . . . . . . . . 9
6661, 65eqeq12d 2449 . . . . . . . 8 mrClsSubGrp
6756, 66anbi12d 692 . . . . . . 7 Cntz mrClsSubGrp
6867ralbidv 2717 . . . . . 6 Cntz mrClsSubGrp
6950, 68anbi12d 692 . . . . 5 SubGrp Cntz mrClsSubGrp SubGrp
7069abbidv 2549 . . . 4 SubGrp Cntz mrClsSubGrp SubGrp
7170opeliunxp2 5005 . . 3 SubGrp Cntz mrClsSubGrp SubGrp
7234, 47, 713bitri 263 . 2 DProd SubGrp
73 3anass 940 . 2 SubGrp SubGrp
7433, 72, 733bitr4g 280 1 DProd SubGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  cab 2421  wral 2697  crab 2701  cvv 2948  wsbc 3153   cdif 3309   cin 3311   wss 3312  csn 3806  cop 3809  cuni 4007  ciun 4085   class class class wbr 4204   cmpt 4258   cxp 4868  ccnv 4869   cdm 4870   crn 4871  cima 4873  wf 5442  cfv 5446  (class class class)co 6073  cixp 7055  cfn 7101  c0g 13715   g cgsu 13716  mrClscmrc 13800  cgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546 This theorem is referenced by:  dmdprdd  15552  dprdgrp  15555  dprdf  15556  dprdcntz  15558  dprddisj  15559  dprdres  15578  subgdmdprd  15584 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ixp 7056  df-dprd 15548
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