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Theorem eldprdiOLD 17191
 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.) Obsolete version of eldprdi 17184 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0
eldprdiOLD.w
eldprdiOLD.1 DProd
eldprdiOLD.2
eldprdiOLD.3
Assertion
Ref Expression
eldprdiOLD g DProd
Distinct variable groups:   ,   ,,   ,,   ,   ,,
Allowed substitution hints:   (,)   ()   (,)   ()

Proof of Theorem eldprdiOLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.1 . 2 DProd
2 eldprdiOLD.3 . . 3
3 eqid 2457 . . 3 g g
4 oveq2 6304 . . . . 5 g g
54eqeq2d 2471 . . . 4 g g g g
65rspcev 3210 . . 3 g g g g
72, 3, 6sylancl 662 . 2 g g
8 eldprdiOLD.2 . . 3
9 eldprdiOLD.0 . . . 4
10 eldprdiOLD.w . . . 4
119, 10eldprdOLD 17163 . . 3 g DProd DProd g g
128, 11syl 16 . 2 g DProd DProd g g
131, 7, 12mpbir2and 922 1 g DProd
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wrex 2808  crab 2811  cvv 3109   cdif 3468  csn 4032   class class class wbr 4456  ccnv 5007   cdm 5008  cima 5011  cfv 5594  (class class class)co 6296  cixp 7488  cfn 7535  c0g 14856   g cgsu 14857   DProd cdprd 17150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-supp 6918  df-ixp 7489  df-fsupp 7848  df-dprd 17152 This theorem is referenced by:  dprdfsubOLD  17194  dprdf11OLD  17196  dpjidclOLD  17240
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