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Theorem eldprdi 17037
 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.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
eldprdi.0
eldprdi.w finSupp
eldprdi.1 DProd
eldprdi.2
eldprdi.3
Assertion
Ref Expression
eldprdi g DProd
Distinct variable groups:   ,   ,,   ,,   ,   ,,
Allowed substitution hints:   (,)   ()   (,)   ()

Proof of Theorem eldprdi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2 DProd
2 eldprdi.3 . . 3
3 eqid 2443 . . 3 g g
4 oveq2 6289 . . . . 5 g g
54eqeq2d 2457 . . . 4 g g g g
65rspcev 3196 . . 3 g g g g
72, 3, 6sylancl 662 . 2 g g
8 eldprdi.2 . . 3
9 eldprdi.0 . . . 4
10 eldprdi.w . . . 4 finSupp
119, 10eldprd 17014 . . 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 1383   wcel 1804  wrex 2794  crab 2797   class class class wbr 4437   cdm 4989  cfv 5578  (class class class)co 6281  cixp 7471   finSupp cfsupp 7831  c0g 14819   g cgsu 14820   DProd cdprd 17003 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-ixp 7472  df-dprd 17005 This theorem is referenced by:  dprdfsub  17040  dprdf11  17042  dprdsubg  17050  dprdub  17051  dpjidcl  17086
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