Theorem eldprd 16823

Description: A class A is an internal direct product iff it is the (group) sum of an infinite, but finitely supported cartesian product of subgroups (which mutually commute and have trivial intersections). (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)

Hypotheses

Ref	Expression
dprdval.0	|- .0. = ( 0g ` G )
dprdval.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }

Assertion

Ref	Expression
eldprd	|- ( dom S = I -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) ) )

Distinct variable groups:   f, h, i   A, f   f, I, h, i   S, f, h, i   f, G, h, i

Allowed substitution hints:   A( h, i)   W( f, h, i)   .0. ( f, h, i)

Proof of Theorem eldprd

Step	Hyp	Ref	Expression
1		elfvdm 5890	. . . . 5 |- ( A e. ( DProd ` <. G , S >. ) -> <. G , S >. e. dom DProd )
2		df-ov 6285	. . . . 5 |- ( G DProd S ) = ( DProd ` <. G , S >. )
3	1, 2	eleq2s 2575	. . . 4 |- ( A e. ( G DProd S ) -> <. G , S >. e. dom DProd )
4		df-br 4448	. . . 4 |- ( G dom DProd S <-> <. G , S >. e. dom DProd )
5	3, 4	sylibr 212	. . 3 |- ( A e. ( G DProd S ) -> G dom DProd S )
6	5	pm4.71ri 633	. 2 |- ( A e. ( G DProd S ) <-> ( G dom DProd S /\ A e. ( G DProd S ) ) )
7		dprdval.0	. . . . . . 7 |- .0. = ( 0g ` G )
8		dprdval.w	. . . . . . 7 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
9	7, 8	dprdval 16822	. . . . . 6 |- ( ( G dom DProd S /\ dom S = I ) -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) )
10	9	eleq2d 2537	. . . . 5 |- ( ( G dom DProd S /\ dom S = I ) -> ( A e. ( G DProd S ) <-> A e. ran ( f e. W |-> ( G gsumg f ) ) ) )
11		eqid 2467	. . . . . 6 |- ( f e. W |-> ( G gsumg f ) ) = ( f e. W |-> ( G gsumg f ) )
12		ovex 6307	. . . . . 6 |- ( G gsumg f ) e. _V
13	11, 12	elrnmpti 5251	. . . . 5 |- ( A e. ran ( f e. W |-> ( G gsumg f ) ) <-> E. f e. W A = ( G gsumg f ) )
14	10, 13	syl6bb 261	. . . 4 |- ( ( G dom DProd S /\ dom S = I ) -> ( A e. ( G DProd S ) <-> E. f e. W A = ( G gsumg f ) ) )
15	14	ancoms 453	. . 3 |- ( ( dom S = I /\ G dom DProd S ) -> ( A e. ( G DProd S ) <-> E. f e. W A = ( G gsumg f ) ) )
16	15	pm5.32da 641	. 2 |- ( dom S = I -> ( ( G dom DProd S /\ A e. ( G DProd S ) ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) ) )
17	6, 16	syl5bb 257	1 |- ( dom S = I -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   E.wrex 2815   {crab 2818   <.cop 4033   class class class wbr 4447   |-> cmpt 4505   dom cdm 4999   ran crn 5000   ` cfv 5586  (class class class)co 6282   X_cixp 7466   finSupp cfsupp 7825   0gc0g 14688   gsum_g cgsu 14689   DProd cdprd 16812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-ixp 7467  df-dprd 16814

This theorem is referenced by:  dprdssv  16843  eldprdi  16845  dprdsubg  16858  dprdss  16863  dmdprdsplitlem  16871  dprddisj2  16874  dpjidcl  16894