Theorem eldprd 17571

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 5907	. . . . 5 |- ( A e. ( DProd ` <. G , S >. ) -> <. G , S >. e. dom DProd )
2		df-ov 6308	. . . . 5 |- ( G DProd S ) = ( DProd ` <. G , S >. )
3	1, 2	eleq2s 2537	. . . 4 |- ( A e. ( G DProd S ) -> <. G , S >. e. dom DProd )
4		df-br 4427	. . . 4 |- ( G dom DProd S <-> <. G , S >. e. dom DProd )
5	3, 4	sylibr 215	. . 3 |- ( A e. ( G DProd S ) -> G dom DProd S )
6	5	pm4.71ri 637	. 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 17570	. . . . . 6 |- ( ( G dom DProd S /\ dom S = I ) -> ( G DProd S ) = ran ( f e. W |-> ( G gsumg f ) ) )
10	9	eleq2d 2499	. . . . 5 |- ( ( G dom DProd S /\ dom S = I ) -> ( A e. ( G DProd S ) <-> A e. ran ( f e. W |-> ( G gsumg f ) ) ) )
11		eqid 2429	. . . . . 6 |- ( f e. W |-> ( G gsumg f ) ) = ( f e. W |-> ( G gsumg f ) )
12		ovex 6333	. . . . . 6 |- ( G gsumg f ) e. _V
13	11, 12	elrnmpti 5105	. . . . 5 |- ( A e. ran ( f e. W |-> ( G gsumg f ) ) <-> E. f e. W A = ( G gsumg f ) )
14	10, 13	syl6bb 264	. . . 4 |- ( ( G dom DProd S /\ dom S = I ) -> ( A e. ( G DProd S ) <-> E. f e. W A = ( G gsumg f ) ) )
15	14	ancoms 454	. . 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 645	. 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 260	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 187   /\ wa 370   = wceq 1437   e. wcel 1870   E.wrex 2783   {crab 2786   <.cop 4008   class class class wbr 4426   |-> cmpt 4484   dom cdm 4854   ran crn 4855   ` cfv 5601  (class class class)co 6305   X_cixp 7530   finSupp cfsupp 7889   0gc0g 15297   gsum_g cgsu 15298   DProd cdprd 17560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-ixp 7531  df-dprd 17562

This theorem is referenced by:  dprdssv  17584  eldprdi  17586  dprdsubg  17592  dprdss  17597  dmdprdsplitlem  17605  dprddisj2  17607  dpjidcl  17626