Theorem dpjidcl 17739

Description: The key property of projections: the sum of all the projections of A is A. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
dpjfval.1	|- ( ph -> G dom DProd S )
dpjfval.2	|- ( ph -> dom S = I )
dpjfval.p	|- P = ( GdProj S )
dpjidcl.3	|- ( ph -> A e. ( G DProd S ) )
dpjidcl.0	|- .0. = ( 0g ` G )
dpjidcl.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }

Assertion

Ref	Expression
dpjidcl	|- ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) )

Distinct variable groups:   x, h, .0.   h, i, G, x   P, h, x   ph, i, x   h, I, i, x   x, W   A, h, x   S, h, i, x

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

Proof of Theorem dpjidcl

Dummy variables k f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dpjidcl.3	. . . 4 |- ( ph -> A e. ( G DProd S ) )
2		dpjfval.2	. . . . 5 |- ( ph -> dom S = I )
3		dpjidcl.0	. . . . . 6 |- .0. = ( 0g ` G )
4		dpjidcl.w	. . . . . 6 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
5	3, 4	eldprd 17684	. . . . 5 |- ( dom S = I -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) ) )
6	2, 5	syl 17	. . . 4 |- ( ph -> ( A e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) ) )
7	1, 6	mpbid 215	. . 3 |- ( ph -> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) )
8	7	simprd 469	. 2 |- ( ph -> E. f e. W A = ( G gsumg f ) )
9		dpjfval.1	. . . . 5 |- ( ph -> G dom DProd S )
10	9	adantr 471	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> G dom DProd S )
11	2	adantr 471	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> dom S = I )
12	9	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> G dom DProd S )
13	2	ad2antrr 737	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> dom S = I )
14		dpjfval.p	. . . . . 6 |- P = ( GdProj S )
15		simpr 467	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> x e. I )
16	12, 13, 14, 15	dpjf 17738	. . . . 5 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( P ` x ) : ( G DProd S ) --> ( S ` x ) )
17	1	ad2antrr 737	. . . . 5 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> A e. ( G DProd S ) )
18	16, 17	ffvelrnd 6045	. . . 4 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) e. ( S ` x ) )
19	9, 2	dprddomcld 17681	. . . . . . 7 |- ( ph -> I e. _V )
20		mptexg 6159	. . . . . . 7 |- ( I e. _V -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V )
21	19, 20	syl 17	. . . . . 6 |- ( ph -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V )
22	21	adantr 471	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V )
23		funmpt 5636	. . . . . 6 |- Fun ( x e. I |-> ( ( P ` x ) ` A ) )
24	23	a1i 11	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> Fun ( x e. I |-> ( ( P ` x ) ` A ) ) )
25		simprl 769	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f e. W )
26	4, 10, 11, 25	dprdffsupp 17695	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f finSupp .0. )
27		eldifi 3566	. . . . . . . 8 |- ( x e. ( I \ ( f supp .0. ) ) -> x e. I )
28		eqid 2461	. . . . . . . . . 10 |- ( proj1 ` G ) = ( proj1 ` G )
29	12, 13, 14, 28, 15	dpjval 17737	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( P ` x ) = ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) )
30	29	fveq1d 5889	. . . . . . . 8 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) = ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) )
31	27, 30	sylan2 481	. . . . . . 7 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( P ` x ) ` A ) = ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) )
32		simplrr 776	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsumg f ) )
33		eqid 2461	. . . . . . . . . . 11 |- ( Base ` G ) = ( Base ` G )
34		eqid 2461	. . . . . . . . . . 11 |- (Cntz ` G ) = (Cntz ` G )
35		dprdgrp 17685	. . . . . . . . . . . . 13 |- ( G dom DProd S -> G e. Grp )
36		grpmnd 16726	. . . . . . . . . . . . 13 |- ( G e. Grp -> G e. Mnd )
37	10, 35, 36	3syl 18	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> G e. Mnd )
38	37	adantr 471	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> G e. Mnd )
39	19	ad2antrr 737	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> I e. _V )
40	4, 10, 11, 25, 33	dprdff 17693	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f : I --> ( Base ` G ) )
41	40	adantr 471	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> f : I --> ( Base ` G ) )
42	25	adantr 471	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f e. W )
43	4, 12, 13, 42, 34	dprdfcntz 17696	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ran f C_ ( (Cntz ` G ) ` ran f ) )
44	27, 43	sylan2 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ran f C_ ( (Cntz ` G ) ` ran f ) )
45		snssi 4128	. . . . . . . . . . . . 13 |- ( x e. ( I \ ( f supp .0. ) ) -> { x } C_ ( I \ ( f supp .0. ) ) )
46	45	adantl 472	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ ( I \ ( f supp .0. ) ) )
47	46	difss2d 3574	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ I )
48		suppssdm 6953	. . . . . . . . . . . . . . 15 |- ( f supp .0. ) C_ dom f
49		fdm 5755	. . . . . . . . . . . . . . . 16 |- ( f : I --> ( Base ` G ) -> dom f = I )
50	40, 49	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> dom f = I )
51	48, 50	syl5sseq 3491	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( f supp .0. ) C_ I )
52	51	adantr 471	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ I )
53		ssconb 3577	. . . . . . . . . . . . 13 |- ( ( { x } C_ I /\ ( f supp .0. ) C_ I ) -> ( { x } C_ ( I \ ( f supp .0. ) ) <-> ( f supp .0. ) C_ ( I \ { x } ) ) )
54	47, 52, 53	syl2anc 671	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( { x } C_ ( I \ ( f supp .0. ) ) <-> ( f supp .0. ) C_ ( I \ { x } ) ) )
55	46, 54	mpbid 215	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ ( I \ { x } ) )
56	26	adantr 471	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> f finSupp .0. )
57	33, 3, 34, 38, 39, 41, 44, 55, 56	gsumzres 17591	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( G gsumg ( f |` ( I \ { x } ) ) ) = ( G gsumg f ) )
58	32, 57	eqtr4d 2498	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsumg ( f |` ( I \ { x } ) ) ) )
59		eqid 2461	. . . . . . . . . . 11 |- { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } = { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. }
60		difss 3571	. . . . . . . . . . . . . 14 |- ( I \ { x } ) C_ I
61	60	a1i 11	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( I \ { x } ) C_ I )
62	12, 13, 61	dprdres 17709	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G dom DProd ( S |` ( I \ { x } ) ) /\ ( G DProd ( S |` ( I \ { x } ) ) ) C_ ( G DProd S ) ) )
63	62	simpld 465	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> G dom DProd ( S |` ( I \ { x } ) ) )
64	12, 13	dprdf2 17687	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> S : I --> (SubGrp ` G ) )
65		fssres 5771	. . . . . . . . . . . . 13 |- ( ( S : I --> (SubGrp ` G ) /\ ( I \ { x } ) C_ I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> (SubGrp ` G ) )
66	64, 60, 65	sylancl 673	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> (SubGrp ` G ) )
67		fdm 5755	. . . . . . . . . . . 12 |- ( ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> (SubGrp ` G ) -> dom ( S |` ( I \ { x } ) ) = ( I \ { x } ) )
68	66, 67	syl 17	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> dom ( S |` ( I \ { x } ) ) = ( I \ { x } ) )
69	40	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f : I --> ( Base ` G ) )
70	69	feqmptd 5940	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f = ( k e. I |-> ( f ` k ) ) )
71	70	reseq1d 5122	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) )
72		resmpt 5172	. . . . . . . . . . . . . 14 |- ( ( I \ { x } ) C_ I -> ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) )
73	60, 72	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) )
74	71, 73	syl6eq 2511	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) )
75		eldifi 3566	. . . . . . . . . . . . . . 15 |- ( k e. ( I \ { x } ) -> k e. I )
76	4, 12, 13, 42	dprdfcl 17694	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. I ) -> ( f ` k ) e. ( S ` k ) )
77	75, 76	sylan2 481	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( f ` k ) e. ( S ` k ) )
78		fvres 5901	. . . . . . . . . . . . . . 15 |- ( k e. ( I \ { x } ) -> ( ( S |` ( I \ { x } ) ) ` k ) = ( S ` k ) )
79	78	adantl 472	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( ( S |` ( I \ { x } ) ) ` k ) = ( S ` k ) )
80	77, 79	eleqtrrd 2542	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( f ` k ) e. ( ( S |` ( I \ { x } ) ) ` k ) )
81		difexg 4564	. . . . . . . . . . . . . . . . 17 |- ( I e. _V -> ( I \ { x } ) e. _V )
82	19, 81	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( I \ { x } ) e. _V )
83		mptexg 6159	. . . . . . . . . . . . . . . 16 |- ( ( I \ { x } ) e. _V -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V )
84	82, 83	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V )
85	84	ad2antrr 737	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V )
86		funmpt 5636	. . . . . . . . . . . . . . 15 |- Fun ( k e. ( I \ { x } ) |-> ( f ` k ) )
87	86	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> Fun ( k e. ( I \ { x } ) |-> ( f ` k ) ) )
88	26	adantr 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f finSupp .0. )
89		ssdif 3579	. . . . . . . . . . . . . . . . . 18 |- ( ( I \ { x } ) C_ I -> ( ( I \ { x } ) \ ( f supp .0. ) ) C_ ( I \ ( f supp .0. ) ) )
90	60, 89	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( ( I \ { x } ) \ ( f supp .0. ) ) C_ ( I \ ( f supp .0. ) )
91	90	sseli 3439	. . . . . . . . . . . . . . . 16 |- ( k e. ( ( I \ { x } ) \ ( f supp .0. ) ) -> k e. ( I \ ( f supp .0. ) ) )
92		ssid 3462	. . . . . . . . . . . . . . . . . 18 |- ( f supp .0. ) C_ ( f supp .0. )
93	92	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f supp .0. ) C_ ( f supp .0. ) )
94	19	ad2antrr 737	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> I e. _V )
95		fvex 5897	. . . . . . . . . . . . . . . . . . 19 |- ( 0g ` G ) e. _V
96	3, 95	eqeltri 2535	. . . . . . . . . . . . . . . . . 18 |- .0. e. _V
97	96	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> .0. e. _V )
98	69, 93, 94, 97	suppssr 6972	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( I \ ( f supp .0. ) ) ) -> ( f ` k ) = .0. )
99	91, 98	sylan2 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( ( I \ { x } ) \ ( f supp .0. ) ) ) -> ( f ` k ) = .0. )
100	82	ad2antrr 737	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( I \ { x } ) e. _V )
101	99, 100	suppss2 6975	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) supp .0. ) C_ ( f supp .0. ) )
102		fsuppsssupp 7924	. . . . . . . . . . . . . 14 |- ( ( ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V /\ Fun ( k e. ( I \ { x } ) |-> ( f ` k ) ) ) /\ ( f finSupp .0. /\ ( ( k e. ( I \ { x } ) |-> ( f ` k ) ) supp .0. ) C_ ( f supp .0. ) ) ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) finSupp .0. )
103	85, 87, 88, 101, 102	syl22anc 1277	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) finSupp .0. )
104	59, 63, 68, 80, 103	dprdwd 17691	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } )
105	74, 104	eqeltrd 2539	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) e. { h e. X_ i e. ( I \ { x } ) ( ( S |` ( I \ { x } ) ) ` i ) | h finSupp .0. } )
106	3, 59, 63, 68, 105	eldprdi 17699	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg ( f |` ( I \ { x } ) ) ) e. ( G DProd ( S |` ( I \ { x } ) ) ) )
107	27, 106	sylan2 481	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( G gsumg ( f |` ( I \ { x } ) ) ) e. ( G DProd ( S |` ( I \ { x } ) ) ) )
108	58, 107	eqeltrd 2539	. . . . . . . 8 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A e. ( G DProd ( S |` ( I \ { x } ) ) ) )
109		eqid 2461	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
110		eqid 2461	. . . . . . . . . 10 |- ( LSSum ` G ) = ( LSSum ` G )
111	64, 15	ffvelrnd 6045	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
112		dprdsubg 17705	. . . . . . . . . . 11 |- ( G dom DProd ( S |` ( I \ { x } ) ) -> ( G DProd ( S |` ( I \ { x } ) ) ) e. (SubGrp ` G ) )
113	63, 112	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G DProd ( S |` ( I \ { x } ) ) ) e. (SubGrp ` G ) )
114	12, 13, 15, 3	dpjdisj 17734	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( S ` x ) i^i ( G DProd ( S |` ( I \ { x } ) ) ) ) = { .0. } )
115	12, 13, 15, 34	dpjcntz 17733	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( S ` x ) C_ ( (Cntz ` G ) ` ( G DProd ( S |` ( I \ { x } ) ) ) ) )
116	109, 110, 3, 34, 111, 113, 114, 115, 28	pj1rid 17400	. . . . . . . . 9 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ A e. ( G DProd ( S |` ( I \ { x } ) ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. )
117	27, 116	sylanl2 661	. . . . . . . 8 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) /\ A e. ( G DProd ( S |` ( I \ { x } ) ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. )
118	108, 117	mpdan 679	. . . . . . 7 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = .0. )
119	31, 118	eqtrd 2495	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( P ` x ) ` A ) = .0. )
120	19	adantr 471	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> I e. _V )
121	119, 120	suppss2 6975	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) supp .0. ) C_ ( f supp .0. ) )
122		fsuppsssupp 7924	. . . . 5 |- ( ( ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V /\ Fun ( x e. I |-> ( ( P ` x ) ` A ) ) ) /\ ( f finSupp .0. /\ ( ( x e. I |-> ( ( P ` x ) ` A ) ) supp .0. ) C_ ( f supp .0. ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) finSupp .0. )
123	22, 24, 26, 121, 122	syl22anc 1277	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) finSupp .0. )
124	4, 10, 11, 18, 123	dprdwd 17691	. . 3 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. W )
125		simprr 771	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> A = ( G gsumg f ) )
126	40	feqmptd 5940	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f = ( x e. I |-> ( f ` x ) ) )
127		simplrr 776	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> A = ( G gsumg f ) )
128	12, 35, 36	3syl 18	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> G e. Mnd )
129	4, 12, 13, 42	dprdffsupp 17695	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f finSupp .0. )
130		disjdif 3850	. . . . . . . . . . . . 13 |- ( { x } i^i ( I \ { x } ) ) = (/)
131	130	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( { x } i^i ( I \ { x } ) ) = (/) )
132		undif2 3854	. . . . . . . . . . . . 13 |- ( { x } u. ( I \ { x } ) ) = ( { x } u. I )
133	15	snssd 4129	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> { x } C_ I )
134		ssequn1 3615	. . . . . . . . . . . . . 14 |- ( { x } C_ I <-> ( { x } u. I ) = I )
135	133, 134	sylib 201	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( { x } u. I ) = I )
136	132, 135	syl5req 2508	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> I = ( { x } u. ( I \ { x } ) ) )
137	33, 3, 109, 34, 128, 94, 69, 43, 129, 131, 136	gsumzsplit 17608	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg f ) = ( ( G gsumg ( f |` { x } ) ) ( +g ` G ) ( G gsumg ( f |` ( I \ { x } ) ) ) ) )
138	69, 133	feqresmpt 5941	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` { x } ) = ( k e. { x } |-> ( f ` k ) ) )
139	138	oveq2d 6330	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg ( f |` { x } ) ) = ( G gsumg ( k e. { x } |-> ( f ` k ) ) ) )
140	69, 15	ffvelrnd 6045	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f ` x ) e. ( Base ` G ) )
141		fveq2 5887	. . . . . . . . . . . . . . 15 |- ( k = x -> ( f ` k ) = ( f ` x ) )
142	33, 141	gsumsn 17635	. . . . . . . . . . . . . 14 |- ( ( G e. Mnd /\ x e. I /\ ( f ` x ) e. ( Base ` G ) ) -> ( G gsumg ( k e. { x } |-> ( f ` k ) ) ) = ( f ` x ) )
143	128, 15, 140, 142	syl3anc 1276	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg ( k e. { x } |-> ( f ` k ) ) ) = ( f ` x ) )
144	139, 143	eqtrd 2495	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg ( f |` { x } ) ) = ( f ` x ) )
145	144	oveq1d 6329	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( G gsumg ( f |` { x } ) ) ( +g ` G ) ( G gsumg ( f |` ( I \ { x } ) ) ) ) = ( ( f ` x ) ( +g ` G ) ( G gsumg ( f |` ( I \ { x } ) ) ) ) )
146	127, 137, 145	3eqtrd 2499	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> A = ( ( f ` x ) ( +g ` G ) ( G gsumg ( f |` ( I \ { x } ) ) ) ) )
147	12, 13, 15, 110	dpjlsm 17735	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G DProd S ) = ( ( S ` x ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) )
148	17, 147	eleqtrd 2541	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> A e. ( ( S ` x ) ( LSSum ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) )
149	4, 10, 11, 25	dprdfcl 17694	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f ` x ) e. ( S ` x ) )
150	109, 110, 3, 34, 111, 113, 114, 115, 28, 148, 149, 106	pj1eq 17398	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( A = ( ( f ` x ) ( +g ` G ) ( G gsumg ( f |` ( I \ { x } ) ) ) ) <-> ( ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) /\ ( ( ( G DProd ( S |` ( I \ { x } ) ) ) ( proj1 ` G ) ( S ` x ) ) ` A ) = ( G gsumg ( f |` ( I \ { x } ) ) ) ) ) )
151	146, 150	mpbid 215	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) /\ ( ( ( G DProd ( S |` ( I \ { x } ) ) ) ( proj1 ` G ) ( S ` x ) ) ` A ) = ( G gsumg ( f |` ( I \ { x } ) ) ) ) )
152	151	simpld 465	. . . . . . . 8 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( ( S ` x ) ( proj1 ` G ) ( G DProd ( S |` ( I \ { x } ) ) ) ) ` A ) = ( f ` x ) )
153	30, 152	eqtrd 2495	. . . . . . 7 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) = ( f ` x ) )
154	153	mpteq2dva 4502	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> ( f ` x ) ) )
155	126, 154	eqtr4d 2498	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f = ( x e. I |-> ( ( P ` x ) ` A ) ) )
156	155	oveq2d 6330	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( G gsumg f ) = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) )
157	125, 156	eqtrd 2495	. . 3 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) )
158	124, 157	jca 539	. 2 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) )
159	8, 158	rexlimddv 2894	1 |- ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   E.wrex 2749   {crab 2752   _Vcvv 3056   \ cdif 3412   u. cun 3413   i^i cin 3414   C_ wss 3415   (/)c0 3742   {csn 3979   class class class wbr 4415   |-> cmpt 4474   dom cdm 4852   ran crn 4853   |` cres 4854   Fun wfun 5594   -->wf 5596   ` cfv 5600  (class class class)co 6314   supp csupp 6940   X_cixp 7547   finSupp cfsupp 7908   Basecbs 15169   +g cplusg 15238   0gc0g 15386   gsum_g cgsu 15387   Mndcmnd 16583   Grpcgrp 16717  SubGrpcsubg 16859  Cntzccntz 17017   LSSumclsm 17334   proj1cpj1 17335   DProd cdprd 17673  dProjcdpj 17674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-tpos 6998  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-oi 8050  df-card 8398  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-fzo 11946  df-seq 12245  df-hash 12547  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-0g 15388  df-gsum 15389  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-mhm 16630  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-mulg 16724  df-subg 16862  df-ghm 16929  df-gim 16971  df-cntz 17019  df-oppg 17045  df-lsm 17336  df-pj1 17337  df-cmn 17480  df-dprd 17675  df-dpj 17676

This theorem is referenced by:  dpjeq  17740  dpjid  17741