Theorem dpjidcl 18457

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 18403	. . . . 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 222	. . 3 |- ( ph -> ( G dom DProd S /\ E. f e. W A = ( G gsumg f ) ) )
8	7	simprd 479	. 2 |- ( ph -> E. f e. W A = ( G gsumg f ) )
9		dpjfval.1	. . . . 5 |- ( ph -> G dom DProd S )
10	9	adantr 481	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> G dom DProd S )
11	2	adantr 481	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> dom S = I )
12	9	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> G dom DProd S )
13	2	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> dom S = I )
14		dpjfval.p	. . . . . 6 |- P = ( GdProj S )
15		simpr 477	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> x e. I )
16	12, 13, 14, 15	dpjf 18456	. . . . 5 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( P ` x ) : ( G DProd S ) --> ( S ` x ) )
17	1	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> A e. ( G DProd S ) )
18	16, 17	ffvelrnd 6360	. . . 4 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) e. ( S ` x ) )
19	9, 2	dprddomcld 18400	. . . . . . 7 |- ( ph -> I e. _V )
20		mptexg 6484	. . . . . . 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 481	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. _V )
23		funmpt 5926	. . . . . 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 794	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f e. W )
26	4, 10, 11, 25	dprdffsupp 18413	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f finSupp .0. )
27		eldifi 3732	. . . . . . . 8 |- ( x e. ( I \ ( f supp .0. ) ) -> x e. I )
28		eqid 2622	. . . . . . . . . 10 |- ( proj1 ` G ) = ( proj1 ` G )
29	12, 13, 14, 28, 15	dpjval 18455	. . . . . . . . 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 6193	. . . . . . . 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 491	. . . . . . 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 801	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsumg f ) )
33		eqid 2622	. . . . . . . . . . 11 |- ( Base ` G ) = ( Base ` G )
34		eqid 2622	. . . . . . . . . . 11 |- (Cntz ` G ) = (Cntz ` G )
35		dprdgrp 18404	. . . . . . . . . . . . 13 |- ( G dom DProd S -> G e. Grp )
36		grpmnd 17429	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> G e. Mnd )
39	19	ad2antrr 762	. . . . . . . . . . 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 18411	. . . . . . . . . . . 12 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f : I --> ( Base ` G ) )
41	40	adantr 481	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> f : I --> ( Base ` G ) )
42	25	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f e. W )
43	4, 12, 13, 42, 34	dprdfcntz 18414	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ran f C_ ( (Cntz ` G ) ` ran f ) )
44	27, 43	sylan2 491	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ran f C_ ( (Cntz ` G ) ` ran f ) )
45		snssi 4339	. . . . . . . . . . . . 13 |- ( x e. ( I \ ( f supp .0. ) ) -> { x } C_ ( I \ ( f supp .0. ) ) )
46	45	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ ( I \ ( f supp .0. ) ) )
47	46	difss2d 3740	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> { x } C_ I )
48		suppssdm 7308	. . . . . . . . . . . . . . 15 |- ( f supp .0. ) C_ dom f
49		fdm 6051	. . . . . . . . . . . . . . . 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 3653	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( f supp .0. ) C_ I )
52	51	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ I )
53		ssconb 3743	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 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 222	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( f supp .0. ) C_ ( I \ { x } ) )
56	26	adantr 481	. . . . . . . . . . 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 18310	. . . . . . . . . 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 2659	. . . . . . . . 9 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A = ( G gsumg ( f |` ( I \ { x } ) ) ) )
59		eqid 2622	. . . . . . . . . . 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 3737	. . . . . . . . . . . . . 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 18427	. . . . . . . . . . . 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 475	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> G dom DProd ( S |` ( I \ { x } ) ) )
64	12, 13	dprdf2 18406	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> S : I --> (SubGrp ` G ) )
65		fssres 6070	. . . . . . . . . . . . 13 |- ( ( S : I --> (SubGrp ` G ) /\ ( I \ { x } ) C_ I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> (SubGrp ` G ) )
66	64, 60, 65	sylancl 694	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( S |` ( I \ { x } ) ) : ( I \ { x } ) --> (SubGrp ` G ) )
67		fdm 6051	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f : I --> ( Base ` G ) )
70	69	feqmptd 6249	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f = ( k e. I |-> ( f ` k ) ) )
71	70	reseq1d 5395	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( ( k e. I |-> ( f ` k ) ) |` ( I \ { x } ) ) )
72		resmpt 5449	. . . . . . . . . . . . . 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 2672	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` ( I \ { x } ) ) = ( k e. ( I \ { x } ) |-> ( f ` k ) ) )
75		eldifi 3732	. . . . . . . . . . . . . . 15 |- ( k e. ( I \ { x } ) -> k e. I )
76	4, 12, 13, 42	dprdfcl 18412	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. I ) -> ( f ` k ) e. ( S ` k ) )
77	75, 76	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) /\ k e. ( I \ { x } ) ) -> ( f ` k ) e. ( S ` k ) )
78		fvres 6207	. . . . . . . . . . . . . . 15 |- ( k e. ( I \ { x } ) -> ( ( S |` ( I \ { x } ) ) ` k ) = ( S ` k ) )
79	78	adantl 482	. . . . . . . . . . . . . 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 2704	. . . . . . . . . . . . 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 4808	. . . . . . . . . . . . . . . . 17 |- ( I e. _V -> ( I \ { x } ) e. _V )
82	19, 81	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( I \ { x } ) e. _V )
83		mptexg 6484	. . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( k e. ( I \ { x } ) |-> ( f ` k ) ) e. _V )
86		funmpt 5926	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f finSupp .0. )
89		ssdif 3745	. . . . . . . . . . . . . . . . . 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 3599	. . . . . . . . . . . . . . . 16 |- ( k e. ( ( I \ { x } ) \ ( f supp .0. ) ) -> k e. ( I \ ( f supp .0. ) ) )
92		ssid 3624	. . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> I e. _V )
95		fvex 6201	. . . . . . . . . . . . . . . . . . 19 |- ( 0g ` G ) e. _V
96	3, 95	eqeltri 2697	. . . . . . . . . . . . . . . . . 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 7326	. . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( I \ { x } ) e. _V )
101	99, 100	suppss2 7329	. . . . . . . . . . . . . 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 8291	. . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . 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 18410	. . . . . . . . . . . 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 2701	. . . . . . . . . . 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 18417	. . . . . . . . . 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 491	. . . . . . . . 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 2701	. . . . . . . 8 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> A e. ( G DProd ( S |` ( I \ { x } ) ) ) )
109		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
110		eqid 2622	. . . . . . . . . 10 |- ( LSSum ` G ) = ( LSSum ` G )
111	64, 15	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
112		dprdsubg 18423	. . . . . . . . . . 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 18452	. . . . . . . . . 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 18451	. . . . . . . . . 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 18115	. . . . . . . . 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 683	. . . . . . . 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 702	. . . . . . 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 2656	. . . . . 6 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. ( I \ ( f supp .0. ) ) ) -> ( ( P ` x ) ` A ) = .0. )
120	19	adantr 481	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> I e. _V )
121	119, 120	suppss2 7329	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) supp .0. ) C_ ( f supp .0. ) )
122		fsuppsssupp 8291	. . . . 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 1327	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) finSupp .0. )
124	4, 10, 11, 18, 123	dprdwd 18410	. . 3 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) e. W )
125		simprr 796	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> A = ( G gsumg f ) )
126	40	feqmptd 6249	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f = ( x e. I |-> ( f ` x ) ) )
127		simplrr 801	. . . . . . . . . . 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 18413	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> f finSupp .0. )
130		disjdif 4040	. . . . . . . . . . . . 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 4044	. . . . . . . . . . . . 13 |- ( { x } u. ( I \ { x } ) ) = ( { x } u. I )
133	15	snssd 4340	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> { x } C_ I )
134		ssequn1 3783	. . . . . . . . . . . . . 14 |- ( { x } C_ I <-> ( { x } u. I ) = I )
135	133, 134	sylib 208	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( { x } u. I ) = I )
136	132, 135	syl5req 2669	. . . . . . . . . . . 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 18327	. . . . . . . . . . 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 6250	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f |` { x } ) = ( k e. { x } |-> ( f ` k ) ) )
139	138	oveq2d 6666	. . . . . . . . . . . . 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 6360	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( f ` x ) e. ( Base ` G ) )
141		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( k = x -> ( f ` k ) = ( f ` x ) )
142	33, 141	gsumsn 18354	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 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 2656	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( G gsumg ( f |` { x } ) ) = ( f ` x ) )
145	144	oveq1d 6665	. . . . . . . . . . 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 2660	. . . . . . . . . 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 18453	. . . . . . . . . . . 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 2703	. . . . . . . . . . 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 18412	. . . . . . . . . . 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 18113	. . . . . . . . . 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 222	. . . . . . . . 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 475	. . . . . . . 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 2656	. . . . . . 7 |- ( ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) /\ x e. I ) -> ( ( P ` x ) ` A ) = ( f ` x ) )
154	153	mpteq2dva 4744	. . . . . 6 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( x e. I |-> ( ( P ` x ) ` A ) ) = ( x e. I |-> ( f ` x ) ) )
155	126, 154	eqtr4d 2659	. . . . 5 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> f = ( x e. I |-> ( ( P ` x ) ` A ) ) )
156	155	oveq2d 6666	. . . 4 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> ( G gsumg f ) = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) )
157	125, 156	eqtrd 2656	. . 3 |- ( ( ph /\ ( f e. W /\ A = ( G gsumg f ) ) ) -> A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) )
158	124, 157	jca 554	. 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 3035	1 |- ( ph -> ( ( x e. I |-> ( ( P ` x ) ` A ) ) e. W /\ A = ( G gsumg ( x e. I |-> ( ( P ` x ) ` A ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   supp csupp 7295   X_cixp 7908   finSupp cfsupp 8275   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   proj1cpj1 18050   DProd cdprd 18392  dProjcdpj 18393

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-gim 17701  df-cntz 17750  df-oppg 17776  df-lsm 18051  df-pj1 18052  df-cmn 18195  df-dprd 18394  df-dpj 18395

This theorem is referenced by:  dpjeq  18458  dpjid  18459