Theorem dprdfadd 18419

Description: Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.)

Hypotheses

Ref	Expression
eldprdi.0	|- .0. = ( 0g ` G )
eldprdi.w	|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
eldprdi.1	|- ( ph -> G dom DProd S )
eldprdi.2	|- ( ph -> dom S = I )
eldprdi.3	|- ( ph -> F e. W )
dprdfadd.4	|- ( ph -> H e. W )
dprdfadd.b	|- .+ = ( +g ` G )

Assertion

Ref	Expression
dprdfadd	|- ( ph -> ( ( F oF .+ H ) e. W /\ ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) ) )

Distinct variable groups:   .+ , h   h, F   h, H   h, i, G   h, I, i   .0. , h   S, h, i

Allowed substitution hints:   ph( h, i)   .+ ( i)   F( i)   H( i)   W( h, i)   .0. ( i)

Proof of Theorem dprdfadd

Dummy variables k x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldprdi.1	. . . . 5 |- ( ph -> G dom DProd S )
2		eldprdi.2	. . . . 5 |- ( ph -> dom S = I )
3	1, 2	dprddomcld 18400	. . . 4 |- ( ph -> I e. _V )
4		eldprdi.w	. . . . 5 |- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. }
5		eldprdi.3	. . . . 5 |- ( ph -> F e. W )
6	4, 1, 2, 5	dprdfcl 18412	. . . 4 |- ( ( ph /\ x e. I ) -> ( F ` x ) e. ( S ` x ) )
7		dprdfadd.4	. . . . 5 |- ( ph -> H e. W )
8	4, 1, 2, 7	dprdfcl 18412	. . . 4 |- ( ( ph /\ x e. I ) -> ( H ` x ) e. ( S ` x ) )
9		eqid 2622	. . . . . 6 |- ( Base ` G ) = ( Base ` G )
10	4, 1, 2, 5, 9	dprdff 18411	. . . . 5 |- ( ph -> F : I --> ( Base ` G ) )
11	10	feqmptd 6249	. . . 4 |- ( ph -> F = ( x e. I |-> ( F ` x ) ) )
12	4, 1, 2, 7, 9	dprdff 18411	. . . . 5 |- ( ph -> H : I --> ( Base ` G ) )
13	12	feqmptd 6249	. . . 4 |- ( ph -> H = ( x e. I |-> ( H ` x ) ) )
14	3, 6, 8, 11, 13	offval2 6914	. . 3 |- ( ph -> ( F oF .+ H ) = ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) )
15	1, 2	dprdf2 18406	. . . . . 6 |- ( ph -> S : I --> (SubGrp ` G ) )
16	15	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ x e. I ) -> ( S ` x ) e. (SubGrp ` G ) )
17		dprdfadd.b	. . . . . 6 |- .+ = ( +g ` G )
18	17	subgcl 17604	. . . . 5 |- ( ( ( S ` x ) e. (SubGrp ` G ) /\ ( F ` x ) e. ( S ` x ) /\ ( H ` x ) e. ( S ` x ) ) -> ( ( F ` x ) .+ ( H ` x ) ) e. ( S ` x ) )
19	16, 6, 8, 18	syl3anc 1326	. . . 4 |- ( ( ph /\ x e. I ) -> ( ( F ` x ) .+ ( H ` x ) ) e. ( S ` x ) )
20	4, 1, 2, 5	dprdffsupp 18413	. . . . . . 7 |- ( ph -> F finSupp .0. )
21	4, 1, 2, 7	dprdffsupp 18413	. . . . . . 7 |- ( ph -> H finSupp .0. )
22	20, 21	fsuppunfi 8295	. . . . . 6 |- ( ph -> ( ( F supp .0. ) u. ( H supp .0. ) ) e. Fin )
23		ssun1 3776	. . . . . . . . . . 11 |- ( F supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) )
24	23	a1i 11	. . . . . . . . . 10 |- ( ph -> ( F supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) ) )
25		eldprdi.0	. . . . . . . . . . . 12 |- .0. = ( 0g ` G )
26		fvex 6201	. . . . . . . . . . . 12 |- ( 0g ` G ) e. _V
27	25, 26	eqeltri 2697	. . . . . . . . . . 11 |- .0. e. _V
28	27	a1i 11	. . . . . . . . . 10 |- ( ph -> .0. e. _V )
29	10, 24, 3, 28	suppssr 7326	. . . . . . . . 9 |- ( ( ph /\ x e. ( I \ ( ( F supp .0. ) u. ( H supp .0. ) ) ) ) -> ( F ` x ) = .0. )
30		ssun2 3777	. . . . . . . . . . 11 |- ( H supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) )
31	30	a1i 11	. . . . . . . . . 10 |- ( ph -> ( H supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) ) )
32	12, 31, 3, 28	suppssr 7326	. . . . . . . . 9 |- ( ( ph /\ x e. ( I \ ( ( F supp .0. ) u. ( H supp .0. ) ) ) ) -> ( H ` x ) = .0. )
33	29, 32	oveq12d 6668	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ ( ( F supp .0. ) u. ( H supp .0. ) ) ) ) -> ( ( F ` x ) .+ ( H ` x ) ) = ( .0. .+ .0. ) )
34		dprdgrp 18404	. . . . . . . . . . 11 |- ( G dom DProd S -> G e. Grp )
35	1, 34	syl 17	. . . . . . . . . 10 |- ( ph -> G e. Grp )
36	9, 25	grpidcl 17450	. . . . . . . . . . 11 |- ( G e. Grp -> .0. e. ( Base ` G ) )
37	35, 36	syl 17	. . . . . . . . . 10 |- ( ph -> .0. e. ( Base ` G ) )
38	9, 17, 25	grplid 17452	. . . . . . . . . 10 |- ( ( G e. Grp /\ .0. e. ( Base ` G ) ) -> ( .0. .+ .0. ) = .0. )
39	35, 37, 38	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( .0. .+ .0. ) = .0. )
40	39	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ ( ( F supp .0. ) u. ( H supp .0. ) ) ) ) -> ( .0. .+ .0. ) = .0. )
41	33, 40	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ x e. ( I \ ( ( F supp .0. ) u. ( H supp .0. ) ) ) ) -> ( ( F ` x ) .+ ( H ` x ) ) = .0. )
42	41, 3	suppss2 7329	. . . . . 6 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) ) )
43		ssfi 8180	. . . . . 6 |- ( ( ( ( F supp .0. ) u. ( H supp .0. ) ) e. Fin /\ ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) C_ ( ( F supp .0. ) u. ( H supp .0. ) ) ) -> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) e. Fin )
44	22, 42, 43	syl2anc 693	. . . . 5 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) e. Fin )
45		funmpt 5926	. . . . . . 7 |- Fun ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) )
46	45	a1i 11	. . . . . 6 |- ( ph -> Fun ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) )
47		mptexg 6484	. . . . . . 7 |- ( I e. _V -> ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) e. _V )
48	3, 47	syl 17	. . . . . 6 |- ( ph -> ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) e. _V )
49		funisfsupp 8280	. . . . . 6 |- ( ( Fun ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) /\ ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) e. _V /\ .0. e. _V ) -> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) finSupp .0. <-> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) e. Fin ) )
50	46, 48, 28, 49	syl3anc 1326	. . . . 5 |- ( ph -> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) finSupp .0. <-> ( ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) supp .0. ) e. Fin ) )
51	44, 50	mpbird 247	. . . 4 |- ( ph -> ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) finSupp .0. )
52	4, 1, 2, 19, 51	dprdwd 18410	. . 3 |- ( ph -> ( x e. I |-> ( ( F ` x ) .+ ( H ` x ) ) ) e. W )
53	14, 52	eqeltrd 2701	. 2 |- ( ph -> ( F oF .+ H ) e. W )
54		eqid 2622	. . 3 |- (Cntz ` G ) = (Cntz ` G )
55		grpmnd 17429	. . . 4 |- ( G e. Grp -> G e. Mnd )
56	35, 55	syl 17	. . 3 |- ( ph -> G e. Mnd )
57		eqid 2622	. . 3 |- ( ( F u. H ) supp .0. ) = ( ( F u. H ) supp .0. )
58	4, 1, 2, 5, 54	dprdfcntz 18414	. . 3 |- ( ph -> ran F C_ ( (Cntz ` G ) ` ran F ) )
59	4, 1, 2, 7, 54	dprdfcntz 18414	. . 3 |- ( ph -> ran H C_ ( (Cntz ` G ) ` ran H ) )
60	4, 1, 2, 53, 54	dprdfcntz 18414	. . 3 |- ( ph -> ran ( F oF .+ H ) C_ ( (Cntz ` G ) ` ran ( F oF .+ H ) ) )
61	56	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> G e. Mnd )
62		vex 3203	. . . . . . . 8 |- x e. _V
63	62	a1i 11	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> x e. _V )
64		eldifi 3732	. . . . . . . . . . 11 |- ( k e. ( I \ x ) -> k e. I )
65	64	adantl 482	. . . . . . . . . 10 |- ( ( x C_ I /\ k e. ( I \ x ) ) -> k e. I )
66		ffvelrn 6357	. . . . . . . . . 10 |- ( ( F : I --> ( Base ` G ) /\ k e. I ) -> ( F ` k ) e. ( Base ` G ) )
67	10, 65, 66	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( F ` k ) e. ( Base ` G ) )
68	67	snssd 4340	. . . . . . . 8 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> { ( F ` k ) } C_ ( Base ` G ) )
69	9, 54	cntzsubm 17768	. . . . . . . 8 |- ( ( G e. Mnd /\ { ( F ` k ) } C_ ( Base ` G ) ) -> ( (Cntz ` G ) ` { ( F ` k ) } ) e. (SubMnd ` G ) )
70	61, 68, 69	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( (Cntz ` G ) ` { ( F ` k ) } ) e. (SubMnd ` G ) )
71	12	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> H : I --> ( Base ` G ) )
72		ffn 6045	. . . . . . . . . 10 |- ( H : I --> ( Base ` G ) -> H Fn I )
73	71, 72	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> H Fn I )
74		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> x C_ I )
75		fnssres 6004	. . . . . . . . 9 |- ( ( H Fn I /\ x C_ I ) -> ( H |` x ) Fn x )
76	73, 74, 75	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( H |` x ) Fn x )
77		fvres 6207	. . . . . . . . . . 11 |- ( y e. x -> ( ( H |` x ) ` y ) = ( H ` y ) )
78	77	adantl 482	. . . . . . . . . 10 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( ( H |` x ) ` y ) = ( H ` y ) )
79	1	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> G dom DProd S )
80	2	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> dom S = I )
81	79, 80	dprdf2 18406	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> S : I --> (SubGrp ` G ) )
82	65	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> k e. I )
83	81, 82	ffvelrnd 6360	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( S ` k ) e. (SubGrp ` G ) )
84	9	subgss 17595	. . . . . . . . . . . . 13 |- ( ( S ` k ) e. (SubGrp ` G ) -> ( S ` k ) C_ ( Base ` G ) )
85	83, 84	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( S ` k ) C_ ( Base ` G ) )
86	5	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> F e. W )
87	4, 79, 80, 86	dprdfcl 18412	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) /\ k e. I ) -> ( F ` k ) e. ( S ` k ) )
88	82, 87	mpdan 702	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( F ` k ) e. ( S ` k ) )
89	88	snssd 4340	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> { ( F ` k ) } C_ ( S ` k ) )
90	9, 54	cntz2ss 17765	. . . . . . . . . . . 12 |- ( ( ( S ` k ) C_ ( Base ` G ) /\ { ( F ` k ) } C_ ( S ` k ) ) -> ( (Cntz ` G ) ` ( S ` k ) ) C_ ( (Cntz ` G ) ` { ( F ` k ) } ) )
91	85, 89, 90	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( (Cntz ` G ) ` ( S ` k ) ) C_ ( (Cntz ` G ) ` { ( F ` k ) } ) )
92	74	sselda 3603	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> y e. I )
93		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> y e. x )
94		simplrr 801	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> k e. ( I \ x ) )
95	94	eldifbd 3587	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> -. k e. x )
96		nelne2 2891	. . . . . . . . . . . . . 14 |- ( ( y e. x /\ -. k e. x ) -> y =/= k )
97	93, 95, 96	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> y =/= k )
98	79, 80, 92, 82, 97, 54	dprdcntz 18407	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( S ` y ) C_ ( (Cntz ` G ) ` ( S ` k ) ) )
99	7	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> H e. W )
100	4, 79, 80, 99	dprdfcl 18412	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) /\ y e. I ) -> ( H ` y ) e. ( S ` y ) )
101	92, 100	mpdan 702	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( H ` y ) e. ( S ` y ) )
102	98, 101	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( H ` y ) e. ( (Cntz ` G ) ` ( S ` k ) ) )
103	91, 102	sseldd 3604	. . . . . . . . . 10 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( H ` y ) e. ( (Cntz ` G ) ` { ( F ` k ) } ) )
104	78, 103	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) /\ y e. x ) -> ( ( H |` x ) ` y ) e. ( (Cntz ` G ) ` { ( F ` k ) } ) )
105	104	ralrimiva 2966	. . . . . . . 8 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> A. y e. x ( ( H |` x ) ` y ) e. ( (Cntz ` G ) ` { ( F ` k ) } ) )
106		ffnfv 6388	. . . . . . . 8 |- ( ( H |` x ) : x --> ( (Cntz ` G ) ` { ( F ` k ) } ) <-> ( ( H |` x ) Fn x /\ A. y e. x ( ( H |` x ) ` y ) e. ( (Cntz ` G ) ` { ( F ` k ) } ) ) )
107	76, 105, 106	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( H |` x ) : x --> ( (Cntz ` G ) ` { ( F ` k ) } ) )
108		resss 5422	. . . . . . . . . 10 |- ( H |` x ) C_ H
109		rnss 5354	. . . . . . . . . 10 |- ( ( H |` x ) C_ H -> ran ( H |` x ) C_ ran H )
110	108, 109	ax-mp 5	. . . . . . . . 9 |- ran ( H |` x ) C_ ran H
111	54	cntzidss 17770	. . . . . . . . 9 |- ( ( ran H C_ ( (Cntz ` G ) ` ran H ) /\ ran ( H |` x ) C_ ran H ) -> ran ( H |` x ) C_ ( (Cntz ` G ) ` ran ( H |` x ) ) )
112	59, 110, 111	sylancl 694	. . . . . . . 8 |- ( ph -> ran ( H |` x ) C_ ( (Cntz ` G ) ` ran ( H |` x ) ) )
113	112	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ran ( H |` x ) C_ ( (Cntz ` G ) ` ran ( H |` x ) ) )
114	21, 28	fsuppres 8300	. . . . . . . 8 |- ( ph -> ( H |` x ) finSupp .0. )
115	114	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( H |` x ) finSupp .0. )
116	25, 54, 61, 63, 70, 107, 113, 115	gsumzsubmcl 18318	. . . . . 6 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( G gsumg ( H |` x ) ) e. ( (Cntz ` G ) ` { ( F ` k ) } ) )
117	116	snssd 4340	. . . . 5 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> { ( G gsumg ( H |` x ) ) } C_ ( (Cntz ` G ) ` { ( F ` k ) } ) )
118	71, 74	fssresd 6071	. . . . . . . 8 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( H |` x ) : x --> ( Base ` G ) )
119	9, 25, 54, 61, 63, 118, 113, 115	gsumzcl 18312	. . . . . . 7 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( G gsumg ( H |` x ) ) e. ( Base ` G ) )
120	119	snssd 4340	. . . . . 6 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> { ( G gsumg ( H |` x ) ) } C_ ( Base ` G ) )
121	9, 54	cntzrec 17766	. . . . . 6 |- ( ( { ( G gsumg ( H |` x ) ) } C_ ( Base ` G ) /\ { ( F ` k ) } C_ ( Base ` G ) ) -> ( { ( G gsumg ( H |` x ) ) } C_ ( (Cntz ` G ) ` { ( F ` k ) } ) <-> { ( F ` k ) } C_ ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) ) )
122	120, 68, 121	syl2anc 693	. . . . 5 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( { ( G gsumg ( H |` x ) ) } C_ ( (Cntz ` G ) ` { ( F ` k ) } ) <-> { ( F ` k ) } C_ ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) ) )
123	117, 122	mpbid 222	. . . 4 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> { ( F ` k ) } C_ ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) )
124		fvex 6201	. . . . 5 |- ( F ` k ) e. _V
125	124	snss 4316	. . . 4 |- ( ( F ` k ) e. ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) <-> { ( F ` k ) } C_ ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) )
126	123, 125	sylibr 224	. . 3 |- ( ( ph /\ ( x C_ I /\ k e. ( I \ x ) ) ) -> ( F ` k ) e. ( (Cntz ` G ) ` { ( G gsumg ( H |` x ) ) } ) )
127	9, 25, 17, 54, 56, 3, 20, 21, 57, 10, 12, 58, 59, 60, 126	gsumzaddlem 18321	. 2 |- ( ph -> ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )
128	53, 127	jca 554	1 |- ( ph -> ( ( F oF .+ H ) e. W /\ ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   X_cixp 7908   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   DProd cdprd 18392

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  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-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-subg 17591  df-cntz 17750  df-dprd 18394

This theorem is referenced by:  dprdfsub  18420