Theorem gsumzsplit 18327

Description: Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)

Hypotheses

Ref	Expression
gsumzsplit.b	|- B = ( Base ` G )
gsumzsplit.0	|- .0. = ( 0g ` G )
gsumzsplit.p	|- .+ = ( +g ` G )
gsumzsplit.z	|- Z = (Cntz ` G )
gsumzsplit.g	|- ( ph -> G e. Mnd )
gsumzsplit.a	|- ( ph -> A e. V )
gsumzsplit.f	|- ( ph -> F : A --> B )
gsumzsplit.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumzsplit.w	|- ( ph -> F finSupp .0. )
gsumzsplit.i	|- ( ph -> ( C i^i D ) = (/) )
gsumzsplit.u	|- ( ph -> A = ( C u. D ) )

Assertion

Ref	Expression
gsumzsplit	|- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` C ) ) .+ ( G gsumg ( F |` D ) ) ) )

Proof of Theorem gsumzsplit

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzsplit.b	. . 3 |- B = ( Base ` G )
2		gsumzsplit.0	. . 3 |- .0. = ( 0g ` G )
3		gsumzsplit.p	. . 3 |- .+ = ( +g ` G )
4		gsumzsplit.z	. . 3 |- Z = (Cntz ` G )
5		gsumzsplit.g	. . 3 |- ( ph -> G e. Mnd )
6		gsumzsplit.a	. . 3 |- ( ph -> A e. V )
7		gsumzsplit.f	. . . 4 |- ( ph -> F : A --> B )
8		fvex 6201	. . . . . 6 |- ( 0g ` G ) e. _V
9	2, 8	eqeltri 2697	. . . . 5 |- .0. e. _V
10	9	a1i 11	. . . 4 |- ( ph -> .0. e. _V )
11		gsumzsplit.w	. . . 4 |- ( ph -> F finSupp .0. )
12	7, 6, 10, 11	fsuppmptif 8305	. . 3 |- ( ph -> ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) finSupp .0. )
13	7, 6, 10, 11	fsuppmptif 8305	. . 3 |- ( ph -> ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) finSupp .0. )
14	1	submacs 17365	. . . . 5 |- ( G e. Mnd -> (SubMnd ` G ) e. (ACS ` B ) )
15		acsmre 16313	. . . . 5 |- ( (SubMnd ` G ) e. (ACS ` B ) -> (SubMnd ` G ) e. (Moore ` B ) )
16	5, 14, 15	3syl 18	. . . 4 |- ( ph -> (SubMnd ` G ) e. (Moore ` B ) )
17		frn 6053	. . . . 5 |- ( F : A --> B -> ran F C_ B )
18	7, 17	syl 17	. . . 4 |- ( ph -> ran F C_ B )
19		eqid 2622	. . . . 5 |- (mrCls ` (SubMnd ` G ) ) = (mrCls ` (SubMnd ` G ) )
20	19	mrccl 16271	. . . 4 |- ( ( (SubMnd ` G ) e. (Moore ` B ) /\ ran F C_ B ) -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) )
21	16, 18, 20	syl2anc 693	. . 3 |- ( ph -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) )
22		gsumzsplit.c	. . . . 5 |- ( ph -> ran F C_ ( Z ` ran F ) )
23		eqid 2622	. . . . . 6 |- ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) = ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
24	4, 19, 23	cntzspan 18247	. . . . 5 |- ( ( G e. Mnd /\ ran F C_ ( Z ` ran F ) ) -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd )
25	5, 22, 24	syl2anc 693	. . . 4 |- ( ph -> ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd )
26	23, 4	submcmn2 18244	. . . . 5 |- ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) )
27	21, 26	syl 17	. . . 4 |- ( ph -> ( ( G ↾s ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) e. CMnd <-> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) ) )
28	25, 27	mpbid 222	. . 3 |- ( ph -> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) )
29	16, 19, 18	mrcssidd 16285	. . . . . . 7 |- ( ph -> ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
30	29	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> ran F C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
31		ffn 6045	. . . . . . . 8 |- ( F : A --> B -> F Fn A )
32	7, 31	syl 17	. . . . . . 7 |- ( ph -> F Fn A )
33		fnfvelrn 6356	. . . . . . 7 |- ( ( F Fn A /\ k e. A ) -> ( F ` k ) e. ran F )
34	32, 33	sylan 488	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. ran F )
35	30, 34	sseldd 3604	. . . . 5 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
36	2	subm0cl 17352	. . . . . . 7 |- ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) e. (SubMnd ` G ) -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
37	21, 36	syl 17	. . . . . 6 |- ( ph -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ k e. A ) -> .0. e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
39	35, 38	ifcld 4131	. . . 4 |- ( ( ph /\ k e. A ) -> if ( k e. C , ( F ` k ) , .0. ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
40		eqid 2622	. . . 4 |- ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) = ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) )
41	39, 40	fmptd 6385	. . 3 |- ( ph -> ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
42	35, 38	ifcld 4131	. . . 4 |- ( ( ph /\ k e. A ) -> if ( k e. D , ( F ` k ) , .0. ) e. ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
43		eqid 2622	. . . 4 |- ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) = ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) )
44	42, 43	fmptd 6385	. . 3 |- ( ph -> ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
45	1, 2, 3, 4, 5, 6, 12, 13, 21, 28, 41, 44	gsumzadd 18322	. 2 |- ( ph -> ( G gsumg ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) oF .+ ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) ) = ( ( G gsumg ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) .+ ( G gsumg ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) ) )
46	7	feqmptd 6249	. . . . 5 |- ( ph -> F = ( k e. A |-> ( F ` k ) ) )
47		iftrue 4092	. . . . . . . . . 10 |- ( k e. C -> if ( k e. C , ( F ` k ) , .0. ) = ( F ` k ) )
48	47	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> if ( k e. C , ( F ` k ) , .0. ) = ( F ` k ) )
49		gsumzsplit.i	. . . . . . . . . . . . . . 15 |- ( ph -> ( C i^i D ) = (/) )
50		noel 3919	. . . . . . . . . . . . . . . 16 |- -. k e. (/)
51		eleq2 2690	. . . . . . . . . . . . . . . 16 |- ( ( C i^i D ) = (/) -> ( k e. ( C i^i D ) <-> k e. (/) ) )
52	50, 51	mtbiri 317	. . . . . . . . . . . . . . 15 |- ( ( C i^i D ) = (/) -> -. k e. ( C i^i D ) )
53	49, 52	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> -. k e. ( C i^i D ) )
54	53	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ k e. A ) -> -. k e. ( C i^i D ) )
55		elin 3796	. . . . . . . . . . . . 13 |- ( k e. ( C i^i D ) <-> ( k e. C /\ k e. D ) )
56	54, 55	sylnib 318	. . . . . . . . . . . 12 |- ( ( ph /\ k e. A ) -> -. ( k e. C /\ k e. D ) )
57		imnan 438	. . . . . . . . . . . 12 |- ( ( k e. C -> -. k e. D ) <-> -. ( k e. C /\ k e. D ) )
58	56, 57	sylibr 224	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( k e. C -> -. k e. D ) )
59	58	imp 445	. . . . . . . . . 10 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> -. k e. D )
60	59	iffalsed 4097	. . . . . . . . 9 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> if ( k e. D , ( F ` k ) , .0. ) = .0. )
61	48, 60	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) = ( ( F ` k ) .+ .0. ) )
62	7	ffvelrnda 6359	. . . . . . . . . 10 |- ( ( ph /\ k e. A ) -> ( F ` k ) e. B )
63	1, 3, 2	mndrid 17312	. . . . . . . . . . 11 |- ( ( G e. Mnd /\ ( F ` k ) e. B ) -> ( ( F ` k ) .+ .0. ) = ( F ` k ) )
64	5, 63	sylan 488	. . . . . . . . . 10 |- ( ( ph /\ ( F ` k ) e. B ) -> ( ( F ` k ) .+ .0. ) = ( F ` k ) )
65	62, 64	syldan 487	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> ( ( F ` k ) .+ .0. ) = ( F ` k ) )
66	65	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> ( ( F ` k ) .+ .0. ) = ( F ` k ) )
67	61, 66	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ k e. A ) /\ k e. C ) -> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) = ( F ` k ) )
68	58	con2d 129	. . . . . . . . . . 11 |- ( ( ph /\ k e. A ) -> ( k e. D -> -. k e. C ) )
69	68	imp 445	. . . . . . . . . 10 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> -. k e. C )
70	69	iffalsed 4097	. . . . . . . . 9 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> if ( k e. C , ( F ` k ) , .0. ) = .0. )
71		iftrue 4092	. . . . . . . . . 10 |- ( k e. D -> if ( k e. D , ( F ` k ) , .0. ) = ( F ` k ) )
72	71	adantl 482	. . . . . . . . 9 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> if ( k e. D , ( F ` k ) , .0. ) = ( F ` k ) )
73	70, 72	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) = ( .0. .+ ( F ` k ) ) )
74	1, 3, 2	mndlid 17311	. . . . . . . . . . 11 |- ( ( G e. Mnd /\ ( F ` k ) e. B ) -> ( .0. .+ ( F ` k ) ) = ( F ` k ) )
75	5, 74	sylan 488	. . . . . . . . . 10 |- ( ( ph /\ ( F ` k ) e. B ) -> ( .0. .+ ( F ` k ) ) = ( F ` k ) )
76	62, 75	syldan 487	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> ( .0. .+ ( F ` k ) ) = ( F ` k ) )
77	76	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> ( .0. .+ ( F ` k ) ) = ( F ` k ) )
78	73, 77	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ k e. A ) /\ k e. D ) -> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) = ( F ` k ) )
79		gsumzsplit.u	. . . . . . . . . 10 |- ( ph -> A = ( C u. D ) )
80	79	eleq2d 2687	. . . . . . . . 9 |- ( ph -> ( k e. A <-> k e. ( C u. D ) ) )
81		elun 3753	. . . . . . . . 9 |- ( k e. ( C u. D ) <-> ( k e. C \/ k e. D ) )
82	80, 81	syl6bb 276	. . . . . . . 8 |- ( ph -> ( k e. A <-> ( k e. C \/ k e. D ) ) )
83	82	biimpa 501	. . . . . . 7 |- ( ( ph /\ k e. A ) -> ( k e. C \/ k e. D ) )
84	67, 78, 83	mpjaodan 827	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) = ( F ` k ) )
85	84	mpteq2dva 4744	. . . . 5 |- ( ph -> ( k e. A |-> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) ) = ( k e. A |-> ( F ` k ) ) )
86	46, 85	eqtr4d 2659	. . . 4 |- ( ph -> F = ( k e. A |-> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) ) )
87	1, 2	mndidcl 17308	. . . . . . . 8 |- ( G e. Mnd -> .0. e. B )
88	5, 87	syl 17	. . . . . . 7 |- ( ph -> .0. e. B )
89	88	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> .0. e. B )
90	62, 89	ifcld 4131	. . . . 5 |- ( ( ph /\ k e. A ) -> if ( k e. C , ( F ` k ) , .0. ) e. B )
91	62, 89	ifcld 4131	. . . . 5 |- ( ( ph /\ k e. A ) -> if ( k e. D , ( F ` k ) , .0. ) e. B )
92		eqidd 2623	. . . . 5 |- ( ph -> ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) = ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) )
93		eqidd 2623	. . . . 5 |- ( ph -> ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) = ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) )
94	6, 90, 91, 92, 93	offval2 6914	. . . 4 |- ( ph -> ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) oF .+ ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) = ( k e. A |-> ( if ( k e. C , ( F ` k ) , .0. ) .+ if ( k e. D , ( F ` k ) , .0. ) ) ) )
95	86, 94	eqtr4d 2659	. . 3 |- ( ph -> F = ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) oF .+ ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) )
96	95	oveq2d 6666	. 2 |- ( ph -> ( G gsumg F ) = ( G gsumg ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) oF .+ ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) ) )
97	46	reseq1d 5395	. . . . . 6 |- ( ph -> ( F |` C ) = ( ( k e. A |-> ( F ` k ) ) |` C ) )
98		ssun1 3776	. . . . . . . 8 |- C C_ ( C u. D )
99	98, 79	syl5sseqr 3654	. . . . . . 7 |- ( ph -> C C_ A )
100	47	mpteq2ia 4740	. . . . . . . 8 |- ( k e. C |-> if ( k e. C , ( F ` k ) , .0. ) ) = ( k e. C |-> ( F ` k ) )
101		resmpt 5449	. . . . . . . 8 |- ( C C_ A -> ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) = ( k e. C |-> if ( k e. C , ( F ` k ) , .0. ) ) )
102		resmpt 5449	. . . . . . . 8 |- ( C C_ A -> ( ( k e. A |-> ( F ` k ) ) |` C ) = ( k e. C |-> ( F ` k ) ) )
103	100, 101, 102	3eqtr4a 2682	. . . . . . 7 |- ( C C_ A -> ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) = ( ( k e. A |-> ( F ` k ) ) |` C ) )
104	99, 103	syl 17	. . . . . 6 |- ( ph -> ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) = ( ( k e. A |-> ( F ` k ) ) |` C ) )
105	97, 104	eqtr4d 2659	. . . . 5 |- ( ph -> ( F |` C ) = ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) )
106	105	oveq2d 6666	. . . 4 |- ( ph -> ( G gsumg ( F |` C ) ) = ( G gsumg ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) ) )
107	90, 40	fmptd 6385	. . . . 5 |- ( ph -> ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) : A --> B )
108		frn 6053	. . . . . . 7 |- ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) -> ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
109	41, 108	syl 17	. . . . . 6 |- ( ph -> ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
110	4	cntzidss 17770	. . . . . 6 |- ( ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) /\ ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) C_ ( Z ` ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) )
111	28, 109, 110	syl2anc 693	. . . . 5 |- ( ph -> ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) C_ ( Z ` ran ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) )
112		eldifn 3733	. . . . . . . 8 |- ( k e. ( A \ C ) -> -. k e. C )
113	112	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( A \ C ) ) -> -. k e. C )
114	113	iffalsed 4097	. . . . . 6 |- ( ( ph /\ k e. ( A \ C ) ) -> if ( k e. C , ( F ` k ) , .0. ) = .0. )
115	114, 6	suppss2 7329	. . . . 5 |- ( ph -> ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) supp .0. ) C_ C )
116	1, 2, 4, 5, 6, 107, 111, 115, 12	gsumzres 18310	. . . 4 |- ( ph -> ( G gsumg ( ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) |` C ) ) = ( G gsumg ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) )
117	106, 116	eqtrd 2656	. . 3 |- ( ph -> ( G gsumg ( F |` C ) ) = ( G gsumg ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) )
118	46	reseq1d 5395	. . . . . 6 |- ( ph -> ( F |` D ) = ( ( k e. A |-> ( F ` k ) ) |` D ) )
119		ssun2 3777	. . . . . . . 8 |- D C_ ( C u. D )
120	119, 79	syl5sseqr 3654	. . . . . . 7 |- ( ph -> D C_ A )
121	71	mpteq2ia 4740	. . . . . . . 8 |- ( k e. D |-> if ( k e. D , ( F ` k ) , .0. ) ) = ( k e. D |-> ( F ` k ) )
122		resmpt 5449	. . . . . . . 8 |- ( D C_ A -> ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) = ( k e. D |-> if ( k e. D , ( F ` k ) , .0. ) ) )
123		resmpt 5449	. . . . . . . 8 |- ( D C_ A -> ( ( k e. A |-> ( F ` k ) ) |` D ) = ( k e. D |-> ( F ` k ) ) )
124	121, 122, 123	3eqtr4a 2682	. . . . . . 7 |- ( D C_ A -> ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) = ( ( k e. A |-> ( F ` k ) ) |` D ) )
125	120, 124	syl 17	. . . . . 6 |- ( ph -> ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) = ( ( k e. A |-> ( F ` k ) ) |` D ) )
126	118, 125	eqtr4d 2659	. . . . 5 |- ( ph -> ( F |` D ) = ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) )
127	126	oveq2d 6666	. . . 4 |- ( ph -> ( G gsumg ( F |` D ) ) = ( G gsumg ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) ) )
128	91, 43	fmptd 6385	. . . . 5 |- ( ph -> ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) : A --> B )
129		frn 6053	. . . . . . 7 |- ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) : A --> ( (mrCls ` (SubMnd ` G ) ) ` ran F ) -> ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
130	44, 129	syl 17	. . . . . 6 |- ( ph -> ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) )
131	4	cntzidss 17770	. . . . . 6 |- ( ( ( (mrCls ` (SubMnd ` G ) ) ` ran F ) C_ ( Z ` ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) /\ ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) C_ ( (mrCls ` (SubMnd ` G ) ) ` ran F ) ) -> ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) C_ ( Z ` ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) )
132	28, 130, 131	syl2anc 693	. . . . 5 |- ( ph -> ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) C_ ( Z ` ran ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) )
133		eldifn 3733	. . . . . . . 8 |- ( k e. ( A \ D ) -> -. k e. D )
134	133	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( A \ D ) ) -> -. k e. D )
135	134	iffalsed 4097	. . . . . 6 |- ( ( ph /\ k e. ( A \ D ) ) -> if ( k e. D , ( F ` k ) , .0. ) = .0. )
136	135, 6	suppss2 7329	. . . . 5 |- ( ph -> ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) supp .0. ) C_ D )
137	1, 2, 4, 5, 6, 128, 132, 136, 13	gsumzres 18310	. . . 4 |- ( ph -> ( G gsumg ( ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) |` D ) ) = ( G gsumg ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) )
138	127, 137	eqtrd 2656	. . 3 |- ( ph -> ( G gsumg ( F |` D ) ) = ( G gsumg ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) )
139	117, 138	oveq12d 6668	. 2 |- ( ph -> ( ( G gsumg ( F |` C ) ) .+ ( G gsumg ( F |` D ) ) ) = ( ( G gsumg ( k e. A |-> if ( k e. C , ( F ` k ) , .0. ) ) ) .+ ( G gsumg ( k e. A |-> if ( k e. D , ( F ` k ) , .0. ) ) ) ) )
140	45, 96, 139	3eqtr4d 2666	1 |- ( ph -> ( G gsumg F ) = ( ( G gsumg ( F |` C ) ) .+ ( G gsumg ( F |` D ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   finSupp cfsupp 8275   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748  CMndccmn 18193

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  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-submnd 17336  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  gsumsplit  18328  gsumzunsnd  18355  dpjidcl  18457