Theorem gsumzmhm 18337

Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumzmhm.b	|- B = ( Base ` G )
gsumzmhm.z	|- Z = (Cntz ` G )
gsumzmhm.g	|- ( ph -> G e. Mnd )
gsumzmhm.h	|- ( ph -> H e. Mnd )
gsumzmhm.a	|- ( ph -> A e. V )
gsumzmhm.k	|- ( ph -> K e. ( G MndHom H ) )
gsumzmhm.f	|- ( ph -> F : A --> B )
gsumzmhm.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumzmhm.0	|- .0. = ( 0g ` G )
gsumzmhm.w	|- ( ph -> F finSupp .0. )

Assertion

Ref	Expression
gsumzmhm	|- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Proof of Theorem gsumzmhm

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

Step	Hyp	Ref	Expression
1		gsumzmhm.h	. . . . . . 7 |- ( ph -> H e. Mnd )
2		gsumzmhm.a	. . . . . . 7 |- ( ph -> A e. V )
3		eqid 2622	. . . . . . . 8 |- ( 0g ` H ) = ( 0g ` H )
4	3	gsumz 17374	. . . . . . 7 |- ( ( H e. Mnd /\ A e. V ) -> ( H gsumg ( k e. A |-> ( 0g ` H ) ) ) = ( 0g ` H ) )
5	1, 2, 4	syl2anc 693	. . . . . 6 |- ( ph -> ( H gsumg ( k e. A |-> ( 0g ` H ) ) ) = ( 0g ` H ) )
6	5	adantr 481	. . . . 5 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( H gsumg ( k e. A |-> ( 0g ` H ) ) ) = ( 0g ` H ) )
7		gsumzmhm.k	. . . . . . 7 |- ( ph -> K e. ( G MndHom H ) )
8		gsumzmhm.0	. . . . . . . 8 |- .0. = ( 0g ` G )
9	8, 3	mhm0 17343	. . . . . . 7 |- ( K e. ( G MndHom H ) -> ( K ` .0. ) = ( 0g ` H ) )
10	7, 9	syl 17	. . . . . 6 |- ( ph -> ( K ` .0. ) = ( 0g ` H ) )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( K ` .0. ) = ( 0g ` H ) )
12	6, 11	eqtr4d 2659	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( H gsumg ( k e. A |-> ( 0g ` H ) ) ) = ( K ` .0. ) )
13		gsumzmhm.g	. . . . . . . . 9 |- ( ph -> G e. Mnd )
14		gsumzmhm.b	. . . . . . . . . 10 |- B = ( Base ` G )
15	14, 8	mndidcl 17308	. . . . . . . . 9 |- ( G e. Mnd -> .0. e. B )
16	13, 15	syl 17	. . . . . . . 8 |- ( ph -> .0. e. B )
17	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) /\ k e. A ) -> .0. e. B )
18		gsumzmhm.f	. . . . . . . 8 |- ( ph -> F : A --> B )
19		fvex 6201	. . . . . . . . . 10 |- ( 0g ` G ) e. _V
20	8, 19	eqeltri 2697	. . . . . . . . 9 |- .0. e. _V
21	20	a1i 11	. . . . . . . 8 |- ( ph -> .0. e. _V )
22		fex 6490	. . . . . . . . . . 11 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
23	18, 2, 22	syl2anc 693	. . . . . . . . . 10 |- ( ph -> F e. _V )
24		suppimacnv 7306	. . . . . . . . . 10 |- ( ( F e. _V /\ .0. e. _V ) -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
25	23, 21, 24	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( F supp .0. ) = ( `' F " ( _V \ { .0. } ) ) )
26		ssid 3624	. . . . . . . . 9 |- ( `' F " ( _V \ { .0. } ) ) C_ ( `' F " ( _V \ { .0. } ) )
27	25, 26	syl6eqss 3655	. . . . . . . 8 |- ( ph -> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) )
28	18, 2, 21, 27	gsumcllem 18309	. . . . . . 7 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> F = ( k e. A |-> .0. ) )
29		eqid 2622	. . . . . . . . . . 11 |- ( Base ` H ) = ( Base ` H )
30	14, 29	mhmf 17340	. . . . . . . . . 10 |- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) )
31	7, 30	syl 17	. . . . . . . . 9 |- ( ph -> K : B --> ( Base ` H ) )
32	31	feqmptd 6249	. . . . . . . 8 |- ( ph -> K = ( x e. B |-> ( K ` x ) ) )
33	32	adantr 481	. . . . . . 7 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> K = ( x e. B |-> ( K ` x ) ) )
34		fveq2 6191	. . . . . . 7 |- ( x = .0. -> ( K ` x ) = ( K ` .0. ) )
35	17, 28, 33, 34	fmptco 6396	. . . . . 6 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( K o. F ) = ( k e. A |-> ( K ` .0. ) ) )
36	10	mpteq2dv 4745	. . . . . . 7 |- ( ph -> ( k e. A |-> ( K ` .0. ) ) = ( k e. A |-> ( 0g ` H ) ) )
37	36	adantr 481	. . . . . 6 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( k e. A |-> ( K ` .0. ) ) = ( k e. A |-> ( 0g ` H ) ) )
38	35, 37	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( K o. F ) = ( k e. A |-> ( 0g ` H ) ) )
39	38	oveq2d 6666	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( H gsumg ( K o. F ) ) = ( H gsumg ( k e. A |-> ( 0g ` H ) ) ) )
40	28	oveq2d 6666	. . . . . 6 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( G gsumg F ) = ( G gsumg ( k e. A |-> .0. ) ) )
41	8	gsumz 17374	. . . . . . . 8 |- ( ( G e. Mnd /\ A e. V ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
42	13, 2, 41	syl2anc 693	. . . . . . 7 |- ( ph -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
43	42	adantr 481	. . . . . 6 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( G gsumg ( k e. A |-> .0. ) ) = .0. )
44	40, 43	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( G gsumg F ) = .0. )
45	44	fveq2d 6195	. . . 4 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( K ` ( G gsumg F ) ) = ( K ` .0. ) )
46	12, 39, 45	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( `' F " ( _V \ { .0. } ) ) = (/) ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
47	46	ex 450	. 2 |- ( ph -> ( ( `' F " ( _V \ { .0. } ) ) = (/) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) ) )
48	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> G e. Mnd )
49		eqid 2622	. . . . . . . . . 10 |- ( +g ` G ) = ( +g ` G )
50	14, 49	mndcl 17301	. . . . . . . . 9 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) e. B )
51	50	3expb 1266	. . . . . . . 8 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
52	48, 51	sylan 488	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` G ) y ) e. B )
53	18	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> F : A --> B )
54		f1of1 6136	. . . . . . . . . . . 12 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) )
55	54	ad2antll 765	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) )
56		cnvimass 5485	. . . . . . . . . . . 12 |- ( `' F " ( _V \ { .0. } ) ) C_ dom F
57		fdm 6051	. . . . . . . . . . . . 13 |- ( F : A --> B -> dom F = A )
58	53, 57	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> dom F = A )
59	56, 58	syl5sseq 3653	. . . . . . . . . . 11 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( `' F " ( _V \ { .0. } ) ) C_ A )
60		f1ss 6106	. . . . . . . . . . 11 |- ( ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> ( `' F " ( _V \ { .0. } ) ) /\ ( `' F " ( _V \ { .0. } ) ) C_ A ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A )
61	55, 59, 60	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A )
62		f1f 6101	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A )
63	61, 62	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A )
64		fco 6058	. . . . . . . . 9 |- ( ( F : A --> B /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> A ) -> ( F o. f ) : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> B )
65	53, 63, 64	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F o. f ) : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> B )
66	65	ffvelrnda 6359	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) -> ( ( F o. f ) ` x ) e. B )
67		simprl 794	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN )
68		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
69	67, 68	syl6eleq 2711	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. ( ZZ>= ` 1 ) )
70	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> K e. ( G MndHom H ) )
71		eqid 2622	. . . . . . . . . 10 |- ( +g ` H ) = ( +g ` H )
72	14, 49, 71	mhmlin 17342	. . . . . . . . 9 |- ( ( K e. ( G MndHom H ) /\ x e. B /\ y e. B ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
73	72	3expb 1266	. . . . . . . 8 |- ( ( K e. ( G MndHom H ) /\ ( x e. B /\ y e. B ) ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
74	70, 73	sylan 488	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ ( x e. B /\ y e. B ) ) -> ( K ` ( x ( +g ` G ) y ) ) = ( ( K ` x ) ( +g ` H ) ( K ` y ) ) )
75		coass 5654	. . . . . . . . 9 |- ( ( K o. F ) o. f ) = ( K o. ( F o. f ) )
76	75	fveq1i 6192	. . . . . . . 8 |- ( ( ( K o. F ) o. f ) ` x ) = ( ( K o. ( F o. f ) ) ` x )
77		fvco3 6275	. . . . . . . . 9 |- ( ( ( F o. f ) : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) --> B /\ x e. ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) -> ( ( K o. ( F o. f ) ) ` x ) = ( K ` ( ( F o. f ) ` x ) ) )
78	65, 77	sylan 488	. . . . . . . 8 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) -> ( ( K o. ( F o. f ) ) ` x ) = ( K ` ( ( F o. f ) ` x ) ) )
79	76, 78	syl5req 2669	. . . . . . 7 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) -> ( K ` ( ( F o. f ) ` x ) ) = ( ( ( K o. F ) o. f ) ` x ) )
80	52, 66, 69, 74, 79	seqhomo 12848	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K ` ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) = ( seq 1 ( ( +g ` H ) , ( ( K o. F ) o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
81		gsumzmhm.z	. . . . . . . 8 |- Z = (Cntz ` G )
82	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> A e. V )
83		gsumzmhm.c	. . . . . . . . 9 |- ( ph -> ran F C_ ( Z ` ran F ) )
84	83	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran F C_ ( Z ` ran F ) )
85	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ( `' F " ( _V \ { .0. } ) ) )
86		f1ofo 6144	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -onto-> ( `' F " ( _V \ { .0. } ) ) )
87		forn 6118	. . . . . . . . . . 11 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -onto-> ( `' F " ( _V \ { .0. } ) ) -> ran f = ( `' F " ( _V \ { .0. } ) ) )
88	86, 87	syl 17	. . . . . . . . . 10 |- ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ran f = ( `' F " ( _V \ { .0. } ) ) )
89	88	ad2antll 765	. . . . . . . . 9 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran f = ( `' F " ( _V \ { .0. } ) ) )
90	85, 89	sseqtr4d 3642	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F supp .0. ) C_ ran f )
91		eqid 2622	. . . . . . . 8 |- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. )
92	14, 8, 49, 81, 48, 82, 53, 84, 67, 61, 90, 91	gsumval3 18308	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( G gsumg F ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
93	92	fveq2d 6195	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K ` ( G gsumg F ) ) = ( K ` ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) )
94		eqid 2622	. . . . . . 7 |- (Cntz ` H ) = (Cntz ` H )
95	1	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> H e. Mnd )
96	31	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> K : B --> ( Base ` H ) )
97		fco 6058	. . . . . . . 8 |- ( ( K : B --> ( Base ` H ) /\ F : A --> B ) -> ( K o. F ) : A --> ( Base ` H ) )
98	96, 53, 97	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K o. F ) : A --> ( Base ` H ) )
99	81, 94	cntzmhm2 17772	. . . . . . . . 9 |- ( ( K e. ( G MndHom H ) /\ ran F C_ ( Z ` ran F ) ) -> ( K " ran F ) C_ ( (Cntz ` H ) ` ( K " ran F ) ) )
100	70, 84, 99	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K " ran F ) C_ ( (Cntz ` H ) ` ( K " ran F ) ) )
101		rnco2 5642	. . . . . . . 8 |- ran ( K o. F ) = ( K " ran F )
102	101	fveq2i 6194	. . . . . . . 8 |- ( (Cntz ` H ) ` ran ( K o. F ) ) = ( (Cntz ` H ) ` ( K " ran F ) )
103	100, 101, 102	3sstr4g 3646	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ran ( K o. F ) C_ ( (Cntz ` H ) ` ran ( K o. F ) ) )
104		eldifi 3732	. . . . . . . . . . 11 |- ( x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) -> x e. A )
105		fvco3 6275	. . . . . . . . . . 11 |- ( ( F : A --> B /\ x e. A ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
106	53, 104, 105	syl2an 494	. . . . . . . . . 10 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( K o. F ) ` x ) = ( K ` ( F ` x ) ) )
107	20	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> .0. e. _V )
108	53, 85, 82, 107	suppssr 7326	. . . . . . . . . . 11 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) ) -> ( F ` x ) = .0. )
109	108	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K ` ( F ` x ) ) = ( K ` .0. ) )
110	10	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) ) -> ( K ` .0. ) = ( 0g ` H ) )
111	106, 109, 110	3eqtrd 2660	. . . . . . . . 9 |- ( ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) /\ x e. ( A \ ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( K o. F ) ` x ) = ( 0g ` H ) )
112	98, 111	suppss 7325	. . . . . . . 8 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( K o. F ) supp ( 0g ` H ) ) C_ ( `' F " ( _V \ { .0. } ) ) )
113	112, 89	sseqtr4d 3642	. . . . . . 7 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( ( K o. F ) supp ( 0g ` H ) ) C_ ran f )
114		eqid 2622	. . . . . . 7 |- ( ( ( K o. F ) o. f ) supp ( 0g ` H ) ) = ( ( ( K o. F ) o. f ) supp ( 0g ` H ) )
115	29, 3, 71, 94, 95, 82, 98, 103, 67, 61, 113, 114	gsumval3 18308	. . . . . 6 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( H gsumg ( K o. F ) ) = ( seq 1 ( ( +g ` H ) , ( ( K o. F ) o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) )
116	80, 93, 115	3eqtr4rd 2667	. . . . 5 |- ( ( ph /\ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )
117	116	expr 643	. . . 4 |- ( ( ph /\ ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN ) -> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) ) )
118	117	exlimdv 1861	. . 3 |- ( ( ph /\ ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN ) -> ( E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) ) )
119	118	expimpd 629	. 2 |- ( ph -> ( ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) ) )
120		gsumzmhm.w	. . . . 5 |- ( ph -> F finSupp .0. )
121	120	fsuppimpd 8282	. . . 4 |- ( ph -> ( F supp .0. ) e. Fin )
122	25, 121	eqeltrrd 2702	. . 3 |- ( ph -> ( `' F " ( _V \ { .0. } ) ) e. Fin )
123		fz1f1o 14441	. . 3 |- ( ( `' F " ( _V \ { .0. } ) ) e. Fin -> ( ( `' F " ( _V \ { .0. } ) ) = (/) \/ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) )
124	122, 123	syl 17	. 2 |- ( ph -> ( ( `' F " ( _V \ { .0. } ) ) = (/) \/ ( ( # ` ( `' F " ( _V \ { .0. } ) ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) ) ) )
125	47, 119, 124	mpjaod 396	1 |- ( ph -> ( H gsumg ( K o. F ) ) = ( K ` ( G gsumg F ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   1c1 9937   NNcn 11020   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333  Cntzccntz 17748

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-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-map 7859  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-cntz 17750

This theorem is referenced by:  gsummhm  18338  gsumzinv  18345