Theorem gsumval3lem2 18307

Description: Lemma 2 for gsumval3 18308. (Contributed by AV, 31-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	|- B = ( Base ` G )
gsumval3.0	|- .0. = ( 0g ` G )
gsumval3.p	|- .+ = ( +g ` G )
gsumval3.z	|- Z = (Cntz ` G )
gsumval3.g	|- ( ph -> G e. Mnd )
gsumval3.a	|- ( ph -> A e. V )
gsumval3.f	|- ( ph -> F : A --> B )
gsumval3.c	|- ( ph -> ran F C_ ( Z ` ran F ) )
gsumval3.m	|- ( ph -> M e. NN )
gsumval3.h	|- ( ph -> H : ( 1 ... M ) -1-1-> A )
gsumval3.n	|- ( ph -> ( F supp .0. ) C_ ran H )
gsumval3.w	|- W = ( ( F o. H ) supp .0. )

Assertion

Ref	Expression
gsumval3lem2	|- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) )

Distinct variable groups:   .+ , f   A, f   ph, f   f, G   f, M   B, f   f, F   f, H   f, W

Allowed substitution hints:   V( f)   .0. ( f)   Z( f)

Proof of Theorem gsumval3lem2

Dummy variables g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.h	. . . . . . 7 |- ( ph -> H : ( 1 ... M ) -1-1-> A )
2		f1f 6101	. . . . . . 7 |- ( H : ( 1 ... M ) -1-1-> A -> H : ( 1 ... M ) --> A )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> H : ( 1 ... M ) --> A )
4		fzfid 12772	. . . . . 6 |- ( ph -> ( 1 ... M ) e. Fin )
5		gsumval3.a	. . . . . 6 |- ( ph -> A e. V )
6		fex2 7121	. . . . . 6 |- ( ( H : ( 1 ... M ) --> A /\ ( 1 ... M ) e. Fin /\ A e. V ) -> H e. _V )
7	3, 4, 5, 6	syl3anc 1326	. . . . 5 |- ( ph -> H e. _V )
8		vex 3203	. . . . 5 |- f e. _V
9		coexg 7117	. . . . 5 |- ( ( H e. _V /\ f e. _V ) -> ( H o. f ) e. _V )
10	7, 8, 9	sylancl 694	. . . 4 |- ( ph -> ( H o. f ) e. _V )
11	10	ad2antrr 762	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H o. f ) e. _V )
12		gsumval3.b	. . . . 5 |- B = ( Base ` G )
13		gsumval3.0	. . . . 5 |- .0. = ( 0g ` G )
14		gsumval3.p	. . . . 5 |- .+ = ( +g ` G )
15		gsumval3.z	. . . . 5 |- Z = (Cntz ` G )
16		gsumval3.g	. . . . 5 |- ( ph -> G e. Mnd )
17		gsumval3.f	. . . . 5 |- ( ph -> F : A --> B )
18		gsumval3.c	. . . . 5 |- ( ph -> ran F C_ ( Z ` ran F ) )
19		gsumval3.m	. . . . 5 |- ( ph -> M e. NN )
20		gsumval3.n	. . . . 5 |- ( ph -> ( F supp .0. ) C_ ran H )
21		gsumval3.w	. . . . 5 |- W = ( ( F o. H ) supp .0. )
22	12, 13, 14, 15, 16, 5, 17, 18, 19, 1, 20, 21	gsumval3lem1 18306	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) )
23		resexg 5442	. . . . . . . . 9 |- ( H e. _V -> ( H |` W ) e. _V )
24	7, 23	syl 17	. . . . . . . 8 |- ( ph -> ( H |` W ) e. _V )
25	24	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) e. _V )
26	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> H : ( 1 ... M ) -1-1-> A )
27		suppssdm 7308	. . . . . . . . . . . 12 |- ( ( F o. H ) supp .0. ) C_ dom ( F o. H )
28	21, 27	eqsstri 3635	. . . . . . . . . . 11 |- W C_ dom ( F o. H )
29		fco 6058	. . . . . . . . . . . . 13 |- ( ( F : A --> B /\ H : ( 1 ... M ) --> A ) -> ( F o. H ) : ( 1 ... M ) --> B )
30	17, 3, 29	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( F o. H ) : ( 1 ... M ) --> B )
31		fdm 6051	. . . . . . . . . . . 12 |- ( ( F o. H ) : ( 1 ... M ) --> B -> dom ( F o. H ) = ( 1 ... M ) )
32	30, 31	syl 17	. . . . . . . . . . 11 |- ( ph -> dom ( F o. H ) = ( 1 ... M ) )
33	28, 32	syl5sseq 3653	. . . . . . . . . 10 |- ( ph -> W C_ ( 1 ... M ) )
34	33	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W C_ ( 1 ... M ) )
35		f1ores 6151	. . . . . . . . 9 |- ( ( H : ( 1 ... M ) -1-1-> A /\ W C_ ( 1 ... M ) ) -> ( H |` W ) : W -1-1-onto-> ( H " W ) )
36	26, 34, 35	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) : W -1-1-onto-> ( H " W ) )
37	21	imaeq2i 5464	. . . . . . . . . . 11 |- ( H " W ) = ( H " ( ( F o. H ) supp .0. ) )
38		fex 6490	. . . . . . . . . . . . . . 15 |- ( ( F : A --> B /\ A e. V ) -> F e. _V )
39	17, 5, 38	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> F e. _V )
40		ovex 6678	. . . . . . . . . . . . . . 15 |- ( 1 ... M ) e. _V
41		fex 6490	. . . . . . . . . . . . . . 15 |- ( ( H : ( 1 ... M ) --> A /\ ( 1 ... M ) e. _V ) -> H e. _V )
42	3, 40, 41	sylancl 694	. . . . . . . . . . . . . 14 |- ( ph -> H e. _V )
43	39, 42	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( F e. _V /\ H e. _V ) )
44		f1fun 6103	. . . . . . . . . . . . . . 15 |- ( H : ( 1 ... M ) -1-1-> A -> Fun H )
45	1, 44	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> Fun H )
46	45, 20	jca 554	. . . . . . . . . . . . 13 |- ( ph -> ( Fun H /\ ( F supp .0. ) C_ ran H ) )
47		imacosupp 7335	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ H e. _V ) -> ( ( Fun H /\ ( F supp .0. ) C_ ran H ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) ) )
48	43, 46, 47	sylc 65	. . . . . . . . . . . 12 |- ( ph -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) )
49	48	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ W =/= (/) ) -> ( H " ( ( F o. H ) supp .0. ) ) = ( F supp .0. ) )
50	37, 49	syl5eq 2668	. . . . . . . . . 10 |- ( ( ph /\ W =/= (/) ) -> ( H " W ) = ( F supp .0. ) )
51	50	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H " W ) = ( F supp .0. ) )
52		f1oeq3 6129	. . . . . . . . 9 |- ( ( H " W ) = ( F supp .0. ) -> ( ( H |` W ) : W -1-1-onto-> ( H " W ) <-> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) )
53	51, 52	syl 17	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H |` W ) : W -1-1-onto-> ( H " W ) <-> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) )
54	36, 53	mpbid 222	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( H |` W ) : W -1-1-onto-> ( F supp .0. ) )
55		f1oen3g 7971	. . . . . . 7 |- ( ( ( H |` W ) e. _V /\ ( H |` W ) : W -1-1-onto-> ( F supp .0. ) ) -> W ~~ ( F supp .0. ) )
56	25, 54, 55	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W ~~ ( F supp .0. ) )
57		fzfi 12771	. . . . . . . . 9 |- ( 1 ... M ) e. Fin
58		ssfi 8180	. . . . . . . . 9 |- ( ( ( 1 ... M ) e. Fin /\ W C_ ( 1 ... M ) ) -> W e. Fin )
59	57, 33, 58	sylancr 695	. . . . . . . 8 |- ( ph -> W e. Fin )
60	59	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> W e. Fin )
61		f1f1orn 6148	. . . . . . . . . . . . 13 |- ( H : ( 1 ... M ) -1-1-> A -> H : ( 1 ... M ) -1-1-onto-> ran H )
62	1, 61	syl 17	. . . . . . . . . . . 12 |- ( ph -> H : ( 1 ... M ) -1-1-onto-> ran H )
63		f1oen3g 7971	. . . . . . . . . . . 12 |- ( ( H e. _V /\ H : ( 1 ... M ) -1-1-onto-> ran H ) -> ( 1 ... M ) ~~ ran H )
64	7, 62, 63	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( 1 ... M ) ~~ ran H )
65		enfi 8176	. . . . . . . . . . 11 |- ( ( 1 ... M ) ~~ ran H -> ( ( 1 ... M ) e. Fin <-> ran H e. Fin ) )
66	64, 65	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( 1 ... M ) e. Fin <-> ran H e. Fin ) )
67	57, 66	mpbii 223	. . . . . . . . 9 |- ( ph -> ran H e. Fin )
68		ssfi 8180	. . . . . . . . 9 |- ( ( ran H e. Fin /\ ( F supp .0. ) C_ ran H ) -> ( F supp .0. ) e. Fin )
69	67, 20, 68	syl2anc 693	. . . . . . . 8 |- ( ph -> ( F supp .0. ) e. Fin )
70	69	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( F supp .0. ) e. Fin )
71		hashen 13135	. . . . . . 7 |- ( ( W e. Fin /\ ( F supp .0. ) e. Fin ) -> ( ( # ` W ) = ( # ` ( F supp .0. ) ) <-> W ~~ ( F supp .0. ) ) )
72	60, 70, 71	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( # ` W ) = ( # ` ( F supp .0. ) ) <-> W ~~ ( F supp .0. ) ) )
73	56, 72	mpbird 247	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( # ` W ) = ( # ` ( F supp .0. ) ) )
74	73	fveq2d 6195	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) )
75	22, 74	jca 554	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) ) )
76		f1oeq1 6127	. . . . 5 |- ( g = ( H o. f ) -> ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) <-> ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) )
77		coeq2 5280	. . . . . . . 8 |- ( g = ( H o. f ) -> ( F o. g ) = ( F o. ( H o. f ) ) )
78	77	seqeq3d 12809	. . . . . . 7 |- ( g = ( H o. f ) -> seq 1 ( .+ , ( F o. g ) ) = seq 1 ( .+ , ( F o. ( H o. f ) ) ) )
79	78	fveq1d 6193	. . . . . 6 |- ( g = ( H o. f ) -> ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) )
80	79	eqeq2d 2632	. . . . 5 |- ( g = ( H o. f ) -> ( ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) <-> ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) ) )
81	76, 80	anbi12d 747	. . . 4 |- ( g = ( H o. f ) -> ( ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) ) ) )
82	81	spcegv 3294	. . 3 |- ( ( H o. f ) e. _V -> ( ( ( H o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` ( F supp .0. ) ) ) ) -> E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) )
83	11, 75, 82	sylc 65	. 2 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) )
84	16	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> G e. Mnd )
85	5	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> A e. V )
86	17	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> F : A --> B )
87	18	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ran F C_ ( Z ` ran F ) )
88	21	neeq1i 2858	. . . . . . . . . 10 |- ( W =/= (/) <-> ( ( F o. H ) supp .0. ) =/= (/) )
89		supp0cosupp0 7334	. . . . . . . . . . . 12 |- ( ( F e. _V /\ H e. _V ) -> ( ( F supp .0. ) = (/) -> ( ( F o. H ) supp .0. ) = (/) ) )
90	89	necon3d 2815	. . . . . . . . . . 11 |- ( ( F e. _V /\ H e. _V ) -> ( ( ( F o. H ) supp .0. ) =/= (/) -> ( F supp .0. ) =/= (/) ) )
91	39, 42, 90	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( ( F o. H ) supp .0. ) =/= (/) -> ( F supp .0. ) =/= (/) ) )
92	88, 91	syl5bi 232	. . . . . . . . 9 |- ( ph -> ( W =/= (/) -> ( F supp .0. ) =/= (/) ) )
93	92	imp 445	. . . . . . . 8 |- ( ( ph /\ W =/= (/) ) -> ( F supp .0. ) =/= (/) )
94	93	adantr 481	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( F supp .0. ) =/= (/) )
95	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( F supp .0. ) C_ ran H )
96		frn 6053	. . . . . . . . . 10 |- ( H : ( 1 ... M ) --> A -> ran H C_ A )
97	3, 96	syl 17	. . . . . . . . 9 |- ( ph -> ran H C_ A )
98	97	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ran H C_ A )
99	95, 98	sstrd 3613	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( F supp .0. ) C_ A )
100	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 99	gsumval3eu 18305	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> E! x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) )
101		iota1 5865	. . . . . 6 |- ( E! x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( iota x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) = x ) )
102	100, 101	syl 17	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( iota x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) = x ) )
103		eqid 2622	. . . . . . 7 |- ( F supp .0. ) = ( F supp .0. )
104		simprl 794	. . . . . . 7 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> -. A e. ran ... )
105	12, 13, 14, 15, 84, 85, 86, 87, 70, 94, 103, 104	gsumval3a 18304	. . . . . 6 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( G gsumg F ) = ( iota x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) )
106	105	eqeq1d 2624	. . . . 5 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( ( G gsumg F ) = x <-> ( iota x E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) = x ) )
107	102, 106	bitr4d 271	. . . 4 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = x ) )
108	107	alrimiv 1855	. . 3 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> A. x ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = x ) )
109		fvex 6201	. . . 4 |- ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) e. _V
110		eqeq1 2626	. . . . . . 7 |- ( x = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) -> ( x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) <-> ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) )
111	110	anbi2d 740	. . . . . 6 |- ( x = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) -> ( ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) )
112	111	exbidv 1850	. . . . 5 |- ( x = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) ) )
113		eqeq2 2633	. . . . 5 |- ( x = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) -> ( ( G gsumg F ) = x <-> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) ) )
114	112, 113	bibi12d 335	. . . 4 |- ( x = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) -> ( ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = x ) <-> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) ) ) )
115	109, 114	spcv 3299	. . 3 |- ( A. x ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ x = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = x ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) ) )
116	108, 115	syl 17	. 2 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( E. g ( g : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) /\ ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) = ( seq 1 ( .+ , ( F o. g ) ) ` ( # ` ( F supp .0. ) ) ) ) <-> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) ) )
117	83, 116	mpbid 222	1 |- ( ( ( ph /\ W =/= (/) ) /\ ( -. A e. ran ... /\ f Isom < , < ( ( 1 ... ( # ` W ) ) , W ) ) ) -> ( G gsumg F ) = ( seq 1 ( .+ , ( F o. ( H o. f ) ) ) ` ( # ` W ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   iotacio 5849   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   supp csupp 7295   ~~ cen 7952   Fincfn 7955   1c1 9937   < clt 10074   NNcn 11020   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-cntz 17750

This theorem is referenced by:  gsumval3  18308