Theorem bj-finsumval0 33147

Description: Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)

Hypotheses

Ref	Expression
bj-finsumval0.1	|- ( ph -> A e. CMnd )
bj-finsumval0.2	|- ( ph -> I e. Fin )
bj-finsumval0.3	|- ( ph -> B : I --> ( Base ` A ) )

Assertion

Ref	Expression
bj-finsumval0	|- ( ph -> ( A FinSum B ) = ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )

Distinct variable groups:   A, s, f, m, n   B, f, m, n, s   f, I, n   ph, f, m, s

Allowed substitution hints:   ph( n)   I( m, s)

Proof of Theorem bj-finsumval0

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

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( A FinSum B ) = ( FinSum ` <. A , B >. )
2		df-bj-finsum 33146	. . . 4 |- FinSum = ( x e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } |-> ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) ) )
3	2	a1i 11	. . 3 |- ( ph -> FinSum = ( x e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } |-> ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) ) ) )
4		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ x = <. A , B >. ) -> x = <. A , B >. )
5	4	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = ( 1st ` <. A , B >. ) )
6		bj-finsumval0.1	. . . . . . . . . . 11 |- ( ph -> A e. CMnd )
7	6	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x = <. A , B >. ) -> A e. CMnd )
8		bj-finsumval0.3	. . . . . . . . . . . 12 |- ( ph -> B : I --> ( Base ` A ) )
9		bj-finsumval0.2	. . . . . . . . . . . 12 |- ( ph -> I e. Fin )
10		fex 6490	. . . . . . . . . . . 12 |- ( ( B : I --> ( Base ` A ) /\ I e. Fin ) -> B e. _V )
11	8, 9, 10	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> B e. _V )
12	11	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x = <. A , B >. ) -> B e. _V )
13		op1stg 7180	. . . . . . . . . 10 |- ( ( A e. CMnd /\ B e. _V ) -> ( 1st ` <. A , B >. ) = A )
14	7, 12, 13	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` <. A , B >. ) = A )
15	5, 14	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = A )
16	4	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
17		op2ndg 7181	. . . . . . . . . 10 |- ( ( A e. CMnd /\ B e. _V ) -> ( 2nd ` <. A , B >. ) = B )
18	7, 12, 17	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` <. A , B >. ) = B )
19	16, 18	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = B )
20	19	dmeqd 5326	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = dom B )
21		fdm 6051	. . . . . . . . . . 11 |- ( B : I --> ( Base ` A ) -> dom B = I )
22	8, 21	syl 17	. . . . . . . . . 10 |- ( ph -> dom B = I )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom B = I )
24	20, 23	eqtrd 2656	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = I )
25		f1oeq3 6129	. . . . . . . . . . . . . . 15 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) <-> f : ( 1 ... m ) -1-1-onto-> I ) )
26	25	biimpd 219	. . . . . . . . . . . . . 14 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> f : ( 1 ... m ) -1-1-onto-> I ) )
27	26	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> f : ( 1 ... m ) -1-1-onto-> I ) )
28	27	adantrd 484	. . . . . . . . . . . 12 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) -> f : ( 1 ... m ) -1-1-onto-> I ) )
29	28	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) -> f : ( 1 ... m ) -1-1-onto-> I ) )
30		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> 1 = 1 )
31		simprl 794	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> ( 1st ` x ) = A )
32	31	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> ( +g ` ( 1st ` x ) ) = ( +g ` A ) )
33	32	adantrr 753	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( +g ` ( 1st ` x ) ) = ( +g ` A ) )
34		simprrl 804	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> ( 2nd ` x ) = B )
35	34	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) /\ n e. NN ) -> ( 2nd ` x ) = B )
36	35	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) /\ n e. NN ) -> ( ( 2nd ` x ) ` ( f ` n ) ) = ( B ` ( f ` n ) ) )
37	36	mpteq2dva 4744	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) = ( n e. NN |-> ( B ` ( f ` n ) ) ) )
38	37	adantrr 753	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) = ( n e. NN |-> ( B ` ( f ` n ) ) ) )
39	30, 33, 38	seqeq123d 12810	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) = seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) )
40		simpr 477	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> m e. NN0 )
41		simprr 796	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> dom ( 2nd ` x ) = I )
42	41	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> dom ( 2nd ` x ) = I )
43	40, 42	jca 554	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( m e. NN0 /\ dom ( 2nd ` x ) = I ) )
44		hashfz1 13134	. . . . . . . . . . . . . . . . . . . 20 |- ( m e. NN0 -> ( # ` ( 1 ... m ) ) = m )
45	44	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( m e. NN0 -> m = ( # ` ( 1 ... m ) ) )
46	45	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` ( 1 ... m ) ) )
47		fzfid 12772	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( 1 ... m ) e. Fin )
48		19.8a 2052	. . . . . . . . . . . . . . . . . . . 20 |- ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> E. f f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) )
49	48	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> E. f f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) )
50		hasheqf1oi 13140	. . . . . . . . . . . . . . . . . . 19 |- ( ( 1 ... m ) e. Fin -> ( E. f f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> ( # ` ( 1 ... m ) ) = ( # ` dom ( 2nd ` x ) ) ) )
51	47, 49, 50	sylc 65	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( # ` ( 1 ... m ) ) = ( # ` dom ( 2nd ` x ) ) )
52		simprr 796	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> dom ( 2nd ` x ) = I )
53	52	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( # ` dom ( 2nd ` x ) ) = ( # ` I ) )
54	46, 51, 53	3eqtrd 2660	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` I ) )
55	43, 54	sylan2 491	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> m = ( # ` I ) )
56	39, 55	fveq12d 6197	. . . . . . . . . . . . . . 15 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) )
57	56	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) <-> s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) )
58	57	biimpd 219	. . . . . . . . . . . . 13 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) -> s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) )
59	58	impancom 456	. . . . . . . . . . . 12 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) -> ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) )
60	59	com12 32	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) -> s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) )
61	29, 60	jcad 555	. . . . . . . . . 10 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) -> ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
62	25	biimprd 238	. . . . . . . . . . . . . 14 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> I -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) ) )
63	62	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( f : ( 1 ... m ) -1-1-onto-> I -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) ) )
64	63	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( f : ( 1 ... m ) -1-1-onto-> I -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) ) )
65	64	adantrd 484	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) ) )
66		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> 1 = 1 )
67		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( 1st ` x ) = A )
68		tru 1487	. . . . . . . . . . . . . . . . . . . . 21 |- T.
69	67, 68	jctir 561	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( ( 1st ` x ) = A /\ T. ) )
70	69	ad2antrl 764	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( ( 1st ` x ) = A /\ T. ) )
71		simpl 473	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` x ) = A /\ T. ) -> ( 1st ` x ) = A )
72	71	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 1st ` x ) = A /\ T. ) -> A = ( 1st ` x ) )
73	70, 72	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> A = ( 1st ` x ) )
74	73	fveq2d 6195	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( +g ` A ) = ( +g ` ( 1st ` x ) ) )
75		simpl 473	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) -> ( 2nd ` x ) = B )
76	75	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) -> B = ( 2nd ` x ) )
77	76	ad2antll 765	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> B = ( 2nd ` x ) )
78	77	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) /\ n e. NN ) -> B = ( 2nd ` x ) )
79	78	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) /\ n e. NN ) -> ( B ` ( f ` n ) ) = ( ( 2nd ` x ) ` ( f ` n ) ) )
80	79	adantlrr 757	. . . . . . . . . . . . . . . . . 18 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) /\ n e. NN ) -> ( B ` ( f ` n ) ) = ( ( 2nd ` x ) ` ( f ` n ) ) )
81	80	mpteq2dva 4744	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( n e. NN |-> ( B ` ( f ` n ) ) ) = ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) )
82	66, 74, 81	seqeq123d 12810	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) = seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) )
83	64	impcom 446	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) )
84		simprr 796	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> m e. NN0 )
85	41	ad2antrl 764	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> dom ( 2nd ` x ) = I )
86	83, 84, 85, 54	syl12anc 1324	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> m = ( # ` I ) )
87	86	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( # ` I ) = m )
88	82, 87	fveq12d 6197	. . . . . . . . . . . . . . 15 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) )
89	88	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) <-> s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) )
90	89	biimpd 219	. . . . . . . . . . . . 13 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) -> s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) )
91	90	impancom 456	. . . . . . . . . . . 12 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) -> ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) )
92	91	com12 32	. . . . . . . . . . 11 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) -> s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) )
93	65, 92	jcad 555	. . . . . . . . . 10 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) ) )
94	61, 93	impbid 202	. . . . . . . . 9 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
95	94	ex 450	. . . . . . . 8 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( m e. NN0 -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) ) )
96	15, 19, 24, 95	syl12anc 1324	. . . . . . 7 |- ( ( ph /\ x = <. A , B >. ) -> ( m e. NN0 -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) ) )
97	96	imp 445	. . . . . 6 |- ( ( ( ph /\ x = <. A , B >. ) /\ m e. NN0 ) -> ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
98	97	exbidv 1850	. . . . 5 |- ( ( ( ph /\ x = <. A , B >. ) /\ m e. NN0 ) -> ( E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
99	98	rexbidva 3049	. . . 4 |- ( ( ph /\ x = <. A , B >. ) -> ( E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) <-> E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
100	99	iotabidv 5872	. . 3 |- ( ( ph /\ x = <. A , B >. ) -> ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ s = ( seq 1 ( ( +g ` ( 1st ` x ) ) , ( n e. NN |-> ( ( 2nd ` x ) ` ( f ` n ) ) ) ) ` m ) ) ) = ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
101		eleq1 2689	. . . . . . . . . 10 |- ( t = I -> ( t e. Fin <-> I e. Fin ) )
102		feq2 6027	. . . . . . . . . 10 |- ( t = I -> ( B : t --> ( Base ` A ) <-> B : I --> ( Base ` A ) ) )
103	101, 102	anbi12d 747	. . . . . . . . 9 |- ( t = I -> ( ( t e. Fin /\ B : t --> ( Base ` A ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
104	103	ceqsexgv 3335	. . . . . . . 8 |- ( I e. Fin -> ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
105	9, 104	syl 17	. . . . . . 7 |- ( ph -> ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
106	9, 8, 105	mpbir2and 957	. . . . . 6 |- ( ph -> E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) )
107		exsimpr 1796	. . . . . 6 |- ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) -> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
108	106, 107	syl 17	. . . . 5 |- ( ph -> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
109		df-rex 2918	. . . . 5 |- ( E. t e. Fin B : t --> ( Base ` A ) <-> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
110	108, 109	sylibr 224	. . . 4 |- ( ph -> E. t e. Fin B : t --> ( Base ` A ) )
111		eleq1 2689	. . . . . . 7 |- ( y = A -> ( y e. CMnd <-> A e. CMnd ) )
112		fveq2 6191	. . . . . . . . 9 |- ( y = A -> ( Base ` y ) = ( Base ` A ) )
113	112	feq3d 6032	. . . . . . . 8 |- ( y = A -> ( z : t --> ( Base ` y ) <-> z : t --> ( Base ` A ) ) )
114	113	rexbidv 3052	. . . . . . 7 |- ( y = A -> ( E. t e. Fin z : t --> ( Base ` y ) <-> E. t e. Fin z : t --> ( Base ` A ) ) )
115	111, 114	anbi12d 747	. . . . . 6 |- ( y = A -> ( ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) <-> ( A e. CMnd /\ E. t e. Fin z : t --> ( Base ` A ) ) ) )
116		feq1 6026	. . . . . . . 8 |- ( z = B -> ( z : t --> ( Base ` A ) <-> B : t --> ( Base ` A ) ) )
117	116	rexbidv 3052	. . . . . . 7 |- ( z = B -> ( E. t e. Fin z : t --> ( Base ` A ) <-> E. t e. Fin B : t --> ( Base ` A ) ) )
118	117	anbi2d 740	. . . . . 6 |- ( z = B -> ( ( A e. CMnd /\ E. t e. Fin z : t --> ( Base ` A ) ) <-> ( A e. CMnd /\ E. t e. Fin B : t --> ( Base ` A ) ) ) )
119	115, 118	opelopabg 4993	. . . . 5 |- ( ( A e. CMnd /\ B e. _V ) -> ( <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } <-> ( A e. CMnd /\ E. t e. Fin B : t --> ( Base ` A ) ) ) )
120	6, 11, 119	syl2anc 693	. . . 4 |- ( ph -> ( <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } <-> ( A e. CMnd /\ E. t e. Fin B : t --> ( Base ` A ) ) ) )
121	6, 110, 120	mpbir2and 957	. . 3 |- ( ph -> <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } )
122		iotaex 5868	. . . 4 |- ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) e. _V
123	122	a1i 11	. . 3 |- ( ph -> ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) e. _V )
124	3, 100, 121, 123	fvmptd 6288	. 2 |- ( ph -> ( FinSum ` <. A , B >. ) = ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )
125	1, 124	syl5eq 2668	1 |- ( ph -> ( A FinSum B ) = ( iota s E. m e. NN0 E. f ( f : ( 1 ... m ) -1-1-onto-> I /\ s = ( seq 1 ( ( +g ` A ) , ( n e. NN |-> ( B ` ( f ` n ) ) ) ) ` ( # ` I ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   <.cop 4183   {copab 4712   |-> cmpt 4729   dom cdm 5114   iotacio 5849   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   1c1 9937   NNcn 11020   NN0cn0 11292   ...cfz 12326   seqcseq 12801   #chash 13117   Basecbs 15857   +g cplusg 15941  CMndccmn 18193   FinSum cfinsum 33145

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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  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-seq 12802  df-hash 13118  df-bj-finsum 33146

This theorem is referenced by: (None)