Theorem bj-finsumval0 33735

Description: Value of a finite sum. (Contributed by BJ, 9-Jun-2019.)

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 6285	. 2 |- ( A FinSum B ) = ( FinSum ` <. A , B >. )
2		df-bj-finsum 33734	. . . 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 461	. . . . . . . . . 10 |- ( ( ph /\ x = <. A , B >. ) -> x = <. A , B >. )
5	4	fveq2d 5868	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = ( 1st ` <. A , B >. ) )
6		bj-finsumval0.1	. . . . . . . . . . 11 |- ( ph -> A e. CMnd )
7	6	adantr 465	. . . . . . . . . 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 6131	. . . . . . . . . . . 12 |- ( ( B : I --> ( Base ` A ) /\ I e. Fin ) -> B e. _V )
11	8, 9, 10	syl2anc 661	. . . . . . . . . . 11 |- ( ph -> B e. _V )
12	11	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ x = <. A , B >. ) -> B e. _V )
13		op1stg 6793	. . . . . . . . . 10 |- ( ( A e. CMnd /\ B e. _V ) -> ( 1st ` <. A , B >. ) = A )
14	7, 12, 13	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` <. A , B >. ) = A )
15	5, 14	eqtrd 2508	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = A )
16	4	fveq2d 5868	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
17		op2ndg 6794	. . . . . . . . . 10 |- ( ( A e. CMnd /\ B e. _V ) -> ( 2nd ` <. A , B >. ) = B )
18	7, 12, 17	syl2anc 661	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` <. A , B >. ) = B )
19	16, 18	eqtrd 2508	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = B )
20	19	dmeqd 5203	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = dom B )
21		fdm 5733	. . . . . . . . . . 11 |- ( B : I --> ( Base ` A ) -> dom B = I )
22	8, 21	syl 16	. . . . . . . . . 10 |- ( ph -> dom B = I )
23	22	adantr 465	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom B = I )
24	20, 23	eqtrd 2508	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = I )
25		f1oeq3 5807	. . . . . . . . . . . . . . 15 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) <-> f : ( 1 ... m ) -1-1-onto-> I ) )
26	25	biimpd 207	. . . . . . . . . . . . . 14 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> f : ( 1 ... m ) -1-1-onto-> I ) )
27	26	ad2antll 728	. . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 465	. . . . . . . . . . 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 2468	. . . . . . . . . . . . . . . . 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 755	. . . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 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 763	. . . . . . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . . . . . . 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 5866	. . . . . . . . . . . . . . . . . . 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 4533	. . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . 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 12080	. . . . . . . . . . . . . . . 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 461	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> m e. NN0 )
41		simprr 756	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> dom ( 2nd ` x ) = I )
42	41	adantr 465	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> dom ( 2nd ` x ) = I )
43	40, 42	jca 532	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) -> ( m e. NN0 /\ dom ( 2nd ` x ) = I ) )
44		hashfz1 12383	. . . . . . . . . . . . . . . . . . . . 21 |- ( m e. NN0 -> ( # ` ( 1 ... m ) ) = m )
45	44	eqcomd 2475	. . . . . . . . . . . . . . . . . . . 20 |- ( m e. NN0 -> m = ( # ` ( 1 ... m ) ) )
46	45	ad2antrl 727	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` ( 1 ... m ) ) )
47		fzfid 12047	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> ( 1 ... m ) e. Fin )
48		f1ofo 5821	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> f : ( 1 ... m ) -onto-> dom ( 2nd ` x ) )
49	48	adantr 465	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> f : ( 1 ... m ) -onto-> dom ( 2nd ` x ) )
50		fornex 6750	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 1 ... m ) e. Fin -> ( f : ( 1 ... m ) -onto-> dom ( 2nd ` x ) -> dom ( 2nd ` x ) e. _V ) )
51	47, 49, 50	sylc 60	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> dom ( 2nd ` x ) e. _V )
52	47, 51	jca 532	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> ( ( 1 ... m ) e. Fin /\ dom ( 2nd ` x ) e. _V ) )
53	52	adantrr 716	. . . . . . . . . . . . . . . . . . . 20 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( ( 1 ... m ) e. Fin /\ dom ( 2nd ` x ) e. _V ) )
54		19.8a 1806	. . . . . . . . . . . . . . . . . . . . 21 |- ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> E. f f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) )
55	54	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( 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 ) )
56		hasheqf1oi 12388	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1 ... m ) e. Fin /\ dom ( 2nd ` x ) e. _V ) -> ( E. f f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) -> ( # ` ( 1 ... m ) ) = ( # ` dom ( 2nd ` x ) ) ) )
57	53, 55, 56	sylc 60	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( # ` ( 1 ... m ) ) = ( # ` dom ( 2nd ` x ) ) )
58	46, 57	eqtrd 2508	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` dom ( 2nd ` x ) ) )
59		simprr 756	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> dom ( 2nd ` x ) = I )
60	59	fveq2d 5868	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> ( # ` dom ( 2nd ` x ) ) = ( # ` I ) )
61	58, 60	eqtrd 2508	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` I ) )
62	43, 61	sylan2 474	. . . . . . . . . . . . . . . 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 ) )
63	39, 62	fveq12d 5870	. . . . . . . . . . . . . . 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 ) ) )
64	63	eqeq2d 2481	. . . . . . . . . . . . . 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 ) ) ) )
65	64	biimpd 207	. . . . . . . . . . . . 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 ) ) ) )
66	65	impancom 440	. . . . . . . . . . . 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 ) ) ) )
67	66	com12 31	. . . . . . . . . . 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 ) ) ) )
68	29, 67	jcad 533	. . . . . . . . . 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 ) ) ) ) )
69	25	biimprd 223	. . . . . . . . . . . . . 14 |- ( dom ( 2nd ` x ) = I -> ( f : ( 1 ... m ) -1-1-onto-> I -> f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) ) )
70	69	ad2antll 728	. . . . . . . . . . . . 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 ) ) )
71	70	adantr 465	. . . . . . . . . . . 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 ) ) )
72	71	adantrd 468	. . . . . . . . . . 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 ) ) )
73		eqidd 2468	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> 1 = 1 )
74		simpl 457	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( 1st ` x ) = A )
75		tru 1383	. . . . . . . . . . . . . . . . . . . . 21 |- T.
76	74, 75	jctir 538	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) -> ( ( 1st ` x ) = A /\ T. ) )
77	76	ad2antrl 727	. . . . . . . . . . . . . . . . . . 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. ) )
78		simpl 457	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( 1st ` x ) = A /\ T. ) -> ( 1st ` x ) = A )
79	78	eqcomd 2475	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( 1st ` x ) = A /\ T. ) -> A = ( 1st ` x ) )
80	77, 79	syl 16	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> A = ( 1st ` x ) )
81	80	fveq2d 5868	. . . . . . . . . . . . . . . . 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 ) ) )
82		simpl 457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) -> ( 2nd ` x ) = B )
83	82	eqcomd 2475	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) -> B = ( 2nd ` x ) )
84	83	ad2antll 728	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) -> B = ( 2nd ` x ) )
85	84	adantr 465	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) ) /\ n e. NN ) -> B = ( 2nd ` x ) )
86	85	fveq1d 5866	. . . . . . . . . . . . . . . . . . 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 ) ) )
87	86	adantlrr 720	. . . . . . . . . . . . . . . . . 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 ) ) )
88	87	mpteq2dva 4533	. . . . . . . . . . . . . . . . 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 ) ) ) )
89	73, 81, 88	seqeq123d 12080	. . . . . . . . . . . . . . . 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 ) ) ) ) )
90	71	com12 31	. . . . . . . . . . . . . . . . . . 19 |- ( 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 ) ) )
91	90	imp 429	. . . . . . . . . . . . . . . . . 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 ) )
92		simprr 756	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> m e. NN0 )
93	41	ad2antrl 727	. . . . . . . . . . . . . . . . . 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 )
94	91, 92, 93, 61	syl12anc 1226	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> m = ( # ` I ) )
95	94	eqcomd 2475	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 1 ... m ) -1-1-onto-> I /\ ( ( ( 1st ` x ) = A /\ ( ( 2nd ` x ) = B /\ dom ( 2nd ` x ) = I ) ) /\ m e. NN0 ) ) -> ( # ` I ) = m )
96	89, 95	fveq12d 5870	. . . . . . . . . . . . . . 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 ) )
97	96	eqeq2d 2481	. . . . . . . . . . . . . 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 ) ) )
98	97	biimpd 207	. . . . . . . . . . . . 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 ) ) )
99	98	impancom 440	. . . . . . . . . . . 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 ) ) )
100	99	com12 31	. . . . . . . . . . 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 ) ) )
101	72, 100	jcad 533	. . . . . . . . . 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 ) ) ) )
102	68, 101	impbid 191	. . . . . . . . 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 ) ) ) ) )
103	102	ex 434	. . . . . . . 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 ) ) ) ) ) )
104	15, 19, 24, 103	syl12anc 1226	. . . . . . 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 ) ) ) ) ) )
105	104	imp 429	. . . . . 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 ) ) ) ) )
106	105	exbidv 1690	. . . . 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 ) ) ) ) )
107	106	rexbidva 2970	. . . 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 ) ) ) ) )
108	107	iotabidv 5570	. . 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 ) ) ) ) )
109		eleq1 2539	. . . . . . . . . 10 |- ( t = I -> ( t e. Fin <-> I e. Fin ) )
110		feq2 5712	. . . . . . . . . 10 |- ( t = I -> ( B : t --> ( Base ` A ) <-> B : I --> ( Base ` A ) ) )
111	109, 110	anbi12d 710	. . . . . . . . 9 |- ( t = I -> ( ( t e. Fin /\ B : t --> ( Base ` A ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
112	111	ceqsexgv 3236	. . . . . . . 8 |- ( I e. Fin -> ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
113	9, 112	syl 16	. . . . . . 7 |- ( ph -> ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) <-> ( I e. Fin /\ B : I --> ( Base ` A ) ) ) )
114	9, 8, 113	mpbir2and 920	. . . . . 6 |- ( ph -> E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) )
115		exsimpr 1655	. . . . . 6 |- ( E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) -> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
116	114, 115	syl 16	. . . . 5 |- ( ph -> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
117		df-rex 2820	. . . . 5 |- ( E. t e. Fin B : t --> ( Base ` A ) <-> E. t ( t e. Fin /\ B : t --> ( Base ` A ) ) )
118	116, 117	sylibr 212	. . . 4 |- ( ph -> E. t e. Fin B : t --> ( Base ` A ) )
119		eleq1 2539	. . . . . . 7 |- ( y = A -> ( y e. CMnd <-> A e. CMnd ) )
120		eqidd 2468	. . . . . . . . 9 |- ( y = A -> t = t )
121		fveq2 5864	. . . . . . . . 9 |- ( y = A -> ( Base ` y ) = ( Base ` A ) )
122	120, 121	feq23d 5724	. . . . . . . 8 |- ( y = A -> ( z : t --> ( Base ` y ) <-> z : t --> ( Base ` A ) ) )
123	122	rexbidv 2973	. . . . . . 7 |- ( y = A -> ( E. t e. Fin z : t --> ( Base ` y ) <-> E. t e. Fin z : t --> ( Base ` A ) ) )
124	119, 123	anbi12d 710	. . . . . 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 ) ) ) )
125		feq1 5711	. . . . . . . 8 |- ( z = B -> ( z : t --> ( Base ` A ) <-> B : t --> ( Base ` A ) ) )
126	125	rexbidv 2973	. . . . . . 7 |- ( z = B -> ( E. t e. Fin z : t --> ( Base ` A ) <-> E. t e. Fin B : t --> ( Base ` A ) ) )
127	126	anbi2d 703	. . . . . 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 ) ) ) )
128	124, 127	opelopabg 4765	. . . . 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 ) ) ) )
129	6, 11, 128	syl2anc 661	. . . 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 ) ) ) )
130	6, 118, 129	mpbir2and 920	. . 3 |- ( ph -> <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } )
131		iotaex 5566	. . . 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
132	131	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 )
133	3, 108, 130, 132	fvmptd 5953	. 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 ) ) ) ) )
134	1, 133	syl5eq 2520	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 184   /\ wa 369   = wceq 1379   T. wtru 1380   E.wex 1596   e. wcel 1767   E.wrex 2815   _Vcvv 3113   <.cop 4033   {copab 4504   |-> cmpt 4505   dom cdm 4999   iotacio 5547   -->wf 5582   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   Fincfn 7513   1c1 9489   NNcn 10532   NN0cn0 10791   ...cfz 11668   seqcseq 12071   #chash 12369   Basecbs 14486   +g cplusg 14551  CMndccmn 16594   FinSum cfinsum 33733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12072  df-hash 12370  df-bj-finsum 33734

This theorem is referenced by: (None)