Theorem bj-finsumval0 32588

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 6099	. 2 |- ( A FinSum B ) = ( FinSum ` <. A , B >. )
2		df-bj-finsum 32587	. . . 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 5700	. . . . . . . . 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 5955	. . . . . . . . . . . 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 6594	. . . . . . . . . 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 2475	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = A )
16	4	fveq2d 5700	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
17		op2ndg 6595	. . . . . . . . . 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 2475	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = B )
20	19	dmeqd 5047	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = dom B )
21		fdm 5568	. . . . . . . . . . 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 2475	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = I )
25		f1oeq3 5639	. . . . . . . . . . . . . . 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 2444	. . . . . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . . . . . 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 5698	. . . . . . . . . . . . . . . . . . 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 4383	. . . . . . . . . . . . . . . . . 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 11820	. . . . . . . . . . . . . . . 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 12122	. . . . . . . . . . . . . . . . . . . . 21 |- ( m e. NN0 -> ( # ` ( 1 ... m ) ) = m )
45	44	eqcomd 2448	. . . . . . . . . . . . . . . . . . . 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 11800	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> ( 1 ... m ) e. Fin )
48		f1ofo 5653	. . . . . . . . . . . . . . . . . . . . . . . 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 6551	. . . . . . . . . . . . . . . . . . . . . . 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 1793	. . . . . . . . . . . . . . . . . . . . 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 12127	. . . . . . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . . . . . 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 2475	. . . . . . . . . . . . . . . . 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 5702	. . . . . . . . . . . . . . 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 2454	. . . . . . . . . . . . . 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 2444	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . . . . . . 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 5700	. . . . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . . . . . . . . . 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 5698	. . . . . . . . . . . . . . . . . . 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 4383	. . . . . . . . . . . . . . . . 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 11820	. . . . . . . . . . . . . . . 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 1216	. . . . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . . . 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 5702	. . . . . . . . . . . . . . 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 2454	. . . . . . . . . . . . . 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 1216	. . . . . . 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 1680	. . . . 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 2737	. . . 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 5407	. . 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 2503	. . . . . . . . . 10 |- ( t = I -> ( t e. Fin <-> I e. Fin ) )
110		feq2 5548	. . . . . . . . . 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 3097	. . . . . . . 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 913	. . . . . 6 |- ( ph -> E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) )
115		exsimpr 1645	. . . . . 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 2726	. . . . 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 2503	. . . . . . 7 |- ( y = A -> ( y e. CMnd <-> A e. CMnd ) )
120		eqidd 2444	. . . . . . . . 9 |- ( y = A -> t = t )
121		fveq2 5696	. . . . . . . . 9 |- ( y = A -> ( Base ` y ) = ( Base ` A ) )
122	120, 121	feq23d 5559	. . . . . . . 8 |- ( y = A -> ( z : t --> ( Base ` y ) <-> z : t --> ( Base ` A ) ) )
123	122	rexbidv 2741	. . . . . . 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 5547	. . . . . . . 8 |- ( z = B -> ( z : t --> ( Base ` A ) <-> B : t --> ( Base ` A ) ) )
126	125	rexbidv 2741	. . . . . . 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 4612	. . . . 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 913	. . 3 |- ( ph -> <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } )
131		iotaex 5403	. . . 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 5784	. 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 2487	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 1369   T. wtru 1370   E.wex 1586   e. wcel 1756   E.wrex 2721   _Vcvv 2977   <.cop 3888   {copab 4354   e. cmpt 4355   dom cdm 4845   iotacio 5384   -->wf 5419   -onto->wfo 5421   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581   Fincfn 7315   1c1 9288   NNcn 10327   NN0cn0 10584   ...cfz 11442   seqcseq 11811   #chash 12108   Basecbs 14179   +g cplusg 14243  CMndccmn 16282   FinSum cfinsum 32586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-hash 12109  df-bj-finsum 32587

This theorem is referenced by: (None)