Theorem bj-finsumval0 34764

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 6299	. 2 |- ( A FinSum B ) = ( FinSum ` <. A , B >. )
2		df-bj-finsum 34763	. . . 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 5876	. . . . . . . . 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 6146	. . . . . . . . . . . 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 6811	. . . . . . . . . 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 2498	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 1st ` x ) = A )
16	4	fveq2d 5876	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = ( 2nd ` <. A , B >. ) )
17		op2ndg 6812	. . . . . . . . . 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 2498	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> ( 2nd ` x ) = B )
20	19	dmeqd 5215	. . . . . . . . 9 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = dom B )
21		fdm 5741	. . . . . . . . . . 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 2498	. . . . . . . 8 |- ( ( ph /\ x = <. A , B >. ) -> dom ( 2nd ` x ) = I )
25		f1oeq3 5815	. . . . . . . . . . . . . . 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 2458	. . . . . . . . . . . . . . . . 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 756	. . . . . . . . . . . . . . . . . . 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 5876	. . . . . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . . . 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 5874	. . . . . . . . . . . . . . . . . . 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 4543	. . . . . . . . . . . . . . . . . 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 12118	. . . . . . . . . . . . . . . 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 757	. . . . . . . . . . . . . . . . . . 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 12421	. . . . . . . . . . . . . . . . . . . . 21 |- ( m e. NN0 -> ( # ` ( 1 ... m ) ) = m )
45	44	eqcomd 2465	. . . . . . . . . . . . . . . . . . . 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 12085	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ m e. NN0 ) -> ( 1 ... m ) e. Fin )
48		f1ofo 5829	. . . . . . . . . . . . . . . . . . . . . . . 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 6768	. . . . . . . . . . . . . . . . . . . . . . 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 1858	. . . . . . . . . . . . . . . . . . . . 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 12426	. . . . . . . . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . . . . . . 18 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> m = ( # ` dom ( 2nd ` x ) ) )
59		simprr 757	. . . . . . . . . . . . . . . . . . 19 |- ( ( f : ( 1 ... m ) -1-1-onto-> dom ( 2nd ` x ) /\ ( m e. NN0 /\ dom ( 2nd ` x ) = I ) ) -> dom ( 2nd ` x ) = I )
60	59	fveq2d 5876	. . . . . . . . . . . . . . . . . 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 2498	. . . . . . . . . . . . . . . . 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 5878	. . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . 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 2458	. . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . . . . . 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 2465	. . . . . . . . . . . . . . . . . . 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 5876	. . . . . . . . . . . . . . . . 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 2465	. . . . . . . . . . . . . . . . . . . . . 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 5874	. . . . . . . . . . . . . . . . . . 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 4543	. . . . . . . . . . . . . . . . 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 12118	. . . . . . . . . . . . . . . 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 757	. . . . . . . . . . . . . . . . . 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 2465	. . . . . . . . . . . . . . . 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 5878	. . . . . . . . . . . . . . 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 2471	. . . . . . . . . . . . . 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 1715	. . . . 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 2965	. . . 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 5578	. . 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 2529	. . . . . . . . . 10 |- ( t = I -> ( t e. Fin <-> I e. Fin ) )
110		feq2 5720	. . . . . . . . . 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 3232	. . . . . . . 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 922	. . . . . 6 |- ( ph -> E. t ( t = I /\ ( t e. Fin /\ B : t --> ( Base ` A ) ) ) )
115		exsimpr 1679	. . . . . 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 2813	. . . . 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 2529	. . . . . . 7 |- ( y = A -> ( y e. CMnd <-> A e. CMnd ) )
120		fveq2 5872	. . . . . . . . 9 |- ( y = A -> ( Base ` y ) = ( Base ` A ) )
121	120	feq3d 5725	. . . . . . . 8 |- ( y = A -> ( z : t --> ( Base ` y ) <-> z : t --> ( Base ` A ) ) )
122	121	rexbidv 2968	. . . . . . 7 |- ( y = A -> ( E. t e. Fin z : t --> ( Base ` y ) <-> E. t e. Fin z : t --> ( Base ` A ) ) )
123	119, 122	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 ) ) ) )
124		feq1 5719	. . . . . . . 8 |- ( z = B -> ( z : t --> ( Base ` A ) <-> B : t --> ( Base ` A ) ) )
125	124	rexbidv 2968	. . . . . . 7 |- ( z = B -> ( E. t e. Fin z : t --> ( Base ` A ) <-> E. t e. Fin B : t --> ( Base ` A ) ) )
126	125	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 ) ) ) )
127	123, 126	opelopabg 4774	. . . . 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 ) ) ) )
128	6, 11, 127	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 ) ) ) )
129	6, 118, 128	mpbir2and 922	. . 3 |- ( ph -> <. A , B >. e. { <. y , z >. | ( y e. CMnd /\ E. t e. Fin z : t --> ( Base ` y ) ) } )
130		iotaex 5574	. . . 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
131	130	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 )
132	3, 108, 129, 131	fvmptd 5961	. 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 ) ) ) ) )
133	1, 132	syl5eq 2510	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 1395   T. wtru 1396   E.wex 1613   e. wcel 1819   E.wrex 2808   _Vcvv 3109   <.cop 4038   {copab 4514   |-> cmpt 4515   dom cdm 5008   iotacio 5555   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   Fincfn 7535   1c1 9510   NNcn 10556   NN0cn0 10816   ...cfz 11697   seqcseq 12109   #chash 12407   Basecbs 14643   +g cplusg 14711  CMndccmn 16924   FinSum cfinsum 34762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110  df-hash 12408  df-bj-finsum 34763

This theorem is referenced by: (None)