Theorem prodfrecap 11556

Description: The reciprocal of a finite product. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)

Hypotheses

Ref	Expression
prodfap0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
prodfap0.2	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
prodfap0.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )
prodfrec.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) )
prodfrecap.g	|- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )

Assertion

Ref	Expression
prodfrecap	|- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )

Distinct variable groups:   k, F   k, M   k, N   ph, k   k, G

Proof of Theorem prodfrecap

Dummy variables n v m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfap0.1	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzfz2 10034	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5517	. . . . 5 |- ( m = M -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` M ) )
5		fveq2 5517	. . . . . 6 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
6	5	oveq2d 5893	. . . . 5 |- ( m = M -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
7	4, 6	eqeq12d 2192	. . . 4 |- ( m = M -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) )
8	7	imbi2d 230	. . 3 |- ( m = M -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) ) )
9		fveq2 5517	. . . . 5 |- ( m = n -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` n ) )
10		fveq2 5517	. . . . . 6 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
11	10	oveq2d 5893	. . . . 5 |- ( m = n -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` n ) ) )
12	9, 11	eqeq12d 2192	. . . 4 |- ( m = n -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) )
13	12	imbi2d 230	. . 3 |- ( m = n -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) ) )
14		fveq2 5517	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` ( n + 1 ) ) )
15		fveq2 5517	. . . . . 6 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
16	15	oveq2d 5893	. . . . 5 |- ( m = ( n + 1 ) -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) )
17	14, 16	eqeq12d 2192	. . . 4 |- ( m = ( n + 1 ) -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) )
18	17	imbi2d 230	. . 3 |- ( m = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) )
19		fveq2 5517	. . . . 5 |- ( m = N -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` N ) )
20		fveq2 5517	. . . . . 6 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
21	20	oveq2d 5893	. . . . 5 |- ( m = N -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
22	19, 21	eqeq12d 2192	. . . 4 |- ( m = N -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) )
23	22	imbi2d 230	. . 3 |- ( m = N -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) ) )
24		eluzfz1 10033	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
25	1, 24	syl 14	. . . . . 6 |- ( ph -> M e. ( M ... N ) )
26		fveq2 5517	. . . . . . . . 9 |- ( k = M -> ( G ` k ) = ( G ` M ) )
27		fveq2 5517	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
28	27	oveq2d 5893	. . . . . . . . 9 |- ( k = M -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` M ) ) )
29	26, 28	eqeq12d 2192	. . . . . . . 8 |- ( k = M -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` M ) = ( 1 / ( F ` M ) ) ) )
30	29	imbi2d 230	. . . . . . 7 |- ( k = M -> ( ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) <-> ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) ) )
31		prodfrec.4	. . . . . . . 8 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) )
32	31	expcom 116	. . . . . . 7 |- ( k e. ( M ... N ) -> ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) )
33	30, 32	vtoclga 2805	. . . . . 6 |- ( M e. ( M ... N ) -> ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) )
34	25, 33	mpcom 36	. . . . 5 |- ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) )
35		eluzel2 9535	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . . 6 |- ( ph -> M e. ZZ )
37		prodfrecap.g	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
38		mulcl 7940	. . . . . . 7 |- ( ( k e. CC /\ v e. CC ) -> ( k x. v ) e. CC )
39	38	adantl 277	. . . . . 6 |- ( ( ph /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
40	36, 37, 39	seq3-1 10462	. . . . 5 |- ( ph -> ( seq M ( x. , G ) ` M ) = ( G ` M ) )
41		prodfap0.2	. . . . . . 7 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
42	36, 41, 39	seq3-1 10462	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
43	42	oveq2d 5893	. . . . 5 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` M ) ) = ( 1 / ( F ` M ) ) )
44	34, 40, 43	3eqtr4d 2220	. . . 4 |- ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
45	44	a1i 9	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) )
46		oveq1 5884	. . . . . . . . 9 |- ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) )
47	46	3ad2ant3 1020	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) )
48		fzofzp1 10229	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5517	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
50		fveq2 5517	. . . . . . . . . . . . . . . . 17 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
51	50	oveq2d 5893	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
52	49, 51	eqeq12d 2192	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
53	52	imbi2d 230	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) <-> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) )
54	53, 32	vtoclga 2805	. . . . . . . . . . . . 13 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
55	48, 54	syl 14	. . . . . . . . . . . 12 |- ( n e. ( M..^ N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
56	55	impcom 125	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
57	56	oveq2d 5893	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) )
58		1cnd 7975	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> 1 e. CC )
59		eqid 2177	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
60	59, 36, 41	prodf 11548	. . . . . . . . . . . . . 14 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
61	60	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( M..^ N ) ) -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
62		elfzouz 10153	. . . . . . . . . . . . . 14 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
63	62	adantl 277	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( M..^ N ) ) -> n e. ( ZZ>= ` M ) )
64	61, 63	ffvelcdmd 5654	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
65	50	eleq1d 2246	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
66	65	imbi2d 230	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) )
67		elfzuz 10023	. . . . . . . . . . . . . . . 16 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
68	41	expcom 116	. . . . . . . . . . . . . . . 16 |- ( k e. ( ZZ>= ` M ) -> ( ph -> ( F ` k ) e. CC ) )
69	67, 68	syl 14	. . . . . . . . . . . . . . 15 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) e. CC ) )
70	66, 69	vtoclga 2805	. . . . . . . . . . . . . 14 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
71	48, 70	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
72	71	impcom 125	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC )
73	41	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
74		elfzouz2 10163	. . . . . . . . . . . . . . . . 17 |- ( n e. ( M..^ N ) -> N e. ( ZZ>= ` n ) )
75		fzss2 10066	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` n ) -> ( M ... n ) C_ ( M ... N ) )
76	74, 75	syl 14	. . . . . . . . . . . . . . . 16 |- ( n e. ( M..^ N ) -> ( M ... n ) C_ ( M ... N ) )
77	76	sselda 3157	. . . . . . . . . . . . . . 15 |- ( ( n e. ( M..^ N ) /\ k e. ( M ... n ) ) -> k e. ( M ... N ) )
78		prodfap0.3	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) # 0 )
79	77, 78	sylan2 286	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. ( M..^ N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) # 0 )
80	79	anassrs 400	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) # 0 )
81	63, 73, 80	prodfap0 11555	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) # 0 )
82	50	breq1d 4015	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( ( F ` k ) # 0 <-> ( F ` ( n + 1 ) ) # 0 ) )
83	82	imbi2d 230	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) # 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) # 0 ) ) )
84	78	expcom 116	. . . . . . . . . . . . . . 15 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) # 0 ) )
85	83, 84	vtoclga 2805	. . . . . . . . . . . . . 14 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) # 0 ) )
86	48, 85	syl 14	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) # 0 ) )
87	86	impcom 125	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) # 0 )
88	58, 64, 58, 72, 81, 87	divmuldivapd 8791	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) = ( ( 1 x. 1 ) / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
89		1t1e1 9073	. . . . . . . . . . . 12 |- ( 1 x. 1 ) = 1
90	89	oveq1i 5887	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
91	88, 90	eqtrdi 2226	. . . . . . . . . 10 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
92	57, 91	eqtrd 2210	. . . . . . . . 9 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
93	92	3adant3 1017	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
94	47, 93	eqtrd 2210	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
95	63	3adant3 1017	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> n e. ( ZZ>= ` M ) )
96	37	3ad2antl1 1159	. . . . . . . 8 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
97	38	adantl 277	. . . . . . . 8 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) /\ ( k e. CC /\ v e. CC ) ) -> ( k x. v ) e. CC )
98	95, 96, 97	seq3p1 10464	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) )
99	41	3ad2antl1 1159	. . . . . . . . 9 |- ( ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
100	95, 99, 97	seq3p1 10464	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
101	100	oveq2d 5893	. . . . . . 7 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
102	94, 98, 101	3eqtr4d 2220	. . . . . 6 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) )
103	102	3exp 1202	. . . . 5 |- ( ph -> ( n e. ( M..^ N ) -> ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) )
104	103	com12 30	. . . 4 |- ( n e. ( M..^ N ) -> ( ph -> ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) )
105	104	a2d 26	. . 3 |- ( n e. ( M..^ N ) -> ( ( ph -> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ph -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) )
106	8, 13, 18, 23, 45, 105	fzind2 10241	. 2 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) )
107	3, 106	mpcom 36	1 |- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   C_ wss 3131   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   # cap 8540   / cdiv 8631   ZZcz 9255   ZZ>=cuz 9530   ...cfz 10010  ..^cfzo 10144   seqcseq 10447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145  df-seqfrec 10448

This theorem is referenced by:  prodfdivap  11557