Theorem prodfrecap 12291

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 10415	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5690	. . . . 5 |- ( m = M -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` M ) )
5		fveq2 5690	. . . . . 6 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
6	5	oveq2d 6091	. . . . 5 |- ( m = M -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
7	4, 6	eqeq12d 2253	. . . 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 5690	. . . . 5 |- ( m = n -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` n ) )
10		fveq2 5690	. . . . . 6 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
11	10	oveq2d 6091	. . . . 5 |- ( m = n -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` n ) ) )
12	9, 11	eqeq12d 2253	. . . 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 5690	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` ( n + 1 ) ) )
15		fveq2 5690	. . . . . 6 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
16	15	oveq2d 6091	. . . . 5 |- ( m = ( n + 1 ) -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) )
17	14, 16	eqeq12d 2253	. . . 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 5690	. . . . 5 |- ( m = N -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` N ) )
20		fveq2 5690	. . . . . 6 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
21	20	oveq2d 6091	. . . . 5 |- ( m = N -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
22	19, 21	eqeq12d 2253	. . . 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 10414	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
25	1, 24	syl 14	. . . . . 6 |- ( ph -> M e. ( M ... N ) )
26		fveq2 5690	. . . . . . . . 9 |- ( k = M -> ( G ` k ) = ( G ` M ) )
27		fveq2 5690	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
28	27	oveq2d 6091	. . . . . . . . 9 |- ( k = M -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` M ) ) )
29	26, 28	eqeq12d 2253	. . . . . . . 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 2889	. . . . . 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 9905	. . . . . . 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 8296	. . . . . . 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 10877	. . . . 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 10877	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
43	42	oveq2d 6091	. . . . 5 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` M ) ) = ( 1 / ( F ` M ) ) )
44	34, 40, 43	3eqtr4d 2281	. . . 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 6082	. . . . . . . . 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 1051	. . . . . . . 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 10623	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5690	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
50		fveq2 5690	. . . . . . . . . . . . . . . . 17 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
51	50	oveq2d 6091	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
52	49, 51	eqeq12d 2253	. . . . . . . . . . . . . . 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 2889	. . . . . . . . . . . . 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 6091	. . . . . . . . . 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 8332	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> 1 e. CC )
59		eqid 2238	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
60	59, 36, 41	prodf 12283	. . . . . . . . . . . . . 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 10536	. . . . . . . . . . . . . 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 5835	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
65	50	eleq1d 2307	. . . . . . . . . . . . . . . 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 10403	. . . . . . . . . . . . . . . 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 2889	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
74		elfzouz2 10547	. . . . . . . . . . . . . . . . 17 |- ( n e. ( M..^ N ) -> N e. ( ZZ>= ` n ) )
75		fzss2 10448	. . . . . . . . . . . . . . . . 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 3248	. . . . . . . . . . . . . . 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 404	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) # 0 )
81	63, 73, 80	prodfap0 12290	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) # 0 )
82	50	breq1d 4135	. . . . . . . . . . . . . . . 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 2889	. . . . . . . . . . . . . 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 9152	. . . . . . . . . . 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 9436	. . . . . . . . . . . 12 |- ( 1 x. 1 ) = 1
90	89	oveq1i 6085	. . . . . . . . . . 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 2287	. . . . . . . . . 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 2271	. . . . . . . . 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 1048	. . . . . . . 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 2271	. . . . . . 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 1048	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> n e. ( ZZ>= ` M ) )
96	37	3ad2antl1 1190	. . . . . . . 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 10880	. . . . . . 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 1190	. . . . . . . . 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 10880	. . . . . . . 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 6091	. . . . . . 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 2281	. . . . . 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 1233	. . . . 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 10636	. 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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   # cap 8899   / cdiv 8992   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390  ..^cfzo 10527   seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863

This theorem is referenced by:  prodfdivap  12292