Theorem prodfrecap 12170

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 10312	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 14	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 5648	. . . . 5 |- ( m = M -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` M ) )
5		fveq2 5648	. . . . . 6 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
6	5	oveq2d 6044	. . . . 5 |- ( m = M -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
7	4, 6	eqeq12d 2246	. . . 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 5648	. . . . 5 |- ( m = n -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` n ) )
10		fveq2 5648	. . . . . 6 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
11	10	oveq2d 6044	. . . . 5 |- ( m = n -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` n ) ) )
12	9, 11	eqeq12d 2246	. . . 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 5648	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` ( n + 1 ) ) )
15		fveq2 5648	. . . . . 6 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
16	15	oveq2d 6044	. . . . 5 |- ( m = ( n + 1 ) -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) )
17	14, 16	eqeq12d 2246	. . . 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 5648	. . . . 5 |- ( m = N -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` N ) )
20		fveq2 5648	. . . . . 6 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
21	20	oveq2d 6044	. . . . 5 |- ( m = N -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
22	19, 21	eqeq12d 2246	. . . 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 10311	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
25	1, 24	syl 14	. . . . . 6 |- ( ph -> M e. ( M ... N ) )
26		fveq2 5648	. . . . . . . . 9 |- ( k = M -> ( G ` k ) = ( G ` M ) )
27		fveq2 5648	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
28	27	oveq2d 6044	. . . . . . . . 9 |- ( k = M -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` M ) ) )
29	26, 28	eqeq12d 2246	. . . . . . . 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 2871	. . . . . 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 9804	. . . . . . 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 8202	. . . . . . 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 10770	. . . . 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 10770	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
43	42	oveq2d 6044	. . . . 5 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` M ) ) = ( 1 / ( F ` M ) ) )
44	34, 40, 43	3eqtr4d 2274	. . . 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 6035	. . . . . . . . 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 1047	. . . . . . . 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 10518	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
49		fveq2 5648	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
50		fveq2 5648	. . . . . . . . . . . . . . . . 17 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
51	50	oveq2d 6044	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
52	49, 51	eqeq12d 2246	. . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . 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 6044	. . . . . . . . . 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 8238	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> 1 e. CC )
59		eqid 2231	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
60	59, 36, 41	prodf 12162	. . . . . . . . . . . . . 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 10431	. . . . . . . . . . . . . 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 5791	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
65	50	eleq1d 2300	. . . . . . . . . . . . . . . 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 10301	. . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 10442	. . . . . . . . . . . . . . . . 17 |- ( n e. ( M..^ N ) -> N e. ( ZZ>= ` n ) )
75		fzss2 10344	. . . . . . . . . . . . . . . . 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 3228	. . . . . . . . . . . . . . 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 12169	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) # 0 )
82	50	breq1d 4103	. . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . 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 9054	. . . . . . . . . . 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 9338	. . . . . . . . . . . 12 |- ( 1 x. 1 ) = 1
90	89	oveq1i 6038	. . . . . . . . . . 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 2280	. . . . . . . . . 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 2264	. . . . . . . . 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 1044	. . . . . . . 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 2264	. . . . . . 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 1044	. . . . . . . 8 |- ( ( ph /\ n e. ( M..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> n e. ( ZZ>= ` M ) )
96	37	3ad2antl1 1186	. . . . . . . 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 10773	. . . . . . 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 1186	. . . . . . . . 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 10773	. . . . . . . 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 6044	. . . . . . 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 2274	. . . . . 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 1229	. . . . 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 10531	. 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 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   # cap 8803   / cdiv 8894   ZZcz 9523   ZZ>=cuz 9799   ...cfz 10288  ..^cfzo 10422   seqcseq 10755

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-seqfrec 10756

This theorem is referenced by:  prodfdivap  12171