Theorem prodfrec 14627

Description: The reciprocal of an infinite product. (Contributed by Scott Fenton, 15-Jan-2018.)

Hypotheses

Ref	Expression
prodfn0.1	|- ( ph -> N e. ( ZZ>= ` M ) )
prodfn0.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
prodfn0.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) =/= 0 )
prodfrec.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) )

Assertion

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

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

Proof of Theorem prodfrec

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

Step	Hyp	Ref	Expression
1		prodfn0.1	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
2		eluzfz2 12349	. . 3 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
3	1, 2	syl 17	. 2 |- ( ph -> N e. ( M ... N ) )
4		fveq2 6191	. . . . 5 |- ( m = M -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` M ) )
5		fveq2 6191	. . . . . 6 |- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) )
6	5	oveq2d 6666	. . . . 5 |- ( m = M -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
7	4, 6	eqeq12d 2637	. . . 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 330	. . 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 6191	. . . . 5 |- ( m = n -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` n ) )
10		fveq2 6191	. . . . . 6 |- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) )
11	10	oveq2d 6666	. . . . 5 |- ( m = n -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` n ) ) )
12	9, 11	eqeq12d 2637	. . . 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 330	. . 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 6191	. . . . 5 |- ( m = ( n + 1 ) -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` ( n + 1 ) ) )
15		fveq2 6191	. . . . . 6 |- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) )
16	15	oveq2d 6666	. . . . 5 |- ( m = ( n + 1 ) -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) )
17	14, 16	eqeq12d 2637	. . . 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 330	. . 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 6191	. . . . 5 |- ( m = N -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` N ) )
20		fveq2 6191	. . . . . 6 |- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) )
21	20	oveq2d 6666	. . . . 5 |- ( m = N -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )
22	19, 21	eqeq12d 2637	. . . 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 330	. . 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 12348	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) )
25	1, 24	syl 17	. . . . . 6 |- ( ph -> M e. ( M ... N ) )
26		fveq2 6191	. . . . . . . . 9 |- ( k = M -> ( G ` k ) = ( G ` M ) )
27		fveq2 6191	. . . . . . . . . 10 |- ( k = M -> ( F ` k ) = ( F ` M ) )
28	27	oveq2d 6666	. . . . . . . . 9 |- ( k = M -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` M ) ) )
29	26, 28	eqeq12d 2637	. . . . . . . 8 |- ( k = M -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` M ) = ( 1 / ( F ` M ) ) ) )
30	29	imbi2d 330	. . . . . . 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 451	. . . . . . 7 |- ( k e. ( M ... N ) -> ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) )
33	30, 32	vtoclga 3272	. . . . . 6 |- ( M e. ( M ... N ) -> ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) )
34	25, 33	mpcom 38	. . . . 5 |- ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) )
35		eluzel2 11692	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 17	. . . . . 6 |- ( ph -> M e. ZZ )
37		seq1 12814	. . . . . 6 |- ( M e. ZZ -> ( seq M ( x. , G ) ` M ) = ( G ` M ) )
38	36, 37	syl 17	. . . . 5 |- ( ph -> ( seq M ( x. , G ) ` M ) = ( G ` M ) )
39		seq1 12814	. . . . . . 7 |- ( M e. ZZ -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
40	36, 39	syl 17	. . . . . 6 |- ( ph -> ( seq M ( x. , F ) ` M ) = ( F ` M ) )
41	40	oveq2d 6666	. . . . 5 |- ( ph -> ( 1 / ( seq M ( x. , F ) ` M ) ) = ( 1 / ( F ` M ) ) )
42	34, 38, 41	3eqtr4d 2666	. . . 4 |- ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) )
43	42	a1i 11	. . 3 |- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) )
44		oveq1 6657	. . . . . . . . 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 ) ) ) )
45	44	3ad2ant3 1084	. . . . . . . 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 ) ) ) )
46		fzofzp1 12565	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( n + 1 ) e. ( M ... N ) )
47		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) )
48		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) )
49	48	oveq2d 6666	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
50	47, 49	eqeq12d 2637	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
51	50	imbi2d 330	. . . . . . . . . . . . . 14 |- ( k = ( n + 1 ) -> ( ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) <-> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) )
52	51, 32	vtoclga 3272	. . . . . . . . . . . . 13 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
53	46, 52	syl 17	. . . . . . . . . . . 12 |- ( n e. ( M..^ N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) )
54	53	impcom 446	. . . . . . . . . . 11 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) )
55	54	oveq2d 6666	. . . . . . . . . 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 ) ) ) ) )
56		1cnd 10056	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> 1 e. CC )
57		elfzouz 12474	. . . . . . . . . . . . . 14 |- ( n e. ( M..^ N ) -> n e. ( ZZ>= ` M ) )
58	57	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ n e. ( M..^ N ) ) -> n e. ( ZZ>= ` M ) )
59		elfzouz2 12484	. . . . . . . . . . . . . . . . 17 |- ( n e. ( M..^ N ) -> N e. ( ZZ>= ` n ) )
60		fzss2 12381	. . . . . . . . . . . . . . . . 17 |- ( N e. ( ZZ>= ` n ) -> ( M ... n ) C_ ( M ... N ) )
61	59, 60	syl 17	. . . . . . . . . . . . . . . 16 |- ( n e. ( M..^ N ) -> ( M ... n ) C_ ( M ... N ) )
62	61	sselda 3603	. . . . . . . . . . . . . . 15 |- ( ( n e. ( M..^ N ) /\ k e. ( M ... n ) ) -> k e. ( M ... N ) )
63		prodfn0.2	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
64	62, 63	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. ( M..^ N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) e. CC )
65	64	anassrs 680	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC )
66		mulcl 10020	. . . . . . . . . . . . . 14 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
67	66	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
68	58, 65, 67	seqcl 12821	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC )
69	48	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) )
70	69	imbi2d 330	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) )
71	63	expcom 451	. . . . . . . . . . . . . . 15 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) e. CC ) )
72	70, 71	vtoclga 3272	. . . . . . . . . . . . . 14 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
73	46, 72	syl 17	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) )
74	73	impcom 446	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC )
75		prodfn0.3	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) =/= 0 )
76	62, 75	sylan2 491	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( n e. ( M..^ N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) =/= 0 )
77	76	anassrs 680	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n e. ( M..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) =/= 0 )
78	58, 65, 77	prodfn0 14626	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( seq M ( x. , F ) ` n ) =/= 0 )
79	48	neeq1d 2853	. . . . . . . . . . . . . . . 16 |- ( k = ( n + 1 ) -> ( ( F ` k ) =/= 0 <-> ( F ` ( n + 1 ) ) =/= 0 ) )
80	79	imbi2d 330	. . . . . . . . . . . . . . 15 |- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) =/= 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) ) )
81	75	expcom 451	. . . . . . . . . . . . . . 15 |- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) =/= 0 ) )
82	80, 81	vtoclga 3272	. . . . . . . . . . . . . 14 |- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) )
83	46, 82	syl 17	. . . . . . . . . . . . 13 |- ( n e. ( M..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) )
84	83	impcom 446	. . . . . . . . . . . 12 |- ( ( ph /\ n e. ( M..^ N ) ) -> ( F ` ( n + 1 ) ) =/= 0 )
85	56, 68, 56, 74, 78, 84	divmuldivd 10842	. . . . . . . . . . 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 ) ) ) ) )
86		1t1e1 11175	. . . . . . . . . . . 12 |- ( 1 x. 1 ) = 1
87	86	oveq1i 6660	. . . . . . . . . . 11 |- ( ( 1 x. 1 ) / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
88	85, 87	syl6eq 2672	. . . . . . . . . 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 ) ) ) ) )
89	55, 88	eqtrd 2656	. . . . . . . . 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 ) ) ) ) )
90	89	3adant3 1081	. . . . . . . 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 ) ) ) ) )
91	45, 90	eqtrd 2656	. . . . . . 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 ) ) ) ) )
92		seqp1 12816	. . . . . . . . 9 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) )
93	57, 92	syl 17	. . . . . . . 8 |- ( n e. ( M..^ N ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) )
94	93	3ad2ant2 1083	. . . . . . 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 ) ) ) )
95		seqp1 12816	. . . . . . . . . 10 |- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
96	57, 95	syl 17	. . . . . . . . 9 |- ( n e. ( M..^ N ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) )
97	96	oveq2d 6666	. . . . . . . 8 |- ( n e. ( M..^ N ) -> ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) )
98	97	3ad2ant2 1083	. . . . . . 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 ) ) ) ) )
99	91, 94, 98	3eqtr4d 2666	. . . . . 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 ) ) ) )
100	99	3exp 1264	. . . . 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 ) ) ) ) ) )
101	100	com12 32	. . . 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 ) ) ) ) ) )
102	101	a2d 29	. . 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 ) ) ) ) ) )
103	8, 13, 18, 23, 43, 102	fzind2 12586	. 2 |- ( N e. ( M ... N ) -> ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) )
104	3, 103	mpcom 38	1 |- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802

This theorem is referenced by:  prodfdiv  14628