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Theorem prodfrec 28594
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.
StepHypRef Expression
1 prodfn0.1 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 11685 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5859 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  M
) )
5 fveq2 5859 . . . . . 6  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
65oveq2d 6293 . . . . 5  |-  ( m  =  M  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
74, 6eqeq12d 2484 . . . 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 )
) ) )
87imbi2d 316 . . 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 5859 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  n
) )
10 fveq2 5859 . . . . . 6  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
1110oveq2d 6293 . . . . 5  |-  ( m  =  n  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )
129, 11eqeq12d 2484 . . . 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 )
) ) )
1312imbi2d 316 . . 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 5859 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  (
n  +  1 ) ) )
15 fveq2 5859 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1615oveq2d 6293 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) ) )
1714, 16eqeq12d 2484 . . . 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 ) ) ) ) )
1817imbi2d 316 . . 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 5859 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  N
) )
20 fveq2 5859 . . . . . 6  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2120oveq2d 6293 . . . . 5  |-  ( m  =  N  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
2219, 21eqeq12d 2484 . . . 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 )
) ) )
2322imbi2d 316 . . 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 11684 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
251, 24syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ( M ... N ) )
26 fveq2 5859 . . . . . . . . 9  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
27 fveq2 5859 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2827oveq2d 6293 . . . . . . . . 9  |-  ( k  =  M  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  M
) ) )
2926, 28eqeq12d 2484 . . . . . . . 8  |-  ( k  =  M  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3029imbi2d 316 . . . . . . 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 ) ) )
3231expcom 435 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( G `  k )  =  ( 1  /  ( F `
 k ) ) ) )
3330, 32vtoclga 3172 . . . . . 6  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3425, 33mpcom 36 . . . . 5  |-  ( ph  ->  ( G `  M
)  =  ( 1  /  ( F `  M ) ) )
35 eluzel2 11078 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 16 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
37 seq1 12078 . . . . . 6  |-  ( M  e.  ZZ  ->  (  seq M (  x.  ,  G ) `  M
)  =  ( G `
 M ) )
3836, 37syl 16 . . . . 5  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( G `  M
) )
39 seq1 12078 . . . . . . 7  |-  ( M  e.  ZZ  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
4036, 39syl 16 . . . . . 6  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 M )  =  ( F `  M
) )
4140oveq2d 6293 . . . . 5  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  M
) )  =  ( 1  /  ( F `
 M ) ) )
4234, 38, 413eqtr4d 2513 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
4342a1i 11 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) ) )
44 oveq1 6284 . . . . . . . . 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 ) ) ) )
45443ad2ant3 1014 . . . . . . . 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 11868 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
47 fveq2 5859 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
48 fveq2 5859 . . . . . . . . . . . . . . . . 17  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4948oveq2d 6293 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  (
n  +  1 ) ) ) )
5047, 49eqeq12d 2484 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  ( n  +  1 ) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5150imbi2d 316 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( G `
 k )  =  ( 1  /  ( F `  k )
) )  <->  ( ph  ->  ( G `  (
n  +  1 ) )  =  ( 1  /  ( F `  ( n  +  1
) ) ) ) ) )
5251, 32vtoclga 3172 . . . . . . . . . . . . 13  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5346, 52syl 16 . . . . . . . . . . . 12  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( G `  (
n  +  1 ) )  =  ( 1  /  ( F `  ( n  +  1
) ) ) ) )
5453impcom 430 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) )
5554oveq2d 6293 . . . . . . . . . 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 ax-1cn 9541 . . . . . . . . . . . . 13  |-  1  e.  CC
5756a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  1  e.  CC )
58 elfzouz 11792 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
5958adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  n  e.  (
ZZ>= `  M ) )
60 elfzouz2 11801 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( M..^ N
)  ->  N  e.  ( ZZ>= `  n )
)
61 fzss2 11714 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
6260, 61syl 16 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( M..^ N
)  ->  ( M ... n )  C_  ( M ... N ) )
6362sselda 3499 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n
) )  ->  k  e.  ( M ... N
) )
64 prodfn0.2 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  CC )
6563, 64sylan2 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  e.  CC )
6665anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
67 mulcl 9567 . . . . . . . . . . . . . 14  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
6867adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
6959, 66, 68seqcl 12085 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
7048eleq1d 2531 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
7170imbi2d 316 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
7264expcom 435 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
7371, 72vtoclga 3172 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  e.  CC ) )
7446, 73syl 16 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) )
7574impcom 430 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
76 prodfn0.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
7763, 76sylan2 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  =/=  0 )
7877anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  =/=  0 )
7959, 66, 78prodfn0 28593 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  =/=  0 )
8048neeq1d 2739 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =/=  0  <->  ( F `  ( n  +  1 ) )  =/=  0 ) )
8180imbi2d 316 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) ) )
8276expcom 435 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  =/=  0
) )
8381, 82vtoclga 3172 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  =/=  0
) )
8446, 83syl 16 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) )
8584impcom 430 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  =/=  0
)
8657, 69, 57, 75, 79, 85divmuldivd 10352 . . . . . . . . . . 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 ) ) ) ) )
87 1t1e1 10674 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
8887oveq1i 6287 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
8986, 88syl6eq 2519 . . . . . . . . . 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 ) ) ) ) )
9055, 89eqtrd 2503 . . . . . . . . 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 ) ) ) ) )
91903adant3 1011 . . . . . . . 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 ) ) ) ) )
9245, 91eqtrd 2503 . . . . . . 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 ) ) ) ) )
93 seqp1 12080 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  G ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  G ) `  n )  x.  ( G `  ( n  +  1 ) ) ) )
9458, 93syl 16 . . . . . . . 8  |-  ( n  e.  ( M..^ N
)  ->  (  seq M (  x.  ,  G ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  G ) `  n )  x.  ( G `  ( n  +  1 ) ) ) )
95943ad2ant2 1013 . . . . . . 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 ) ) ) )
96 seqp1 12080 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
9758, 96syl 16 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
9897oveq2d 6293 . . . . . . . 8  |-  ( n  e.  ( M..^ N
)  ->  ( 1  /  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) ) )
99983ad2ant2 1013 . . . . . . 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 ) ) ) ) )
10092, 95, 993eqtr4d 2513 . . . . . 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 ) ) ) )
1011003exp 1190 . . . . 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 ) ) ) ) ) )
102101com12 31 . . . 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 ) ) ) ) ) )
103102a2d 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 ) ) ) ) ) )
1048, 13, 18, 23, 43, 103fzind2 11883 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  G
) `  N )  =  ( 1  / 
(  seq M (  x.  ,  F ) `  N ) ) ) )
1053, 104mpcom 36 1  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2657    C_ wss 3471   ` cfv 5581  (class class class)co 6277   CCcc 9481   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    / cdiv 10197   ZZcz 10855   ZZ>=cuz 11073   ...cfz 11663  ..^cfzo 11783    seqcseq 12065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-fzo 11784  df-seq 12066
This theorem is referenced by:  prodfdiv  28595
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