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Theorem prodfrec 13858
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 11750 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 17 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5851 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  M
) )
5 fveq2 5851 . . . . . 6  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
65oveq2d 6296 . . . . 5  |-  ( m  =  M  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
74, 6eqeq12d 2426 . . . 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 5851 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  n
) )
10 fveq2 5851 . . . . . 6  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
1110oveq2d 6296 . . . . 5  |-  ( m  =  n  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )
129, 11eqeq12d 2426 . . . 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 5851 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  (
n  +  1 ) ) )
15 fveq2 5851 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1615oveq2d 6296 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) ) )
1714, 16eqeq12d 2426 . . . 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 5851 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  N
) )
20 fveq2 5851 . . . . . 6  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2120oveq2d 6296 . . . . 5  |-  ( m  =  N  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
2219, 21eqeq12d 2426 . . . 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 11749 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
251, 24syl 17 . . . . . 6  |-  ( ph  ->  M  e.  ( M ... N ) )
26 fveq2 5851 . . . . . . . . 9  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
27 fveq2 5851 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2827oveq2d 6296 . . . . . . . . 9  |-  ( k  =  M  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  M
) ) )
2926, 28eqeq12d 2426 . . . . . . . 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 3125 . . . . . 6  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3425, 33mpcom 36 . . . . 5  |-  ( ph  ->  ( G `  M
)  =  ( 1  /  ( F `  M ) ) )
35 eluzel2 11134 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 17 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
37 seq1 12166 . . . . . 6  |-  ( M  e.  ZZ  ->  (  seq M (  x.  ,  G ) `  M
)  =  ( G `
 M ) )
3836, 37syl 17 . . . . 5  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( G `  M
) )
39 seq1 12166 . . . . . . 7  |-  ( M  e.  ZZ  ->  (  seq M (  x.  ,  F ) `  M
)  =  ( F `
 M ) )
4036, 39syl 17 . . . . . 6  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 M )  =  ( F `  M
) )
4140oveq2d 6296 . . . . 5  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  M
) )  =  ( 1  /  ( F `
 M ) ) )
4234, 38, 413eqtr4d 2455 . . . 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 6287 . . . . . . . . 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 1022 . . . . . . . 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 11948 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
47 fveq2 5851 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
48 fveq2 5851 . . . . . . . . . . . . . . . . 17  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
4948oveq2d 6296 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  (
n  +  1 ) ) ) )
5047, 49eqeq12d 2426 . . . . . . . . . . . . . . 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 3125 . . . . . . . . . . . . 13  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5346, 52syl 17 . . . . . . . . . . . 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 6296 . . . . . . . . . 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 9644 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  1  e.  CC )
57 elfzouz 11865 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
5857adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  n  e.  (
ZZ>= `  M ) )
59 elfzouz2 11875 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( M..^ N
)  ->  N  e.  ( ZZ>= `  n )
)
60 fzss2 11780 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
6159, 60syl 17 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( M..^ N
)  ->  ( M ... n )  C_  ( M ... N ) )
6261sselda 3444 . . . . . . . . . . . . . . 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 )
6462, 63sylan2 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  e.  CC )
6564anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  e.  CC )
66 mulcl 9608 . . . . . . . . . . . . . 14  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
6766adantl 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  ( k  e.  CC  /\  x  e.  CC ) )  -> 
( k  x.  x
)  e.  CC )
6858, 65, 67seqcl 12173 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
6948eleq1d 2473 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
7069imbi2d 316 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
7163expcom 435 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
7270, 71vtoclga 3125 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  e.  CC ) )
7346, 72syl 17 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) )
7473impcom 430 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
75 prodfn0.3 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  =/=  0
)
7662, 75sylan2 474 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k )  =/=  0 )
7776anassrs 648 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k )  =/=  0 )
7858, 65, 77prodfn0 13857 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  =/=  0 )
7948neeq1d 2682 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  =/=  0  <->  ( F `  ( n  +  1 ) )  =/=  0 ) )
8079imbi2d 316 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  =/=  0 )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) ) )
8175expcom 435 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  =/=  0
) )
8280, 81vtoclga 3125 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  =/=  0
) )
8346, 82syl 17 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  =/=  0 ) )
8483impcom 430 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  =/=  0
)
8556, 68, 56, 74, 78, 84divmuldivd 10404 . . . . . . . . . . 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 10726 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
8786oveq1i 6290 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
8885, 87syl6eq 2461 . . . . . . . . . 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 ) ) ) ) )
8955, 88eqtrd 2445 . . . . . . . . 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 ) ) ) ) )
90893adant3 1019 . . . . . . . 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 ) ) ) ) )
9145, 90eqtrd 2445 . . . . . . 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 12168 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  G ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  G ) `  n )  x.  ( G `  ( n  +  1 ) ) ) )
9357, 92syl 17 . . . . . . . 8  |-  ( n  e.  ( M..^ N
)  ->  (  seq M (  x.  ,  G ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  G ) `  n )  x.  ( G `  ( n  +  1 ) ) ) )
94933ad2ant2 1021 . . . . . . 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 12168 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
9657, 95syl 17 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
9796oveq2d 6296 . . . . . . . 8  |-  ( n  e.  ( M..^ N
)  ->  ( 1  /  (  seq M
(  x.  ,  F
) `  ( n  +  1 ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) ) )
98973ad2ant2 1021 . . . . . . 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 ) ) ) ) )
9991, 94, 983eqtr4d 2455 . . . . . 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 ) ) ) )
100993exp 1198 . . . . 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 ) ) ) ) ) )
101100com12 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 ) ) ) ) ) )
102101a2d 28 . . 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 ) ) ) ) ) )
1038, 13, 18, 23, 43, 102fzind2 11963 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  G
) `  N )  =  ( 1  / 
(  seq M (  x.  ,  F ) `  N ) ) ) )
1043, 103mpcom 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 976    = wceq 1407    e. wcel 1844    =/= wne 2600    C_ wss 3416   ` cfv 5571  (class class class)co 6280   CCcc 9522   0cc0 9524   1c1 9525    + caddc 9527    x. cmul 9529    / cdiv 10249   ZZcz 10907   ZZ>=cuz 11129   ...cfz 11728  ..^cfzo 11856    seqcseq 12153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-seq 12154
This theorem is referenced by:  prodfdiv  13859
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