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Theorem prodfrecap 11554
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.
StepHypRef Expression
1 prodfap0.1 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 10032 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5516 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  M
) )
5 fveq2 5516 . . . . . 6  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
65oveq2d 5891 . . . . 5  |-  ( m  =  M  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
74, 6eqeq12d 2192 . . . 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 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 5516 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  n
) )
10 fveq2 5516 . . . . . 6  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
1110oveq2d 5891 . . . . 5  |-  ( m  =  n  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )
129, 11eqeq12d 2192 . . . 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 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 5516 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  (
n  +  1 ) ) )
15 fveq2 5516 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1615oveq2d 5891 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) ) )
1714, 16eqeq12d 2192 . . . 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 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 5516 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  N
) )
20 fveq2 5516 . . . . . 6  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2120oveq2d 5891 . . . . 5  |-  ( m  =  N  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
2219, 21eqeq12d 2192 . . . 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 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 10031 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
251, 24syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ( M ... N ) )
26 fveq2 5516 . . . . . . . . 9  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
27 fveq2 5516 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2827oveq2d 5891 . . . . . . . . 9  |-  ( k  =  M  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  M
) ) )
2926, 28eqeq12d 2192 . . . . . . . 8  |-  ( k  =  M  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3029imbi2d 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 ) ) )
3231expcom 116 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( G `  k )  =  ( 1  /  ( F `
 k ) ) ) )
3330, 32vtoclga 2804 . . . . . 6  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3425, 33mpcom 36 . . . . 5  |-  ( ph  ->  ( G `  M
)  =  ( 1  /  ( F `  M ) ) )
35 eluzel2 9533 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
361, 35syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
37 prodfrecap.g . . . . . 6  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
38 mulcl 7938 . . . . . . 7  |-  ( ( k  e.  CC  /\  v  e.  CC )  ->  ( k  x.  v
)  e.  CC )
3938adantl 277 . . . . . 6  |-  ( (
ph  /\  ( k  e.  CC  /\  v  e.  CC ) )  -> 
( k  x.  v
)  e.  CC )
4036, 37, 39seq3-1 10460 . . . . 5  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( G `  M
) )
41 prodfap0.2 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
4236, 41, 39seq3-1 10460 . . . . . 6  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 M )  =  ( F `  M
) )
4342oveq2d 5891 . . . . 5  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  M
) )  =  ( 1  /  ( F `
 M ) ) )
4434, 40, 433eqtr4d 2220 . . . 4  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
4544a1i 9 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq M (  x.  ,  G ) `
 M )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) ) )
46 oveq1 5882 . . . . . . . . 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 ) ) ) )
47463ad2ant3 1020 . . . . . . . 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 10227 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
49 fveq2 5516 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
50 fveq2 5516 . . . . . . . . . . . . . . . . 17  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5150oveq2d 5891 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  (
n  +  1 ) ) ) )
5249, 51eqeq12d 2192 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  ( n  +  1 ) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5352imbi2d 230 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( G `
 k )  =  ( 1  /  ( F `  k )
) )  <->  ( ph  ->  ( G `  (
n  +  1 ) )  =  ( 1  /  ( F `  ( n  +  1
) ) ) ) ) )
5453, 32vtoclga 2804 . . . . . . . . . . . . 13  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5548, 54syl 14 . . . . . . . . . . . 12  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( G `  (
n  +  1 ) )  =  ( 1  /  ( F `  ( n  +  1
) ) ) ) )
5655impcom 125 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) )
5756oveq2d 5891 . . . . . . . . . 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 7973 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  1  e.  CC )
59 eqid 2177 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
6059, 36, 41prodf 11546 . . . . . . . . . . . . . 14  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
6160adantr 276 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
62 elfzouz 10151 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
6362adantl 277 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  n  e.  (
ZZ>= `  M ) )
6461, 63ffvelcdmd 5653 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
6550eleq1d 2246 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
6665imbi2d 230 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
67 elfzuz 10021 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
6841expcom 116 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ZZ>= `  M
)  ->  ( ph  ->  ( F `  k
)  e.  CC ) )
6967, 68syl 14 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k )  e.  CC ) )
7066, 69vtoclga 2804 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) )  e.  CC ) )
7148, 70syl 14 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) )
7271impcom 125 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
7341adantlr 477 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
74 elfzouz2 10161 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( M..^ N
)  ->  N  e.  ( ZZ>= `  n )
)
75 fzss2 10064 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
7674, 75syl 14 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( M..^ N
)  ->  ( M ... n )  C_  ( M ... N ) )
7776sselda 3156 . . . . . . . . . . . . . . 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 )
7977, 78sylan2 286 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k ) #  0 )
8079anassrs 400 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k ) #  0 )
8163, 73, 80prodfap0 11553 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n ) #  0 )
8250breq1d 4014 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
) #  0  <->  ( F `  ( n  +  1 ) ) #  0 ) )
8382imbi2d 230 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  ( n  +  1 ) ) #  0 ) ) )
8478expcom 116 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k ) #  0 ) )
8583, 84vtoclga 2804 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( ph  ->  ( F `  ( n  +  1
) ) #  0 ) )
8648, 85syl 14 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( F `  (
n  +  1 ) ) #  0 ) )
8786impcom 125 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) ) #  0 )
8858, 64, 58, 72, 81, 87divmuldivapd 8789 . . . . . . . . . . 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 9071 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
9089oveq1i 5885 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
9188, 90eqtrdi 2226 . . . . . . . . . 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 ) ) ) ) )
9257, 91eqtrd 2210 . . . . . . . . 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 ) ) ) ) )
93923adant3 1017 . . . . . . . 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 ) ) ) ) )
9447, 93eqtrd 2210 . . . . . . 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 ) ) ) ) )
95633adant3 1017 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  G ) `  n )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )  ->  n  e.  ( ZZ>= `  M ) )
96373ad2antl1 1159 . . . . . . . 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 )
9738adantl 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 )
9895, 96, 97seq3p1 10462 . . . . . . 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 ) ) ) )
99413ad2antl1 1159 . . . . . . . . 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 )
10095, 99, 97seq3p1 10462 . . . . . . . 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 ) ) ) )
101100oveq2d 5891 . . . . . . 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 ) ) ) ) )
10294, 98, 1013eqtr4d 2220 . . . . . 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 ) ) ) )
1031023exp 1202 . . . . 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 ) ) ) ) ) )
104103com12 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 ) ) ) ) ) )
105104a2d 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 ) ) ) ) ) )
1068, 13, 18, 23, 45, 105fzind2 10239 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq M
(  x.  ,  G
) `  N )  =  ( 1  / 
(  seq M (  x.  ,  F ) `  N ) ) ) )
1073, 106mpcom 36 1  |-  ( ph  ->  (  seq M (  x.  ,  G ) `
 N )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148    C_ wss 3130   class class class wbr 4004   -->wf 5213   ` cfv 5217  (class class class)co 5875   CCcc 7809   0cc0 7811   1c1 7812    + caddc 7814    x. cmul 7816   # cap 8538    / cdiv 8629   ZZcz 9253   ZZ>=cuz 9528   ...cfz 10008  ..^cfzo 10142    seqcseq 10445
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009  df-fzo 10143  df-seqfrec 10446
This theorem is referenced by:  prodfdivap  11555
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