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Theorem prodfrecap 11322
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 9819 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5421 . . . . 5  |-  ( m  =  M  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  M
) )
5 fveq2 5421 . . . . . 6  |-  ( m  =  M  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  M
) )
65oveq2d 5790 . . . . 5  |-  ( m  =  M  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  M
) ) )
74, 6eqeq12d 2154 . . . 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 229 . . 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 5421 . . . . 5  |-  ( m  =  n  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  n
) )
10 fveq2 5421 . . . . . 6  |-  ( m  =  n  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  n
) )
1110oveq2d 5790 . . . . 5  |-  ( m  =  n  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )
129, 11eqeq12d 2154 . . . 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 229 . . 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 5421 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  (
n  +  1 ) ) )
15 fveq2 5421 . . . . . 6  |-  ( m  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
1615oveq2d 5790 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) ) )
1714, 16eqeq12d 2154 . . . 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 229 . . 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 5421 . . . . 5  |-  ( m  =  N  ->  (  seq M (  x.  ,  G ) `  m
)  =  (  seq M (  x.  ,  G ) `  N
) )
20 fveq2 5421 . . . . . 6  |-  ( m  =  N  ->  (  seq M (  x.  ,  F ) `  m
)  =  (  seq M (  x.  ,  F ) `  N
) )
2120oveq2d 5790 . . . . 5  |-  ( m  =  N  ->  (
1  /  (  seq M (  x.  ,  F ) `  m
) )  =  ( 1  /  (  seq M (  x.  ,  F ) `  N
) ) )
2219, 21eqeq12d 2154 . . . 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 229 . . 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 9818 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
251, 24syl 14 . . . . . 6  |-  ( ph  ->  M  e.  ( M ... N ) )
26 fveq2 5421 . . . . . . . . 9  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
27 fveq2 5421 . . . . . . . . . 10  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
2827oveq2d 5790 . . . . . . . . 9  |-  ( k  =  M  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  M
) ) )
2926, 28eqeq12d 2154 . . . . . . . 8  |-  ( k  =  M  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3029imbi2d 229 . . . . . . 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 115 . . . . . . 7  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( G `  k )  =  ( 1  /  ( F `
 k ) ) ) )
3330, 32vtoclga 2752 . . . . . 6  |-  ( M  e.  ( M ... N )  ->  ( ph  ->  ( G `  M )  =  ( 1  /  ( F `
 M ) ) ) )
3425, 33mpcom 36 . . . . 5  |-  ( ph  ->  ( G `  M
)  =  ( 1  /  ( F `  M ) ) )
35 eluzel2 9338 . . . . . . 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 7754 . . . . . . 7  |-  ( ( k  e.  CC  /\  v  e.  CC )  ->  ( k  x.  v
)  e.  CC )
3938adantl 275 . . . . . 6  |-  ( (
ph  /\  ( k  e.  CC  /\  v  e.  CC ) )  -> 
( k  x.  v
)  e.  CC )
4036, 37, 39seq3-1 10240 . . . . 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 10240 . . . . . 6  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 M )  =  ( F `  M
) )
4342oveq2d 5790 . . . . 5  |-  ( ph  ->  ( 1  /  (  seq M (  x.  ,  F ) `  M
) )  =  ( 1  /  ( F `
 M ) ) )
4434, 40, 433eqtr4d 2182 . . . 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 5781 . . . . . . . . 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 1004 . . . . . . . 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 10011 . . . . . . . . . . . . 13  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
49 fveq2 5421 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
50 fveq2 5421 . . . . . . . . . . . . . . . . 17  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
5150oveq2d 5790 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
1  /  ( F `
 k ) )  =  ( 1  / 
( F `  (
n  +  1 ) ) ) )
5249, 51eqeq12d 2154 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( G `  k
)  =  ( 1  /  ( F `  k ) )  <->  ( G `  ( n  +  1 ) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) ) )
5352imbi2d 229 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( G `
 k )  =  ( 1  /  ( F `  k )
) )  <->  ( ph  ->  ( G `  (
n  +  1 ) )  =  ( 1  /  ( F `  ( n  +  1
) ) ) ) ) )
5453, 32vtoclga 2752 . . . . . . . . . . . . 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 124 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( G `  ( n  +  1
) )  =  ( 1  /  ( F `
 ( n  + 
1 ) ) ) )
5756oveq2d 5790 . . . . . . . . . 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 7789 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  1  e.  CC )
59 eqid 2139 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
6059, 36, 41prodf 11314 . . . . . . . . . . . . . 14  |-  ( ph  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
6160adantr 274 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  seq M (  x.  ,  F ) : ( ZZ>= `  M ) --> CC )
62 elfzouz 9935 . . . . . . . . . . . . . 14  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
6362adantl 275 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  n  e.  (
ZZ>= `  M ) )
6461, 63ffvelrnd 5556 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n )  e.  CC )
6550eleq1d 2208 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
6665imbi2d 229 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k )  e.  CC )  <->  ( ph  ->  ( F `  (
n  +  1 ) )  e.  CC ) ) )
67 elfzuz 9809 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( M ... N )  ->  k  e.  ( ZZ>= `  M )
)
6841expcom 115 . . . . . . . . . . . . . . . 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 2752 . . . . . . . . . . . . . 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 124 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) )  e.  CC )
7341adantlr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
74 elfzouz2 9945 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( M..^ N
)  ->  N  e.  ( ZZ>= `  n )
)
75 fzss2 9851 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ( ZZ>= `  n
)  ->  ( M ... n )  C_  ( M ... N ) )
7674, 75syl 14 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( M..^ N
)  ->  ( M ... n )  C_  ( M ... N ) )
7776sselda 3097 . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( M..^ N )  /\  k  e.  ( M ... n ) ) )  ->  ( F `  k ) #  0 )
8079anassrs 397 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  e.  ( M..^ N ) )  /\  k  e.  ( M ... n
) )  ->  ( F `  k ) #  0 )
8163, 73, 80prodfap0 11321 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq M
(  x.  ,  F
) `  n ) #  0 )
8250breq1d 3939 . . . . . . . . . . . . . . . 16  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
) #  0  <->  ( F `  ( n  +  1 ) ) #  0 ) )
8382imbi2d 229 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  (
( ph  ->  ( F `
 k ) #  0 )  <->  ( ph  ->  ( F `  ( n  +  1 ) ) #  0 ) ) )
8478expcom 115 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( M ... N )  ->  ( ph  ->  ( F `  k ) #  0 ) )
8583, 84vtoclga 2752 . . . . . . . . . . . . . 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 124 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( F `  ( n  +  1
) ) #  0 )
8858, 64, 58, 72, 81, 87divmuldivapd 8599 . . . . . . . . . . 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 8879 . . . . . . . . . . . 12  |-  ( 1  x.  1 )  =  1
9089oveq1i 5784 . . . . . . . . . . 11  |-  ( ( 1  x.  1 )  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )  =  ( 1  /  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
9188, 90syl6eq 2188 . . . . . . . . . 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 2172 . . . . . . . . 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 1001 . . . . . . . 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 2172 . . . . . . 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 1001 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N )  /\  (  seq M (  x.  ,  G ) `  n )  =  ( 1  /  (  seq M (  x.  ,  F ) `  n
) ) )  ->  n  e.  ( ZZ>= `  M ) )
96373ad2antl1 1143 . . . . . . . 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 275 . . . . . . . 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 10242 . . . . . . 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 1143 . . . . . . . . 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 10242 . . . . . . . 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 5790 . . . . . . 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 2182 . . . . . 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 1180 . . . . 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 10023 . 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480    C_ wss 3071   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    + caddc 7630    x. cmul 7632   # cap 8350    / cdiv 8439   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797  ..^cfzo 9926    seqcseq 10225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-fzo 9927  df-seqfrec 10226
This theorem is referenced by:  prodfdivap  11323
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