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Theorem clim2prod 11315
Description: The limit of an infinite product with an initial segment added. (Contributed by Scott Fenton, 18-Dec-2017.)
Hypotheses
Ref Expression
clim2prod.1  |-  Z  =  ( ZZ>= `  M )
clim2prod.2  |-  ( ph  ->  N  e.  Z )
clim2prod.3  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
clim2prod.4  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  A )
Assertion
Ref Expression
clim2prod  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  ( (  seq M (  x.  ,  F ) `  N )  x.  A
) )
Distinct variable groups:    A, k    k, F    ph, k    k, M   
k, N    k, Z

Proof of Theorem clim2prod
Dummy variables  v  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2139 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2prod.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
3 uzssz 9352 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
42, 3eqsstri 3129 . . . 4  |-  Z  C_  ZZ
5 clim2prod.2 . . . 4  |-  ( ph  ->  N  e.  Z )
64, 5sseldi 3095 . . 3  |-  ( ph  ->  N  e.  ZZ )
76peano2zd 9183 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
8 clim2prod.4 . 2  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F )  ~~>  A )
95, 2eleqtrdi 2232 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
10 eluzel2 9338 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
119, 10syl 14 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2prod.3 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
132, 11, 12prodf 11314 . . 3  |-  ( ph  ->  seq M (  x.  ,  F ) : Z --> CC )
1413, 5ffvelrnd 5556 . 2  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 N )  e.  CC )
15 seqex 10227 . . 3  |-  seq M
(  x.  ,  F
)  e.  _V
1615a1i 9 . 2  |-  ( ph  ->  seq M (  x.  ,  F )  e. 
_V )
17 peano2uz 9385 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
18 uzss 9353 . . . . . . . 8  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  ( N  +  1
) )  C_  ( ZZ>=
`  M ) )
199, 17, 183syl 17 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  ( N  +  1 ) ) 
C_  ( ZZ>= `  M
) )
2019, 2sseqtrrdi 3146 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( N  +  1 ) ) 
C_  Z )
2120sselda 3097 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
2221, 12syldan 280 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
231, 7, 22prodf 11314 . . 3  |-  ( ph  ->  seq ( N  + 
1 ) (  x.  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
2423ffvelrnda 5555 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  x.  ,  F ) `  k
)  e.  CC )
25 fveq2 5421 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  (  seq M (  x.  ,  F ) `  x
)  =  (  seq M (  x.  ,  F ) `  ( N  +  1 ) ) )
26 fveq2 5421 . . . . . . 7  |-  ( x  =  ( N  + 
1 )  ->  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) ) )
2726oveq2d 5790 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( N  + 
1 ) ) ) )
2825, 27eqeq12d 2154 . . . . 5  |-  ( x  =  ( N  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  x )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 x ) )  <-> 
(  seq M (  x.  ,  F ) `  ( N  +  1
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( N  + 
1 ) ) ) ) )
2928imbi2d 229 . . . 4  |-  ( x  =  ( N  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  x
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) ) )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( N  + 
1 ) )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  ( N  +  1 ) ) ) ) ) )
30 fveq2 5421 . . . . . 6  |-  ( x  =  n  ->  (  seq M (  x.  ,  F ) `  x
)  =  (  seq M (  x.  ,  F ) `  n
) )
31 fveq2 5421 . . . . . . 7  |-  ( x  =  n  ->  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )
3231oveq2d 5790 . . . . . 6  |-  ( x  =  n  ->  (
(  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 n ) ) )
3330, 32eqeq12d 2154 . . . . 5  |-  ( x  =  n  ->  (
(  seq M (  x.  ,  F ) `  x )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 x ) )  <-> 
(  seq M (  x.  ,  F ) `  n )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 n ) ) ) )
3433imbi2d 229 . . . 4  |-  ( x  =  n  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  x
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) ) )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 n )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  n )
) ) ) )
35 fveq2 5421 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (  seq M (  x.  ,  F ) `  x
)  =  (  seq M (  x.  ,  F ) `  (
n  +  1 ) ) )
36 fveq2 5421 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) )
3736oveq2d 5790 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( n  + 
1 ) ) ) )
3835, 37eqeq12d 2154 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq M (  x.  ,  F ) `  x )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 x ) )  <-> 
(  seq M (  x.  ,  F ) `  ( n  +  1
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( n  + 
1 ) ) ) ) )
3938imbi2d 229 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  x
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) ) )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( n  + 
1 ) )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  ( n  +  1 ) ) ) ) ) )
40 fveq2 5421 . . . . . 6  |-  ( x  =  k  ->  (  seq M (  x.  ,  F ) `  x
)  =  (  seq M (  x.  ,  F ) `  k
) )
41 fveq2 5421 . . . . . . 7  |-  ( x  =  k  ->  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq ( N  +  1 ) (  x.  ,  F ) `  k
) )
4241oveq2d 5790 . . . . . 6  |-  ( x  =  k  ->  (
(  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 k ) ) )
4340, 42eqeq12d 2154 . . . . 5  |-  ( x  =  k  ->  (
(  seq M (  x.  ,  F ) `  x )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 x ) )  <-> 
(  seq M (  x.  ,  F ) `  k )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 k ) ) ) )
4443imbi2d 229 . . . 4  |-  ( x  =  k  ->  (
( ph  ->  (  seq M (  x.  ,  F ) `  x
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  x
) ) )  <->  ( ph  ->  (  seq M (  x.  ,  F ) `
 k )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  k )
) ) ) )
452eleq2i 2206 . . . . . . . 8  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
4645, 12sylan2br 286 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
47 mulcl 7754 . . . . . . . 8  |-  ( ( k  e.  CC  /\  v  e.  CC )  ->  ( k  x.  v
)  e.  CC )
4847adantl 275 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  CC  /\  v  e.  CC ) )  -> 
( k  x.  v
)  e.  CC )
499, 46, 48seq3p1 10242 . . . . . 6  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( N  + 
1 ) )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  ( F `  ( N  +  1 ) ) ) )
507, 22, 48seq3-1 10240 . . . . . . 7  |-  ( ph  ->  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( N  + 
1 ) )  =  ( F `  ( N  +  1 ) ) )
5150oveq2d 5790 . . . . . 6  |-  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  ( N  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  ( F `  ( N  +  1 ) ) ) )
5249, 51eqtr4d 2175 . . . . 5  |-  ( ph  ->  (  seq M (  x.  ,  F ) `
 ( N  + 
1 ) )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  ( N  +  1 ) ) ) )
5352a1i 9 . . . 4  |-  ( ( N  +  1 )  e.  ZZ  ->  ( ph  ->  (  seq M
(  x.  ,  F
) `  ( N  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) ) ) ) )
5419sselda 3097 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  n  e.  ( ZZ>= `  M )
)
5546adantlr 468 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
5647adantl 275 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  ( k  e.  CC  /\  v  e.  CC ) )  -> 
( k  x.  v
)  e.  CC )
5754, 55, 56seq3p1 10242 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
5857adantr 274 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
(  seq M (  x.  ,  F ) `  ( n  +  1
) )  =  ( (  seq M (  x.  ,  F ) `
 n )  x.  ( F `  (
n  +  1 ) ) ) )
59 oveq1 5781 . . . . . . . . 9  |-  ( (  seq M (  x.  ,  F ) `  n )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 n ) )  ->  ( (  seq M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) )  =  ( ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  x.  ( F `  ( n  +  1 ) ) ) )
6059adantl 275 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
( (  seq M
(  x.  ,  F
) `  n )  x.  ( F `  (
n  +  1 ) ) )  =  ( ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  n )
)  x.  ( F `
 ( n  + 
1 ) ) ) )
6114adantr 274 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  N
)  e.  CC )
6223ffvelrnda 5555 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  x.  ,  F ) `  n
)  e.  CC )
63 peano2uz 9385 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
6463, 2eleqtrrdi 2233 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
6554, 64syl 14 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( n  +  1 )  e.  Z )
6612ralrimiva 2505 . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
67 fveq2 5421 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
6867eleq1d 2208 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
6968rspcv 2785 . . . . . . . . . . . . 13  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  ( n  +  1 ) )  e.  CC ) )
7066, 69mpan9 279 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  ( F `  ( n  +  1 ) )  e.  CC )
7165, 70syldan 280 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  ( n  +  1 ) )  e.  CC )
7261, 62, 71mulassd 7796 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (
(  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  x.  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (
(  seq ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) ) )
7372adantr 274 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
( ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  x.  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (
(  seq ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) ) )
74 simpr 109 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  n  e.  ( ZZ>= `  ( N  +  1 ) ) )
7522adantlr 468 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
7674, 75, 56seq3p1 10242 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq ( N  +  1
) (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
7776oveq2d 5790 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  ( (  seq ( N  +  1 ) (  x.  ,  F
) `  n )  x.  ( F `  (
n  +  1 ) ) ) ) )
7877adantr 274 . . . . . . . . 9  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  ( n  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (
(  seq ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) ) )
7973, 78eqtr4d 2175 . . . . . . . 8  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
( ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  x.  ( F `  ( n  +  1 ) ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) ) )
8058, 60, 793eqtrd 2176 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  /\  (  seq M (  x.  ,  F ) `  n
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
(  seq M (  x.  ,  F ) `  ( n  +  1
) )  =  ( (  seq M (  x.  ,  F ) `
 N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `
 ( n  + 
1 ) ) ) )
8180exp31 361 . . . . . 6  |-  ( ph  ->  ( n  e.  (
ZZ>= `  ( N  + 
1 ) )  -> 
( (  seq M
(  x.  ,  F
) `  n )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) ) ) ) )
8281com12 30 . . . . 5  |-  ( n  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( ph  ->  ( (  seq M
(  x.  ,  F
) `  n )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) )  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) ) ) ) )
8382a2d 26 . . . 4  |-  ( n  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( ( ph  ->  (  seq M
(  x.  ,  F
) `  n )  =  ( (  seq M (  x.  ,  F ) `  N
)  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  n
) ) )  -> 
( ph  ->  (  seq M (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) ) ) ) )
8429, 34, 39, 44, 53, 83uzind4 9390 . . 3  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( ph  ->  (  seq M (  x.  ,  F ) `
 k )  =  ( (  seq M
(  x.  ,  F
) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F
) `  k )
) ) )
8584impcom 124 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq M (  x.  ,  F ) `  k
)  =  ( (  seq M (  x.  ,  F ) `  N )  x.  (  seq ( N  +  1 ) (  x.  ,  F ) `  k
) ) )
861, 7, 8, 14, 16, 24, 85climmulc2 11107 1  |-  ( ph  ->  seq M (  x.  ,  F )  ~~>  ( (  seq M (  x.  ,  F ) `  N )  x.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   _Vcvv 2686    C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   1c1 7628    + caddc 7630    x. cmul 7632   ZZcz 9061   ZZ>=cuz 9333    seqcseq 10225    ~~> cli 11054
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  ax-arch 7746  ax-caucvg 7747
This theorem depends on definitions:  df-bi 116  df-dc 820  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-if 3475  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-2 8786  df-3 8787  df-4 8788  df-n0 8985  df-z 9062  df-uz 9334  df-rp 9449  df-seqfrec 10226  df-exp 10300  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055
This theorem is referenced by:  ntrivcvgap  11324
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