Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clim2prod Unicode version

Theorem clim2prod 25169
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  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . 2  |-  ( ZZ>= `  ( N  +  1
) )  =  (
ZZ>= `  ( N  + 
1 ) )
2 clim2prod.1 . . . . 5  |-  Z  =  ( ZZ>= `  M )
3 uzssz 10461 . . . . 5  |-  ( ZZ>= `  M )  C_  ZZ
42, 3eqsstri 3338 . . . 4  |-  Z  C_  ZZ
5 clim2prod.2 . . . 4  |-  ( ph  ->  N  e.  Z )
64, 5sseldi 3306 . . 3  |-  ( ph  ->  N  e.  ZZ )
76peano2zd 10334 . 2  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
8 clim2prod.4 . 2  |-  ( ph  ->  seq  ( N  + 
1 ) (  x.  ,  F )  ~~>  A )
95, 2syl6eleq 2494 . . . . 5  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
10 eluzel2 10449 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
119, 10syl 16 . . . 4  |-  ( ph  ->  M  e.  ZZ )
12 clim2prod.3 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
132, 11, 12prodf 25168 . . 3  |-  ( ph  ->  seq  M (  x.  ,  F ) : Z --> CC )
1413, 5ffvelrnd 5830 . 2  |-  ( ph  ->  (  seq  M (  x.  ,  F ) `
 N )  e.  CC )
15 seqex 11280 . . 3  |-  seq  M
(  x.  ,  F
)  e.  _V
1615a1i 11 . 2  |-  ( ph  ->  seq  M (  x.  ,  F )  e. 
_V )
17 peano2uz 10486 . . . . . . . . 9  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( N  +  1 )  e.  ( ZZ>= `  M )
)
189, 17syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  ( ZZ>= `  M ) )
19 uzss 10462 . . . . . . . 8  |-  ( ( N  +  1 )  e.  ( ZZ>= `  M
)  ->  ( ZZ>= `  ( N  +  1
) )  C_  ( ZZ>=
`  M ) )
2018, 19syl 16 . . . . . . 7  |-  ( ph  ->  ( ZZ>= `  ( N  +  1 ) ) 
C_  ( ZZ>= `  M
) )
2120, 2syl6sseqr 3355 . . . . . 6  |-  ( ph  ->  ( ZZ>= `  ( N  +  1 ) ) 
C_  Z )
2221sselda 3308 . . . . 5  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  Z )
2322, 12syldan 457 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  k )  e.  CC )
241, 7, 23prodf 25168 . . 3  |-  ( ph  ->  seq  ( N  + 
1 ) (  x.  ,  F ) : ( ZZ>= `  ( N  +  1 ) ) --> CC )
2524ffvelrnda 5829 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  x.  ,  F ) `  k
)  e.  CC )
26 fveq2 5687 . . . . . 6  |-  ( x  =  ( N  + 
1 )  ->  (  seq  M (  x.  ,  F ) `  x
)  =  (  seq 
M (  x.  ,  F ) `  ( N  +  1 ) ) )
27 fveq2 5687 . . . . . . 7  |-  ( x  =  ( N  + 
1 )  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq  ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) ) )
2827oveq2d 6056 . . . . . 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 ) ) ) )
2926, 28eqeq12d 2418 . . . . 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 ) ) ) ) )
3029imbi2d 308 . . . 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 ) ) ) ) ) )
31 fveq2 5687 . . . . . 6  |-  ( x  =  n  ->  (  seq  M (  x.  ,  F ) `  x
)  =  (  seq 
M (  x.  ,  F ) `  n
) )
32 fveq2 5687 . . . . . . 7  |-  ( x  =  n  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq  ( N  +  1 ) (  x.  ,  F ) `  n
) )
3332oveq2d 6056 . . . . . 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 ) ) )
3431, 33eqeq12d 2418 . . . . 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 ) ) ) )
3534imbi2d 308 . . . 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 ) ) ) ) )
36 fveq2 5687 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (  seq  M (  x.  ,  F ) `  x
)  =  (  seq 
M (  x.  ,  F ) `  (
n  +  1 ) ) )
37 fveq2 5687 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq  ( N  +  1 ) (  x.  ,  F ) `  (
n  +  1 ) ) )
3837oveq2d 6056 . . . . . 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 ) ) ) )
3936, 38eqeq12d 2418 . . . . 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 ) ) ) ) )
4039imbi2d 308 . . . 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 ) ) ) ) ) )
41 fveq2 5687 . . . . . 6  |-  ( x  =  k  ->  (  seq  M (  x.  ,  F ) `  x
)  =  (  seq 
M (  x.  ,  F ) `  k
) )
42 fveq2 5687 . . . . . . 7  |-  ( x  =  k  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  x
)  =  (  seq  ( N  +  1 ) (  x.  ,  F ) `  k
) )
4342oveq2d 6056 . . . . . 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 ) ) )
4441, 43eqeq12d 2418 . . . . 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 ) ) ) )
4544imbi2d 308 . . . 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 ) ) ) ) )
469adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( N  +  1 )  e.  ZZ )  ->  N  e.  ( ZZ>= `  M )
)
47 seqp1 11293 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  (  seq  M (  x.  ,  F
) `  ( N  +  1 ) )  =  ( (  seq 
M (  x.  ,  F ) `  N
)  x.  ( F `
 ( N  + 
1 ) ) ) )
4846, 47syl 16 . . . . . 6  |-  ( (
ph  /\  ( N  +  1 )  e.  ZZ )  ->  (  seq  M (  x.  ,  F ) `  ( N  +  1 ) )  =  ( (  seq  M (  x.  ,  F ) `  N )  x.  ( F `  ( N  +  1 ) ) ) )
49 seq1 11291 . . . . . . . 8  |-  ( ( N  +  1 )  e.  ZZ  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) )  =  ( F `
 ( N  + 
1 ) ) )
5049adantl 453 . . . . . . 7  |-  ( (
ph  /\  ( N  +  1 )  e.  ZZ )  ->  (  seq  ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) )  =  ( F `
 ( N  + 
1 ) ) )
5150oveq2d 6056 . . . . . 6  |-  ( (
ph  /\  ( N  +  1 )  e.  ZZ )  ->  (
(  seq  M (  x.  ,  F ) `  N )  x.  (  seq  ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) ) )  =  ( (  seq  M (  x.  ,  F ) `
 N )  x.  ( F `  ( N  +  1 ) ) ) )
5248, 51eqtr4d 2439 . . . . 5  |-  ( (
ph  /\  ( N  +  1 )  e.  ZZ )  ->  (  seq  M (  x.  ,  F ) `  ( N  +  1 ) )  =  ( (  seq  M (  x.  ,  F ) `  N )  x.  (  seq  ( N  +  1 ) (  x.  ,  F ) `  ( N  +  1 ) ) ) )
5352expcom 425 . . . 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 ) ) ) ) )
5420sselda 3308 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  n  e.  ( ZZ>= `  M )
)
55 seqp1 11293 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  x.  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
5654, 55syl 16 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  x.  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  x.  ,  F ) `  n
)  x.  ( F `
 ( n  + 
1 ) ) ) )
5756adantr 452 . . . . . . . 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 ) ) ) )
58 oveq1 6047 . . . . . . . . 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 ) ) ) )
5958adantl 453 . . . . . . . 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
) ) ) )
6014adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  x.  ,  F
) `  N )  e.  CC )
6124ffvelrnda 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  x.  ,  F ) `  n
)  e.  CC )
62 peano2uz 10486 . . . . . . . . . . . . . 14  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  ( ZZ>= `  M )
)
6362, 2syl6eleqr 2495 . . . . . . . . . . . . 13  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( n  +  1 )  e.  Z )
6454, 63syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( n  +  1 )  e.  Z )
6512ralrimiva 2749 . . . . . . . . . . . . 13  |-  ( ph  ->  A. k  e.  Z  ( F `  k )  e.  CC )
66 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
6766eleq1d 2470 . . . . . . . . . . . . . 14  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
)  e.  CC  <->  ( F `  ( n  +  1 ) )  e.  CC ) )
6867rspcv 3008 . . . . . . . . . . . . 13  |-  ( ( n  +  1 )  e.  Z  ->  ( A. k  e.  Z  ( F `  k )  e.  CC  ->  ( F `  ( n  +  1 ) )  e.  CC ) )
6965, 68mpan9 456 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  +  1 )  e.  Z )  ->  ( F `  ( n  +  1 ) )  e.  CC )
7064, 69syldan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( F `  ( n  +  1 ) )  e.  CC )
7160, 61, 70mulassd 9067 . . . . . . . . . 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 ) ) ) ) )
7271adantr 452 . . . . . . . . 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 ) ) ) ) )
73 seqp1 11293 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  ( N  +  1 ) )  ->  (  seq  ( N  +  1
) (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq  ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
7473adantl 453 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  ( N  +  1
) (  x.  ,  F ) `  (
n  +  1 ) )  =  ( (  seq  ( N  + 
1 ) (  x.  ,  F ) `  n )  x.  ( F `  ( n  +  1 ) ) ) )
7574oveq2d 6056 . . . . . . . . . 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 ) ) ) ) )
7675adantr 452 . . . . . . . . 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 ) ) ) ) )
7772, 76eqtr4d 2439 . . . . . . . 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 ) ) ) )
7857, 59, 773eqtrd 2440 . . . . . . 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 ) ) ) )
7978exp31 588 . . . . . 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 ) ) ) ) ) )
8079com12 29 . . . . 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 ) ) ) ) ) )
8180a2d 24 . . . 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 ) ) ) ) ) )
8230, 35, 40, 45, 53, 81uzind4 10490 . . 3  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( ph  ->  (  seq  M (  x.  ,  F ) `
 k )  =  ( (  seq  M
(  x.  ,  F
) `  N )  x.  (  seq  ( N  +  1 ) (  x.  ,  F ) `
 k ) ) ) )
8382impcom 420 . 2  |-  ( (
ph  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) )  ->  (  seq  M (  x.  ,  F
) `  k )  =  ( (  seq 
M (  x.  ,  F ) `  N
)  x.  (  seq  ( N  +  1 ) (  x.  ,  F ) `  k
) ) )
841, 7, 8, 14, 16, 25, 83climmulc2 12385 1  |-  ( ph  ->  seq  M (  x.  ,  F )  ~~>  ( (  seq  M (  x.  ,  F ) `  N )  x.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949    x. cmul 8951   ZZcz 10238   ZZ>=cuz 10444    seq cseq 11278    ~~> cli 12233
This theorem is referenced by:  ntrivcvg  25178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
  Copyright terms: Public domain W3C validator