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Theorem seqcaopr3 11081
Description: Lemma for seqcaopr2 11082. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
seqcaopr3.1  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seqcaopr3.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x Q y )  e.  S )
seqcaopr3.3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
seqcaopr3.4  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( F `  k )  e.  S
)
seqcaopr3.5  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( G `  k )  e.  S
)
seqcaopr3.6  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
) Q ( G `
 k ) ) )
seqcaopr3.7  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( ( (  seq  M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  .+  (
( F `  (
n  +  1 ) ) Q ( G `
 ( n  + 
1 ) ) ) )  =  ( ( (  seq  M ( 
.+  ,  F ) `
 n )  .+  ( F `  ( n  +  1 ) ) ) Q ( (  seq  M (  .+  ,  G ) `  n
)  .+  ( G `  ( n  +  1 ) ) ) ) )
Assertion
Ref Expression
seqcaopr3  |-  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 N )  =  ( (  seq  M
(  .+  ,  F
) `  N ) Q (  seq  M
(  .+  ,  G
) `  N )
) )
Distinct variable groups:    k, n, x, y, F    k, H, n    k, N, n, x, y    ph, k, n, x, y    k, G, n, x, y    k, M, n, x, y    Q, k, n, x, y    .+ , n, x, y    S, k, x, y
Allowed substitution hints:    .+ ( k)    S( n)    H( x, y)

Proof of Theorem seqcaopr3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 seqcaopr3.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
2 eluzfz2 10804 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  N  e.  ( M ... N ) )
31, 2syl 15 . 2  |-  ( ph  ->  N  e.  ( M ... N ) )
4 fveq2 5525 . . . . 5  |-  ( z  =  M  ->  (  seq  M (  .+  ,  H ) `  z
)  =  (  seq 
M (  .+  ,  H ) `  M
) )
5 fveq2 5525 . . . . . 6  |-  ( z  =  M  ->  (  seq  M (  .+  ,  F ) `  z
)  =  (  seq 
M (  .+  ,  F ) `  M
) )
6 fveq2 5525 . . . . . 6  |-  ( z  =  M  ->  (  seq  M (  .+  ,  G ) `  z
)  =  (  seq 
M (  .+  ,  G ) `  M
) )
75, 6oveq12d 5876 . . . . 5  |-  ( z  =  M  ->  (
(  seq  M (  .+  ,  F ) `  z ) Q (  seq  M (  .+  ,  G ) `  z
) )  =  ( (  seq  M ( 
.+  ,  F ) `
 M ) Q (  seq  M ( 
.+  ,  G ) `
 M ) ) )
84, 7eqeq12d 2297 . . . 4  |-  ( z  =  M  ->  (
(  seq  M (  .+  ,  H ) `  z )  =  ( (  seq  M ( 
.+  ,  F ) `
 z ) Q (  seq  M ( 
.+  ,  G ) `
 z ) )  <-> 
(  seq  M (  .+  ,  H ) `  M )  =  ( (  seq  M ( 
.+  ,  F ) `
 M ) Q (  seq  M ( 
.+  ,  G ) `
 M ) ) ) )
98imbi2d 307 . . 3  |-  ( z  =  M  ->  (
( ph  ->  (  seq 
M (  .+  ,  H ) `  z
)  =  ( (  seq  M (  .+  ,  F ) `  z
) Q (  seq 
M (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 M )  =  ( (  seq  M
(  .+  ,  F
) `  M ) Q (  seq  M
(  .+  ,  G
) `  M )
) ) ) )
10 fveq2 5525 . . . . 5  |-  ( z  =  n  ->  (  seq  M (  .+  ,  H ) `  z
)  =  (  seq 
M (  .+  ,  H ) `  n
) )
11 fveq2 5525 . . . . . 6  |-  ( z  =  n  ->  (  seq  M (  .+  ,  F ) `  z
)  =  (  seq 
M (  .+  ,  F ) `  n
) )
12 fveq2 5525 . . . . . 6  |-  ( z  =  n  ->  (  seq  M (  .+  ,  G ) `  z
)  =  (  seq 
M (  .+  ,  G ) `  n
) )
1311, 12oveq12d 5876 . . . . 5  |-  ( z  =  n  ->  (
(  seq  M (  .+  ,  F ) `  z ) Q (  seq  M (  .+  ,  G ) `  z
) )  =  ( (  seq  M ( 
.+  ,  F ) `
 n ) Q (  seq  M ( 
.+  ,  G ) `
 n ) ) )
1410, 13eqeq12d 2297 . . . 4  |-  ( z  =  n  ->  (
(  seq  M (  .+  ,  H ) `  z )  =  ( (  seq  M ( 
.+  ,  F ) `
 z ) Q (  seq  M ( 
.+  ,  G ) `
 z ) )  <-> 
(  seq  M (  .+  ,  H ) `  n )  =  ( (  seq  M ( 
.+  ,  F ) `
 n ) Q (  seq  M ( 
.+  ,  G ) `
 n ) ) ) )
1514imbi2d 307 . . 3  |-  ( z  =  n  ->  (
( ph  ->  (  seq 
M (  .+  ,  H ) `  z
)  =  ( (  seq  M (  .+  ,  F ) `  z
) Q (  seq 
M (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 n )  =  ( (  seq  M
(  .+  ,  F
) `  n ) Q (  seq  M
(  .+  ,  G
) `  n )
) ) ) )
16 fveq2 5525 . . . . 5  |-  ( z  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  H ) `  z
)  =  (  seq 
M (  .+  ,  H ) `  (
n  +  1 ) ) )
17 fveq2 5525 . . . . . 6  |-  ( z  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  F ) `  z
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) )
18 fveq2 5525 . . . . . 6  |-  ( z  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  G ) `  z
)  =  (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) )
1917, 18oveq12d 5876 . . . . 5  |-  ( z  =  ( n  + 
1 )  ->  (
(  seq  M (  .+  ,  F ) `  z ) Q (  seq  M (  .+  ,  G ) `  z
) )  =  ( (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) Q (  seq  M ( 
.+  ,  G ) `
 ( n  + 
1 ) ) ) )
2016, 19eqeq12d 2297 . . . 4  |-  ( z  =  ( n  + 
1 )  ->  (
(  seq  M (  .+  ,  H ) `  z )  =  ( (  seq  M ( 
.+  ,  F ) `
 z ) Q (  seq  M ( 
.+  ,  G ) `
 z ) )  <-> 
(  seq  M (  .+  ,  H ) `  ( n  +  1
) )  =  ( (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) Q (  seq  M ( 
.+  ,  G ) `
 ( n  + 
1 ) ) ) ) )
2120imbi2d 307 . . 3  |-  ( z  =  ( n  + 
1 )  ->  (
( ph  ->  (  seq 
M (  .+  ,  H ) `  z
)  =  ( (  seq  M (  .+  ,  F ) `  z
) Q (  seq 
M (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 ( n  + 
1 ) )  =  ( (  seq  M
(  .+  ,  F
) `  ( n  +  1 ) ) Q (  seq  M
(  .+  ,  G
) `  ( n  +  1 ) ) ) ) ) )
22 fveq2 5525 . . . . 5  |-  ( z  =  N  ->  (  seq  M (  .+  ,  H ) `  z
)  =  (  seq 
M (  .+  ,  H ) `  N
) )
23 fveq2 5525 . . . . . 6  |-  ( z  =  N  ->  (  seq  M (  .+  ,  F ) `  z
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
24 fveq2 5525 . . . . . 6  |-  ( z  =  N  ->  (  seq  M (  .+  ,  G ) `  z
)  =  (  seq 
M (  .+  ,  G ) `  N
) )
2523, 24oveq12d 5876 . . . . 5  |-  ( z  =  N  ->  (
(  seq  M (  .+  ,  F ) `  z ) Q (  seq  M (  .+  ,  G ) `  z
) )  =  ( (  seq  M ( 
.+  ,  F ) `
 N ) Q (  seq  M ( 
.+  ,  G ) `
 N ) ) )
2622, 25eqeq12d 2297 . . . 4  |-  ( z  =  N  ->  (
(  seq  M (  .+  ,  H ) `  z )  =  ( (  seq  M ( 
.+  ,  F ) `
 z ) Q (  seq  M ( 
.+  ,  G ) `
 z ) )  <-> 
(  seq  M (  .+  ,  H ) `  N )  =  ( (  seq  M ( 
.+  ,  F ) `
 N ) Q (  seq  M ( 
.+  ,  G ) `
 N ) ) ) )
2726imbi2d 307 . . 3  |-  ( z  =  N  ->  (
( ph  ->  (  seq 
M (  .+  ,  H ) `  z
)  =  ( (  seq  M (  .+  ,  F ) `  z
) Q (  seq 
M (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 N )  =  ( (  seq  M
(  .+  ,  F
) `  N ) Q (  seq  M
(  .+  ,  G
) `  N )
) ) ) )
28 eluzfz1 10803 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
291, 28syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ( M ... N ) )
30 seqcaopr3.6 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( M ... N ) )  ->  ( H `  k )  =  ( ( F `  k
) Q ( G `
 k ) ) )
3130ralrimiva 2626 . . . . . 6  |-  ( ph  ->  A. k  e.  ( M ... N ) ( H `  k
)  =  ( ( F `  k ) Q ( G `  k ) ) )
32 fveq2 5525 . . . . . . . 8  |-  ( k  =  M  ->  ( H `  k )  =  ( H `  M ) )
33 fveq2 5525 . . . . . . . . 9  |-  ( k  =  M  ->  ( F `  k )  =  ( F `  M ) )
34 fveq2 5525 . . . . . . . . 9  |-  ( k  =  M  ->  ( G `  k )  =  ( G `  M ) )
3533, 34oveq12d 5876 . . . . . . . 8  |-  ( k  =  M  ->  (
( F `  k
) Q ( G `
 k ) )  =  ( ( F `
 M ) Q ( G `  M
) ) )
3632, 35eqeq12d 2297 . . . . . . 7  |-  ( k  =  M  ->  (
( H `  k
)  =  ( ( F `  k ) Q ( G `  k ) )  <->  ( H `  M )  =  ( ( F `  M
) Q ( G `
 M ) ) ) )
3736rspcv 2880 . . . . . 6  |-  ( M  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( H `  k )  =  ( ( F `
 k ) Q ( G `  k
) )  ->  ( H `  M )  =  ( ( F `
 M ) Q ( G `  M
) ) ) )
3829, 31, 37sylc 56 . . . . 5  |-  ( ph  ->  ( H `  M
)  =  ( ( F `  M ) Q ( G `  M ) ) )
39 eluzel2 10235 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
401, 39syl 15 . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
41 seq1 11059 . . . . . 6  |-  ( M  e.  ZZ  ->  (  seq  M (  .+  ,  H ) `  M
)  =  ( H `
 M ) )
4240, 41syl 15 . . . . 5  |-  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 M )  =  ( H `  M
) )
43 seq1 11059 . . . . . . 7  |-  ( M  e.  ZZ  ->  (  seq  M (  .+  ,  F ) `  M
)  =  ( F `
 M ) )
44 seq1 11059 . . . . . . 7  |-  ( M  e.  ZZ  ->  (  seq  M (  .+  ,  G ) `  M
)  =  ( G `
 M ) )
4543, 44oveq12d 5876 . . . . . 6  |-  ( M  e.  ZZ  ->  (
(  seq  M (  .+  ,  F ) `  M ) Q (  seq  M (  .+  ,  G ) `  M
) )  =  ( ( F `  M
) Q ( G `
 M ) ) )
4640, 45syl 15 . . . . 5  |-  ( ph  ->  ( (  seq  M
(  .+  ,  F
) `  M ) Q (  seq  M
(  .+  ,  G
) `  M )
)  =  ( ( F `  M ) Q ( G `  M ) ) )
4738, 42, 463eqtr4d 2325 . . . 4  |-  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 M )  =  ( (  seq  M
(  .+  ,  F
) `  M ) Q (  seq  M
(  .+  ,  G
) `  M )
) )
4847a1i 10 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 M )  =  ( (  seq  M
(  .+  ,  F
) `  M ) Q (  seq  M
(  .+  ,  G
) `  M )
) ) )
49 oveq1 5865 . . . . . 6  |-  ( (  seq  M (  .+  ,  H ) `  n
)  =  ( (  seq  M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  ->  (
(  seq  M (  .+  ,  H ) `  n )  .+  ( H `  ( n  +  1 ) ) )  =  ( ( (  seq  M ( 
.+  ,  F ) `
 n ) Q (  seq  M ( 
.+  ,  G ) `
 n ) ) 
.+  ( H `  ( n  +  1
) ) ) )
50 elfzouz 10879 . . . . . . . . 9  |-  ( n  e.  ( M..^ N
)  ->  n  e.  ( ZZ>= `  M )
)
5150adantl 452 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  n  e.  (
ZZ>= `  M ) )
52 seqp1 11061 . . . . . . . 8  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  .+  ,  H
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  H ) `  n
)  .+  ( H `  ( n  +  1 ) ) ) )
5351, 52syl 15 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  (  seq  M
(  .+  ,  H
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  H ) `  n
)  .+  ( H `  ( n  +  1 ) ) ) )
54 seqcaopr3.7 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( ( (  seq  M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  .+  (
( F `  (
n  +  1 ) ) Q ( G `
 ( n  + 
1 ) ) ) )  =  ( ( (  seq  M ( 
.+  ,  F ) `
 n )  .+  ( F `  ( n  +  1 ) ) ) Q ( (  seq  M (  .+  ,  G ) `  n
)  .+  ( G `  ( n  +  1 ) ) ) ) )
55 fzofzp1 10916 . . . . . . . . . . 11  |-  ( n  e.  ( M..^ N
)  ->  ( n  +  1 )  e.  ( M ... N
) )
5655adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( n  + 
1 )  e.  ( M ... N ) )
5731adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  A. k  e.  ( M ... N ) ( H `  k
)  =  ( ( F `  k ) Q ( G `  k ) ) )
58 fveq2 5525 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  ( H `  k )  =  ( H `  ( n  +  1
) ) )
59 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( F `  k )  =  ( F `  ( n  +  1
) ) )
60 fveq2 5525 . . . . . . . . . . . . 13  |-  ( k  =  ( n  + 
1 )  ->  ( G `  k )  =  ( G `  ( n  +  1
) ) )
6159, 60oveq12d 5876 . . . . . . . . . . . 12  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  k
) Q ( G `
 k ) )  =  ( ( F `
 ( n  + 
1 ) ) Q ( G `  (
n  +  1 ) ) ) )
6258, 61eqeq12d 2297 . . . . . . . . . . 11  |-  ( k  =  ( n  + 
1 )  ->  (
( H `  k
)  =  ( ( F `  k ) Q ( G `  k ) )  <->  ( H `  ( n  +  1 ) )  =  ( ( F `  (
n  +  1 ) ) Q ( G `
 ( n  + 
1 ) ) ) ) )
6362rspcv 2880 . . . . . . . . . 10  |-  ( ( n  +  1 )  e.  ( M ... N )  ->  ( A. k  e.  ( M ... N ) ( H `  k )  =  ( ( F `
 k ) Q ( G `  k
) )  ->  ( H `  ( n  +  1 ) )  =  ( ( F `
 ( n  + 
1 ) ) Q ( G `  (
n  +  1 ) ) ) ) )
6456, 57, 63sylc 56 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( H `  ( n  +  1
) )  =  ( ( F `  (
n  +  1 ) ) Q ( G `
 ( n  + 
1 ) ) ) )
6564oveq2d 5874 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( ( (  seq  M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  .+  ( H `  ( n  +  1 ) ) )  =  ( ( (  seq  M ( 
.+  ,  F ) `
 n ) Q (  seq  M ( 
.+  ,  G ) `
 n ) ) 
.+  ( ( F `
 ( n  + 
1 ) ) Q ( G `  (
n  +  1 ) ) ) ) )
66 seqp1 11061 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  .+  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
67 seqp1 11061 . . . . . . . . . 10  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  .+  ,  G
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  G ) `  n
)  .+  ( G `  ( n  +  1 ) ) ) )
6866, 67oveq12d 5876 . . . . . . . . 9  |-  ( n  e.  ( ZZ>= `  M
)  ->  ( (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) )  =  ( ( (  seq  M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) Q ( (  seq  M ( 
.+  ,  G ) `
 n )  .+  ( G `  ( n  +  1 ) ) ) ) )
6951, 68syl 15 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) )  =  ( ( (  seq  M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) Q ( (  seq  M ( 
.+  ,  G ) `
 n )  .+  ( G `  ( n  +  1 ) ) ) ) )
7054, 65, 693eqtr4rd 2326 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) )  =  ( ( (  seq  M
(  .+  ,  F
) `  n ) Q (  seq  M
(  .+  ,  G
) `  n )
)  .+  ( H `  ( n  +  1 ) ) ) )
7153, 70eqeq12d 2297 . . . . . 6  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( (  seq 
M (  .+  ,  H ) `  (
n  +  1 ) )  =  ( (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) )  <->  ( (  seq  M (  .+  ,  H ) `  n
)  .+  ( H `  ( n  +  1 ) ) )  =  ( ( (  seq 
M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  .+  ( H `  ( n  +  1 ) ) ) ) )
7249, 71syl5ibr 212 . . . . 5  |-  ( (
ph  /\  n  e.  ( M..^ N ) )  ->  ( (  seq 
M (  .+  ,  H ) `  n
)  =  ( (  seq  M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  ->  (  seq  M (  .+  ,  H ) `  (
n  +  1 ) )  =  ( (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) ) ) )
7372expcom 424 . . . 4  |-  ( n  e.  ( M..^ N
)  ->  ( ph  ->  ( (  seq  M
(  .+  ,  H
) `  n )  =  ( (  seq 
M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) )  ->  (  seq  M (  .+  ,  H ) `  (
n  +  1 ) )  =  ( (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) ) ) ) )
7473a2d 23 . . 3  |-  ( n  e.  ( M..^ N
)  ->  ( ( ph  ->  (  seq  M
(  .+  ,  H
) `  n )  =  ( (  seq 
M (  .+  ,  F ) `  n
) Q (  seq 
M (  .+  ,  G ) `  n
) ) )  -> 
( ph  ->  (  seq 
M (  .+  ,  H ) `  (
n  +  1 ) )  =  ( (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) Q (  seq 
M (  .+  ,  G ) `  (
n  +  1 ) ) ) ) ) )
759, 15, 21, 27, 48, 74fzind2 10923 . 2  |-  ( N  e.  ( M ... N )  ->  ( ph  ->  (  seq  M
(  .+  ,  H
) `  N )  =  ( (  seq 
M (  .+  ,  F ) `  N
) Q (  seq 
M (  .+  ,  G ) `  N
) ) ) )
763, 75mpcom 32 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  H ) `
 N )  =  ( (  seq  M
(  .+  ,  F
) `  N ) Q (  seq  M
(  .+  ,  G
) `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782  ..^cfzo 10870    seq cseq 11046
This theorem is referenced by:  seqcaopr2  11082  gsumzaddlem  15203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047
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