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Theorem iseqfveq2 9544
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
iseqfveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
iseqfveq2.2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  ( G `
 K ) )
iseqfveq2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
iseqfveq2.g  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
iseqfveq2.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
iseqfveq2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
iseqfveq2.4  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
iseqfveq2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  (  seq K (  .+  ,  G ,  S ) `  N ) )
Distinct variable groups:    x, k, y, F    k, G, x, y    k, K, x, y    k, N, x, y    ph, k, x, y   
k, M, x, y    .+ , k, x, y    S, k, x, y

Proof of Theorem iseqfveq2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iseqfveq2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 9127 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2142 . . . . . 6  |-  ( z  =  K  ->  (
z  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5209 . . . . . . 7  |-  ( z  =  K  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq M (  .+  ,  F ,  S ) `
 K ) )
6 fveq2 5209 . . . . . . 7  |-  ( z  =  K  ->  (  seq K (  .+  ,  G ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 K ) )
75, 6eqeq12d 2096 . . . . . 6  |-  ( z  =  K  ->  (
(  seq M (  .+  ,  F ,  S ) `
 z )  =  (  seq K ( 
.+  ,  G ,  S ) `  z
)  <->  (  seq M
(  .+  ,  F ,  S ) `  K
)  =  (  seq K (  .+  ,  G ,  S ) `  K ) ) )
84, 7imbi12d 232 . . . . 5  |-  ( z  =  K  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  z
)  =  (  seq K (  .+  ,  G ,  S ) `  z ) )  <->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq K (  .+  ,  G ,  S ) `
 K ) ) ) )
98imbi2d 228 . . . 4  |-  ( z  =  K  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 z ) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 K )  =  (  seq K ( 
.+  ,  G ,  S ) `  K
) ) ) ) )
10 eleq1 2142 . . . . . 6  |-  ( z  =  w  ->  (
z  e.  ( K ... N )  <->  w  e.  ( K ... N ) ) )
11 fveq2 5209 . . . . . . 7  |-  ( z  =  w  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq M (  .+  ,  F ,  S ) `
 w ) )
12 fveq2 5209 . . . . . . 7  |-  ( z  =  w  ->  (  seq K (  .+  ,  G ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 w ) )
1311, 12eqeq12d 2096 . . . . . 6  |-  ( z  =  w  ->  (
(  seq M (  .+  ,  F ,  S ) `
 z )  =  (  seq K ( 
.+  ,  G ,  S ) `  z
)  <->  (  seq M
(  .+  ,  F ,  S ) `  w
)  =  (  seq K (  .+  ,  G ,  S ) `  w ) ) )
1410, 13imbi12d 232 . . . . 5  |-  ( z  =  w  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  z
)  =  (  seq K (  .+  ,  G ,  S ) `  z ) )  <->  ( w  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq K (  .+  ,  G ,  S ) `
 w ) ) ) )
1514imbi2d 228 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 z ) ) )  <->  ( ph  ->  ( w  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
) ) ) ) )
16 eleq1 2142 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
z  e.  ( K ... N )  <->  ( w  +  1 )  e.  ( K ... N
) ) )
17 fveq2 5209 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq M (  .+  ,  F ,  S ) `
 ( w  + 
1 ) ) )
18 fveq2 5209 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq K (  .+  ,  G ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 ( w  + 
1 ) ) )
1917, 18eqeq12d 2096 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 z )  =  (  seq K ( 
.+  ,  G ,  S ) `  z
)  <->  (  seq M
(  .+  ,  F ,  S ) `  (
w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `  ( w  +  1 ) ) ) )
2016, 19imbi12d 232 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  z
)  =  (  seq K (  .+  ,  G ,  S ) `  z ) )  <->  ( (
w  +  1 )  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  ( w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `
 ( w  + 
1 ) ) ) ) )
2120imbi2d 228 . . . 4  |-  ( z  =  ( w  + 
1 )  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 z ) ) )  <->  ( ph  ->  ( ( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( w  + 
1 ) )  =  (  seq K ( 
.+  ,  G ,  S ) `  (
w  +  1 ) ) ) ) ) )
22 eleq1 2142 . . . . . 6  |-  ( z  =  N  ->  (
z  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
23 fveq2 5209 . . . . . . 7  |-  ( z  =  N  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq M (  .+  ,  F ,  S ) `
 N ) )
24 fveq2 5209 . . . . . . 7  |-  ( z  =  N  ->  (  seq K (  .+  ,  G ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 N ) )
2523, 24eqeq12d 2096 . . . . . 6  |-  ( z  =  N  ->  (
(  seq M (  .+  ,  F ,  S ) `
 z )  =  (  seq K ( 
.+  ,  G ,  S ) `  z
)  <->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  (  seq K (  .+  ,  G ,  S ) `  N ) ) )
2622, 25imbi12d 232 . . . . 5  |-  ( z  =  N  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  z
)  =  (  seq K (  .+  ,  G ,  S ) `  z ) )  <->  ( N  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  N )  =  (  seq K (  .+  ,  G ,  S ) `
 N ) ) ) )
2726imbi2d 228 . . . 4  |-  ( z  =  N  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ,  S ) `  z )  =  (  seq K (  .+  ,  G ,  S ) `
 z ) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 N )  =  (  seq K ( 
.+  ,  G ,  S ) `  N
) ) ) ) )
28 iseqfveq2.2 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  ( G `
 K ) )
29 iseqfveq2.1 . . . . . . . 8  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
30 eluzelz 8709 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
3129, 30syl 14 . . . . . . 7  |-  ( ph  ->  K  e.  ZZ )
32 iseqfveq2.g . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
33 iseqfveq2.pl . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3431, 32, 33iseq1 9533 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  G ,  S ) `  K
)  =  ( G `
 K ) )
3528, 34eqtr4d 2117 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  K
)  =  (  seq K (  .+  ,  G ,  S ) `  K ) )
3635a1i13 24 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  K )  =  (  seq K (  .+  ,  G ,  S ) `
 K ) ) ) )
37 peano2fzr 9132 . . . . . . . . . 10  |-  ( ( w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) )  ->  w  e.  ( K ... N ) )
3837adantl 271 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( K ... N ) )
3938expr 367 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  +  1 )  e.  ( K ... N )  ->  w  e.  ( K ... N
) ) )
4039imim1d 74 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
) )  ->  (
( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
) ) ) )
41 oveq1 5550 . . . . . . . . . 10  |-  ( (  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
)  ->  ( (  seq M (  .+  ,  F ,  S ) `  w )  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K (  .+  ,  G ,  S ) `
 w )  .+  ( F `  ( w  +  1 ) ) ) )
42 simprl 498 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  K )
)
4329adantr 270 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
44 uztrn 8716 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  w  e.  ( ZZ>= `  M )
)
4542, 43, 44syl2anc 403 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  M )
)
46 iseqfveq2.f . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4746adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
4833adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4945, 47, 48iseqp1 9538 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ,  S ) `  ( w  +  1 ) )  =  ( (  seq M ( 
.+  ,  F ,  S ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
5032adantlr 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  K ) )  ->  ( G `  x )  e.  S
)
5142, 50, 48iseqp1 9538 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ,  S ) `  ( w  +  1 ) )  =  ( (  seq K ( 
.+  ,  G ,  S ) `  w
)  .+  ( G `  ( w  +  1 ) ) ) )
52 fveq2 5209 . . . . . . . . . . . . . . 15  |-  ( k  =  ( w  + 
1 )  ->  ( F `  k )  =  ( F `  ( w  +  1
) ) )
53 fveq2 5209 . . . . . . . . . . . . . . 15  |-  ( k  =  ( w  + 
1 )  ->  ( G `  k )  =  ( G `  ( w  +  1
) ) )
5452, 53eqeq12d 2096 . . . . . . . . . . . . . 14  |-  ( k  =  ( w  + 
1 )  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) ) )
55 iseqfveq2.4 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
5655ralrimiva 2435 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. k  e.  ( ( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k ) )
5756adantr 270 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  A. k  e.  ( ( K  + 
1 ) ... N
) ( F `  k )  =  ( G `  k ) )
58 eluzp1p1 8725 . . . . . . . . . . . . . . . 16  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
5958ad2antrl 474 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
60 elfzuz3 9118 . . . . . . . . . . . . . . . 16  |-  ( ( w  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
6160ad2antll 475 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
62 elfzuzb 9115 . . . . . . . . . . . . . . 15  |-  ( ( w  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( w  +  1 ) ) ) )
6359, 61, 62sylanbrc 408 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
6454, 57, 63rspcdva 2708 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) )
6564oveq2d 5559 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq K (  .+  ,  G ,  S ) `  w )  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K (  .+  ,  G ,  S ) `
 w )  .+  ( G `  ( w  +  1 ) ) ) )
6651, 65eqtr4d 2117 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ,  S ) `  ( w  +  1 ) )  =  ( (  seq K ( 
.+  ,  G ,  S ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
6749, 66eqeq12d 2096 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  ( w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `
 ( w  + 
1 ) )  <->  ( (  seq M (  .+  ,  F ,  S ) `  w )  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K (  .+  ,  G ,  S ) `
 w )  .+  ( F `  ( w  +  1 ) ) ) ) )
6841, 67syl5ibr 154 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq K (  .+  ,  G ,  S ) `
 w )  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( w  + 
1 ) )  =  (  seq K ( 
.+  ,  G ,  S ) `  (
w  +  1 ) ) ) )
6968expr 367 . . . . . . . 8  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  +  1 )  e.  ( K ... N )  ->  (
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `
 ( w  + 
1 ) ) ) ) )
7069a2d 26 . . . . . . 7  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
) )  ->  (
( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( w  + 
1 ) )  =  (  seq K ( 
.+  ,  G ,  S ) `  (
w  +  1 ) ) ) ) )
7140, 70syld 44 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 w )  =  (  seq K ( 
.+  ,  G ,  S ) `  w
) )  ->  (
( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ,  S ) `
 ( w  + 
1 ) )  =  (  seq K ( 
.+  ,  G ,  S ) `  (
w  +  1 ) ) ) ) )
7271expcom 114 . . . . 5  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( ( w  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq K (  .+  ,  G ,  S ) `
 w ) )  ->  ( ( w  +  1 )  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  ( w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `
 ( w  + 
1 ) ) ) ) ) )
7372a2d 26 . . . 4  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( w  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ,  S ) `  w )  =  (  seq K (  .+  ,  G ,  S ) `
 w ) ) )  ->  ( ph  ->  ( ( w  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  (
w  +  1 ) )  =  (  seq K (  .+  ,  G ,  S ) `  ( w  +  1 ) ) ) ) ) )
749, 15, 21, 27, 36, 73uzind4 8757 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  (  seq K (  .+  ,  G ,  S ) `  N ) ) ) )
751, 74mpcom 36 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F ,  S ) `  N
)  =  (  seq K (  .+  ,  G ,  S ) `  N ) ) )
763, 75mpd 13 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ,  S ) `  N
)  =  (  seq K (  .+  ,  G ,  S ) `  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   A.wral 2349   ` cfv 4932  (class class class)co 5543   1c1 7044    + caddc 7046   ZZcz 8432   ZZ>=cuz 8700   ...cfz 9105    seqcseq 9521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-inn 8107  df-n0 8356  df-z 8433  df-uz 8701  df-fz 9106  df-iseq 9522
This theorem is referenced by:  iseqfeq2  9545  iseqfveq  9546
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