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Theorem seq3fveq2 10536
Description: Equality of sequences. (Contributed by Jim Kingdon, 3-Jun-2020.)
Hypotheses
Ref Expression
seq3fveq2.1  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seq3fveq2.2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
seq3fveq2.f  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
seq3fveq2.g  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
seq3fveq2.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
seq3fveq2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
seq3fveq2.4  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
Assertion
Ref Expression
seq3fveq2  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  G ) `
 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 seq3fveq2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 seq3fveq2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 10088 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2256 . . . . . 6  |-  ( z  =  K  ->  (
z  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5546 . . . . . . 7  |-  ( z  =  K  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  K
) )
6 fveq2 5546 . . . . . . 7  |-  ( z  =  K  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  K
) )
75, 6eqeq12d 2208 . . . . . 6  |-  ( z  =  K  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) )
84, 7imbi12d 234 . . . . 5  |-  ( z  =  K  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )  <-> 
( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) ) )
98imbi2d 230 . . . 4  |-  ( z  =  K  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) ) ) )
10 eleq1 2256 . . . . . 6  |-  ( z  =  w  ->  (
z  e.  ( K ... N )  <->  w  e.  ( K ... N ) ) )
11 fveq2 5546 . . . . . . 7  |-  ( z  =  w  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  w
) )
12 fveq2 5546 . . . . . . 7  |-  ( z  =  w  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  w
) )
1311, 12eqeq12d 2208 . . . . . 6  |-  ( z  =  w  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  w )  =  (  seq K ( 
.+  ,  G ) `
 w ) ) )
1410, 13imbi12d 234 . . . . 5  |-  ( z  =  w  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )  <-> 
( w  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  w )  =  (  seq K ( 
.+  ,  G ) `
 w ) ) ) )
1514imbi2d 230 . . . 4  |-  ( z  =  w  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  ( w  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  w )  =  (  seq K ( 
.+  ,  G ) `
 w ) ) ) ) )
16 eleq1 2256 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
z  e.  ( K ... N )  <->  ( w  +  1 )  e.  ( K ... N
) ) )
17 fveq2 5546 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  (
w  +  1 ) ) )
18 fveq2 5546 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) ) )
1917, 18eqeq12d 2208 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  ( w  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( w  +  1 ) ) ) )
2016, 19imbi12d 234 . . . . 5  |-  ( z  =  ( w  + 
1 )  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )  <-> 
( ( w  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  ( w  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( w  +  1 ) ) ) ) )
2120imbi2d 230 . . . 4  |-  ( z  =  ( w  + 
1 )  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  ( ( w  + 
1 )  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  ( w  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( w  +  1 ) ) ) ) ) )
22 eleq1 2256 . . . . . 6  |-  ( z  =  N  ->  (
z  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
23 fveq2 5546 . . . . . . 7  |-  ( z  =  N  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  N
) )
24 fveq2 5546 . . . . . . 7  |-  ( z  =  N  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  N
) )
2523, 24eqeq12d 2208 . . . . . 6  |-  ( z  =  N  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
2622, 25imbi12d 234 . . . . 5  |-  ( z  =  N  ->  (
( z  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  z )  =  (  seq K ( 
.+  ,  G ) `
 z ) )  <-> 
( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) )
2726imbi2d 230 . . . 4  |-  ( z  =  N  ->  (
( ph  ->  ( z  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) ) )
28 seq3fveq2.2 . . . . . 6  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  ( G `  K
) )
29 seq3fveq2.1 . . . . . . . 8  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
30 eluzelz 9591 . . . . . . . 8  |-  ( K  e.  ( ZZ>= `  M
)  ->  K  e.  ZZ )
3129, 30syl 14 . . . . . . 7  |-  ( ph  ->  K  e.  ZZ )
32 seq3fveq2.g . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ZZ>= `  K )
)  ->  ( G `  x )  e.  S
)
33 seq3fveq2.pl . . . . . . 7  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
3431, 32, 33seq3-1 10523 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  G ) `
 K )  =  ( G `  K
) )
3528, 34eqtr4d 2229 . . . . 5  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 K )  =  (  seq K ( 
.+  ,  G ) `
 K ) )
3635a1i13 24 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  K
)  =  (  seq K (  .+  ,  G ) `  K
) ) ) )
37 peano2fzr 10093 . . . . . . . 8  |-  ( ( w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) )  ->  w  e.  ( K ... N ) )
3837adantl 277 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( K ... N ) )
3938expr 375 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  +  1 )  e.  ( K ... N )  ->  w  e.  ( K ... N
) ) )
4039imim1d 75 . . . . 5  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  w
)  =  (  seq K (  .+  ,  G ) `  w
) )  ->  (
( w  +  1 )  e.  ( K ... N )  -> 
(  seq M (  .+  ,  F ) `  w
)  =  (  seq K (  .+  ,  G ) `  w
) ) ) )
41 oveq1 5917 . . . . . 6  |-  ( (  seq M (  .+  ,  F ) `  w
)  =  (  seq K (  .+  ,  G ) `  w
)  ->  ( (  seq M (  .+  ,  F ) `  w
)  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  w )  .+  ( F `  (
w  +  1 ) ) ) )
42 simprl 529 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  K )
)
4329adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
44 uztrn 9599 . . . . . . . . 9  |-  ( ( w  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  w  e.  ( ZZ>= `  M )
)
4542, 43, 44syl2anc 411 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  M )
)
46 seq3fveq2.f . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ZZ>= `  M )
)  ->  ( F `  x )  e.  S
)
4746adantlr 477 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
4833adantlr 477 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4945, 47, 48seq3p1 10526 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
5032adantlr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  K ) )  ->  ( G `  x )  e.  S
)
5142, 50, 48seq3p1 10526 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ) `  (
w  +  1 ) )  =  ( (  seq K (  .+  ,  G ) `  w
)  .+  ( G `  ( w  +  1 ) ) ) )
52 fveq2 5546 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( F `  k )  =  ( F `  ( w  +  1
) ) )
53 fveq2 5546 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( G `  k )  =  ( G `  ( w  +  1
) ) )
5452, 53eqeq12d 2208 . . . . . . . . . 10  |-  ( k  =  ( w  + 
1 )  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) ) )
55 seq3fveq2.4 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  k )  =  ( G `  k ) )
5655ralrimiva 2567 . . . . . . . . . . 11  |-  ( ph  ->  A. k  e.  ( ( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k ) )
5756adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  A. k  e.  ( ( K  + 
1 ) ... N
) ( F `  k )  =  ( G `  k ) )
58 eluzp1p1 9608 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
5958ad2antrl 490 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
60 elfzuz3 10078 . . . . . . . . . . . 12  |-  ( ( w  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
6160ad2antll 491 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
62 elfzuzb 10075 . . . . . . . . . . 11  |-  ( ( w  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( w  +  1 ) ) ) )
6359, 61, 62sylanbrc 417 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
6454, 57, 63rspcdva 2869 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) )
6564oveq2d 5926 . . . . . . . 8  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq K (  .+  ,  G ) `  w
)  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  w )  .+  ( G `  (
w  +  1 ) ) ) )
6651, 65eqtr4d 2229 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq K (  .+  ,  G ) `  (
w  +  1 ) )  =  ( (  seq K (  .+  ,  G ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
6749, 66eqeq12d 2208 . . . . . 6  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) )  <->  ( (  seq M (  .+  ,  F ) `  w
)  .+  ( F `  ( w  +  1 ) ) )  =  ( (  seq K
(  .+  ,  G
) `  w )  .+  ( F `  (
w  +  1 ) ) ) ) )
6841, 67imbitrrid 156 . . . . 5  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq M (  .+  ,  F ) `  w
)  =  (  seq K (  .+  ,  G ) `  w
)  ->  (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) ) ) )
6940, 68animpimp2impd 559 . . . 4  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( w  e.  ( K ... N
)  ->  (  seq M (  .+  ,  F ) `  w
)  =  (  seq K (  .+  ,  G ) `  w
) ) )  -> 
( ph  ->  ( ( w  +  1 )  e.  ( K ... N )  ->  (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) ) ) ) ) )
709, 15, 21, 27, 36, 69uzind4 9643 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) ) )
711, 70mpcom 36 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
723, 71mpd 13 1  |-  ( ph  ->  (  seq M ( 
.+  ,  F ) `
 N )  =  (  seq K ( 
.+  ,  G ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   A.wral 2472   ` cfv 5246  (class class class)co 5910   1c1 7863    + caddc 7865   ZZcz 9307   ZZ>=cuz 9582   ...cfz 10064    seqcseq 10508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-seqfrec 10509
This theorem is referenced by:  seq3feq2  10537  seq3fveq  10540  gsumsplit1r  12968
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