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Theorem seq3fveq2 9892
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 9446 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2150 . . . . . 6  |-  ( z  =  K  ->  (
z  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5305 . . . . . . 7  |-  ( z  =  K  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  K
) )
6 fveq2 5305 . . . . . . 7  |-  ( z  =  K  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  K
) )
75, 6eqeq12d 2102 . . . . . 6  |-  ( z  =  K  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) )
84, 7imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( z  =  w  ->  (
z  e.  ( K ... N )  <->  w  e.  ( K ... N ) ) )
11 fveq2 5305 . . . . . . 7  |-  ( z  =  w  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  w
) )
12 fveq2 5305 . . . . . . 7  |-  ( z  =  w  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  w
) )
1311, 12eqeq12d 2102 . . . . . 6  |-  ( z  =  w  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  w )  =  (  seq K ( 
.+  ,  G ) `
 w ) ) )
1410, 13imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
z  e.  ( K ... N )  <->  ( w  +  1 )  e.  ( K ... N
) ) )
17 fveq2 5305 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  (
w  +  1 ) ) )
18 fveq2 5305 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) ) )
1917, 18eqeq12d 2102 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  ( w  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( w  +  1 ) ) ) )
2016, 19imbi12d 232 . . . . 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 228 . . . 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 2150 . . . . . 6  |-  ( z  =  N  ->  (
z  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
23 fveq2 5305 . . . . . . 7  |-  ( z  =  N  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  N
) )
24 fveq2 5305 . . . . . . 7  |-  ( z  =  N  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  N
) )
2523, 24eqeq12d 2102 . . . . . 6  |-  ( z  =  N  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
2622, 25imbi12d 232 . . . . 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 228 . . . 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 9028 . . . . . . . 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 9877 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  G ) `
 K )  =  ( G `  K
) )
3528, 34eqtr4d 2123 . . . . 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 9451 . . . . . . . 8  |-  ( ( w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) )  ->  w  e.  ( K ... N ) )
3837adantl 271 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( K ... N ) )
3938expr 367 . . . . . 6  |-  ( (
ph  /\  w  e.  ( ZZ>= `  K )
)  ->  ( (
w  +  1 )  e.  ( K ... N )  ->  w  e.  ( K ... N
) ) )
4039imim1d 74 . . . . 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 5659 . . . . . 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 498 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  K )
)
4329adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
44 uztrn 9035 . . . . . . . . 9  |-  ( ( w  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  w  e.  ( ZZ>= `  M )
)
4542, 43, 44syl2anc 403 . . . . . . . 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 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
4833adantlr 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4945, 47, 48seq3p1 9884 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
5032adantlr 461 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  K ) )  ->  ( G `  x )  e.  S
)
5142, 50, 48seq3p1 9884 . . . . . . . 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 5305 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( F `  k )  =  ( F `  ( w  +  1
) ) )
53 fveq2 5305 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( G `  k )  =  ( G `  ( w  +  1
) ) )
5452, 53eqeq12d 2102 . . . . . . . . . 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 2446 . . . . . . . . . . 11  |-  ( ph  ->  A. k  e.  ( ( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k ) )
5756adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  A. k  e.  ( ( K  + 
1 ) ... N
) ( F `  k )  =  ( G `  k ) )
58 eluzp1p1 9044 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
5958ad2antrl 474 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
60 elfzuz3 9437 . . . . . . . . . . . 12  |-  ( ( w  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
6160ad2antll 475 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
62 elfzuzb 9434 . . . . . . . . . . 11  |-  ( ( w  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( w  +  1 ) ) ) )
6359, 61, 62sylanbrc 408 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
6454, 57, 63rspcdva 2727 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) )
6564oveq2d 5668 . . . . . . . 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 2123 . . . . . . 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 2102 . . . . . 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, 67syl5ibr 154 . . . . 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 526 . . . 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 9076 . . 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 102    = wceq 1289    e. wcel 1438   A.wral 2359   ` cfv 5015  (class class class)co 5652   1c1 7351    + caddc 7353   ZZcz 8750   ZZ>=cuz 9019   ...cfz 9424    seqcseq 9852
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425  df-iseq 9853  df-seq3 9854
This theorem is referenced by:  seq3feq2  9893  seq3fveq  9895
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