ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seq3fveq2 Unicode version

Theorem seq3fveq2 10197
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 9767 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 14 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2180 . . . . . 6  |-  ( z  =  K  ->  (
z  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5389 . . . . . . 7  |-  ( z  =  K  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  K
) )
6 fveq2 5389 . . . . . . 7  |-  ( z  =  K  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  K
) )
75, 6eqeq12d 2132 . . . . . 6  |-  ( z  =  K  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  K )  =  (  seq K ( 
.+  ,  G ) `
 K ) ) )
84, 7imbi12d 233 . . . . 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 229 . . . 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 2180 . . . . . 6  |-  ( z  =  w  ->  (
z  e.  ( K ... N )  <->  w  e.  ( K ... N ) ) )
11 fveq2 5389 . . . . . . 7  |-  ( z  =  w  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  w
) )
12 fveq2 5389 . . . . . . 7  |-  ( z  =  w  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  w
) )
1311, 12eqeq12d 2132 . . . . . 6  |-  ( z  =  w  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  w )  =  (  seq K ( 
.+  ,  G ) `
 w ) ) )
1410, 13imbi12d 233 . . . . 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 229 . . . 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 2180 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
z  e.  ( K ... N )  <->  ( w  +  1 )  e.  ( K ... N
) ) )
17 fveq2 5389 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  (
w  +  1 ) ) )
18 fveq2 5389 . . . . . . 7  |-  ( z  =  ( w  + 
1 )  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  (
w  +  1 ) ) )
1917, 18eqeq12d 2132 . . . . . 6  |-  ( z  =  ( w  + 
1 )  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  ( w  +  1 ) )  =  (  seq K
(  .+  ,  G
) `  ( w  +  1 ) ) ) )
2016, 19imbi12d 233 . . . . 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 229 . . . 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 2180 . . . . . 6  |-  ( z  =  N  ->  (
z  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
23 fveq2 5389 . . . . . . 7  |-  ( z  =  N  ->  (  seq M (  .+  ,  F ) `  z
)  =  (  seq M (  .+  ,  F ) `  N
) )
24 fveq2 5389 . . . . . . 7  |-  ( z  =  N  ->  (  seq K (  .+  ,  G ) `  z
)  =  (  seq K (  .+  ,  G ) `  N
) )
2523, 24eqeq12d 2132 . . . . . 6  |-  ( z  =  N  ->  (
(  seq M (  .+  ,  F ) `  z
)  =  (  seq K (  .+  ,  G ) `  z
)  <->  (  seq M
(  .+  ,  F
) `  N )  =  (  seq K ( 
.+  ,  G ) `
 N ) ) )
2622, 25imbi12d 233 . . . . 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 229 . . . 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 9291 . . . . . . . 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 10188 . . . . . 6  |-  ( ph  ->  (  seq K ( 
.+  ,  G ) `
 K )  =  ( G `  K
) )
3528, 34eqtr4d 2153 . . . . 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 9772 . . . . . . . 8  |-  ( ( w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) )  ->  w  e.  ( K ... N ) )
3837adantl 275 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( K ... N ) )
3938expr 372 . . . . . 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 5749 . . . . . 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 505 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  w  e.  ( ZZ>= `  K )
)
4329adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
44 uztrn 9298 . . . . . . . . 9  |-  ( ( w  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  w  e.  ( ZZ>= `  M )
)
4542, 43, 44syl2anc 408 . . . . . . . 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 468 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  M ) )  ->  ( F `  x )  e.  S
)
4833adantlr 468 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  .+  y
)  e.  S )
4945, 47, 48seq3p1 10190 . . . . . . 7  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  (  seq M (  .+  ,  F ) `  (
w  +  1 ) )  =  ( (  seq M (  .+  ,  F ) `  w
)  .+  ( F `  ( w  +  1 ) ) ) )
5032adantlr 468 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  e.  ( ZZ>= `  K )  /\  (
w  +  1 )  e.  ( K ... N ) ) )  /\  x  e.  (
ZZ>= `  K ) )  ->  ( G `  x )  e.  S
)
5142, 50, 48seq3p1 10190 . . . . . . . 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 5389 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( F `  k )  =  ( F `  ( w  +  1
) ) )
53 fveq2 5389 . . . . . . . . . . 11  |-  ( k  =  ( w  + 
1 )  ->  ( G `  k )  =  ( G `  ( w  +  1
) ) )
5452, 53eqeq12d 2132 . . . . . . . . . 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 2482 . . . . . . . . . . 11  |-  ( ph  ->  A. k  e.  ( ( K  +  1 ) ... N ) ( F `  k
)  =  ( G `
 k ) )
5756adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  A. k  e.  ( ( K  + 
1 ) ... N
) ( F `  k )  =  ( G `  k ) )
58 eluzp1p1 9307 . . . . . . . . . . . 12  |-  ( w  e.  ( ZZ>= `  K
)  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
5958ad2antrl 481 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
60 elfzuz3 9758 . . . . . . . . . . . 12  |-  ( ( w  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
6160ad2antll 482 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( w  +  1 ) ) )
62 elfzuzb 9755 . . . . . . . . . . 11  |-  ( ( w  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
w  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( w  +  1 ) ) ) )
6359, 61, 62sylanbrc 413 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( w  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
6454, 57, 63rspcdva 2768 . . . . . . . . 9  |-  ( (
ph  /\  ( w  e.  ( ZZ>= `  K )  /\  ( w  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( w  +  1 ) )  =  ( G `  ( w  +  1 ) ) )
6564oveq2d 5758 . . . . . . . 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 2153 . . . . . . 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 2132 . . . . . 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 155 . . . . 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 533 . . . 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 9339 . . 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 103    = wceq 1316    e. wcel 1465   A.wral 2393   ` cfv 5093  (class class class)co 5742   1c1 7589    + caddc 7591   ZZcz 9012   ZZ>=cuz 9282   ...cfz 9745    seqcseq 10173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8685  df-n0 8936  df-z 9013  df-uz 9283  df-fz 9746  df-seqfrec 10174
This theorem is referenced by:  seq3feq2  10198  seq3fveq  10199
  Copyright terms: Public domain W3C validator