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Theorem seqid2 11184
Description: The last few terms of a sequence that ends with all zeroes (or whatever the identity  Z is for operation  .+) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
seqid2.1  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
seqid2.2  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
seqid2.3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
seqid2.4  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  e.  S )
seqid2.5  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
Assertion
Ref Expression
seqid2  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) )
Distinct variable groups:    x, F    x, K    x, M    x, N    ph, x    x, S    x, 
.+    x, Z

Proof of Theorem seqid2
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 seqid2.3 . . 3  |-  ( ph  ->  N  e.  ( ZZ>= `  K ) )
2 eluzfz2 10896 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 15 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2418 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5608 . . . . . . 7  |-  ( x  =  K  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
65eqeq2d 2369 . . . . . 6  |-  ( x  =  K  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 311 . . . . 5  |-  ( x  =  K  ->  (
( x  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )  <-> 
( K  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) ) )
87imbi2d 307 . . . 4  |-  ( x  =  K  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( K  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) ) ) )
9 eleq1 2418 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5608 . . . . . . 7  |-  ( x  =  n  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  n
) )
1110eqeq2d 2369 . . . . . 6  |-  ( x  =  n  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 311 . . . . 5  |-  ( x  =  n  ->  (
( x  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )  <-> 
( n  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) ) )
1312imbi2d 307 . . . 4  |-  ( x  =  n  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( n  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) ) ) )
14 eleq1 2418 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5608 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2369 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 311 . . . . 5  |-  ( x  =  ( n  + 
1 )  ->  (
( x  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )  <-> 
( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) )
1817imbi2d 307 . . . 4  |-  ( x  =  ( n  + 
1 )  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) ) )
19 eleq1 2418 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5608 . . . . . . 7  |-  ( x  =  N  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
2120eqeq2d 2369 . . . . . 6  |-  ( x  =  N  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 311 . . . . 5  |-  ( x  =  N  ->  (
( x  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 x ) )  <-> 
( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) ) )
2322imbi2d 307 . . . 4  |-  ( x  =  N  ->  (
( ph  ->  ( x  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  x
) ) )  <->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) ) ) )
24 eqidd 2359 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
2524a1ii 24 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) ) )
26 peano2fzr 10900 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 598 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 69 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  n
) ) ) )
30 oveq1 5952 . . . . . . . . . 10  |-  ( (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  n
)  ->  ( (  seq  M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq  M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) )
31 eluzp1p1 10345 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3231ad2antrl 708 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
33 elfzuz3 10887 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3433ad2antll 709 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
35 elfzuzb 10884 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
3632, 34, 35sylanbrc 645 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ( K  + 
1 ) ... N
) )
37 seqid2.5 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( ( K  + 
1 ) ... N
) )  ->  ( F `  x )  =  Z )
3837ralrimiva 2702 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3938adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
40 fveq2 5608 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
4140eqeq1d 2366 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
4241rspcv 2956 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) ( F `  x
)  =  Z  -> 
( F `  (
n  +  1 ) )  =  Z ) )
4336, 39, 42sylc 56 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4443oveq2d 5961 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq  M
(  .+  ,  F
) `  K )  .+  Z ) )
45 seqid2.4 . . . . . . . . . . . . . 14  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  e.  S )
46 seqid2.1 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  S )  ->  (
x  .+  Z )  =  x )
4746ralrimiva 2702 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 oveq1 5952 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( x  .+  Z
)  =  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )
)
49 id 19 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  x  =  (  seq 
M (  .+  ,  F ) `  K
) )
5048, 49eqeq12d 2372 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) ) )
5150rspcv 2956 . . . . . . . . . . . . . 14  |-  ( (  seq  M (  .+  ,  F ) `  K
)  e.  S  -> 
( A. x  e.  S  ( x  .+  Z )  =  x  ->  ( (  seq 
M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) )
5245, 47, 51sylc 56 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) )
5352adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq  M ( 
.+  ,  F ) `
 K ) )
5444, 53eqtr2d 2391 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( (  seq 
M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
55 simprl 732 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  K )
)
56 seqid2.2 . . . . . . . . . . . . . 14  |-  ( ph  ->  K  e.  ( ZZ>= `  M ) )
5756adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 10336 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5955, 57, 58syl2anc 642 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
60 seqp1 11153 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  .+  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6159, 60syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq  M (  .+  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6254, 61eqeq12d 2372 . . . . . . . . . 10  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) )  <->  ( (  seq 
M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) )  =  ( (  seq  M
(  .+  ,  F
) `  n )  .+  ( F `  (
n  +  1 ) ) ) ) )
6330, 62syl5ibr 212 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  n
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
6463expr 598 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  n
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) )
6564a2d 23 . . . . . . 7  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) )
6629, 65syld 40 . . . . . 6  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  n
) )  ->  (
( n  +  1 )  e.  ( K ... N )  -> 
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  (
n  +  1 ) ) ) ) )
6766expcom 424 . . . . 5  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( ( n  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) )  ->  ( ( n  +  1 )  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) ) )
6867a2d 23 . . . 4  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( ( ph  ->  ( n  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) )  ->  ( ph  ->  ( ( n  + 
1 )  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) ) ) )
698, 13, 18, 23, 25, 68uzind4 10368 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) ) )
701, 69mpcom 32 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) )
713, 70mpd 14 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   ` cfv 5337  (class class class)co 5945   1c1 8828    + caddc 8830   ZZcz 10116   ZZ>=cuz 10322   ...cfz 10874    seq cseq 11138
This theorem is referenced by:  seqcoll  11497  seqcoll2  11498  fsumcvg  12282  ovolicc1  18979  lgsdilem2  20682  fprodcvg  24557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-seq 11139
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