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Theorem seqid2 11374
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 11070 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2498 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5731 . . . . . . 7  |-  ( x  =  K  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
65eqeq2d 2449 . . . . . 6  |-  ( x  =  K  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 313 . . . . 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 309 . . . 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 2498 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5731 . . . . . . 7  |-  ( x  =  n  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  n
) )
1110eqeq2d 2449 . . . . . 6  |-  ( x  =  n  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 313 . . . . 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 309 . . . 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 2498 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5731 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2449 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 313 . . . . 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 309 . . . 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 2498 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5731 . . . . . . 7  |-  ( x  =  N  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
2120eqeq2d 2449 . . . . . 6  |-  ( x  =  N  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 313 . . . . 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 309 . . . 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 2439 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
2524a1ii 26 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) ) )
26 peano2fzr 11074 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 600 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 72 . . . . . . 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 6091 . . . . . . . . . 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 10516 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3231ad2antrl 710 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
33 elfzuz3 11061 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3433ad2antll 711 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
35 elfzuzb 11058 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
3632, 34, 35sylanbrc 647 . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3938adantr 453 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
40 fveq2 5731 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
4140eqeq1d 2446 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
4241rspcv 3050 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) ( F `  x
)  =  Z  -> 
( F `  (
n  +  1 ) )  =  Z ) )
4336, 39, 42sylc 59 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4443oveq2d 6100 . . . . . . . . . . . 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 2791 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 oveq1 6091 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( x  .+  Z
)  =  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )
)
49 id 21 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  x  =  (  seq 
M (  .+  ,  F ) `  K
) )
5048, 49eqeq12d 2452 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) ) )
5150rspcv 3050 . . . . . . . . . . . . . 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 59 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) )
5352adantr 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq  M ( 
.+  ,  F ) `
 K ) )
5444, 53eqtr2d 2471 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( (  seq 
M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
55 simprl 734 . . . . . . . . . . . . 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 453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 10507 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5955, 57, 58syl2anc 644 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
60 seqp1 11343 . . . . . . . . . . . 12  |-  ( n  e.  ( ZZ>= `  M
)  ->  (  seq  M (  .+  ,  F
) `  ( n  +  1 ) )  =  ( (  seq 
M (  .+  ,  F ) `  n
)  .+  ( F `  ( n  +  1 ) ) ) )
6159, 60syl 16 . . . . . . . . . . 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 2452 . . . . . . . . . 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 214 . . . . . . . . 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 600 . . . . . . . 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 25 . . . . . . 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 43 . . . . . 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 426 . . . . 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 25 . . . 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 10539 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) ) )
701, 69mpcom 35 . 2  |-  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) )
713, 70mpd 15 1  |-  ( ph  ->  (  seq  M ( 
.+  ,  F ) `
 K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   ` cfv 5457  (class class class)co 6084   1c1 8996    + caddc 8998   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328
This theorem is referenced by:  seqcoll  11717  seqcoll2  11718  fsumcvg  12511  ovolicc1  19417  lgsdilem2  21120  fprodcvg  25261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-seq 11329
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