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Theorem seqid2 11332
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 11029 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2472 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5695 . . . . . . 7  |-  ( x  =  K  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
65eqeq2d 2423 . . . . . 6  |-  ( x  =  K  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) )
74, 6imbi12d 312 . . . . 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 308 . . . 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 2472 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5695 . . . . . . 7  |-  ( x  =  n  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  n
) )
1110eqeq2d 2423 . . . . . 6  |-  ( x  =  n  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 n ) ) )
129, 11imbi12d 312 . . . . 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 308 . . . 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 2472 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5695 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2423 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 ( n  + 
1 ) ) ) )
1714, 16imbi12d 312 . . . . 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 308 . . . 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 2472 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5695 . . . . . . 7  |-  ( x  =  N  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
2120eqeq2d 2423 . . . . . 6  |-  ( x  =  N  ->  (
(  seq  M (  .+  ,  F ) `  K )  =  (  seq  M (  .+  ,  F ) `  x
)  <->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) )
2219, 21imbi12d 312 . . . . 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 308 . . . 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 2413 . . . . 5  |-  ( K  e.  ( K ... N )  ->  (  seq  M (  .+  ,  F ) `  K
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
2524a1ii 25 . . . 4  |-  ( K  e.  ZZ  ->  ( ph  ->  ( K  e.  ( K ... N
)  ->  (  seq  M (  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 K ) ) ) )
26 peano2fzr 11033 . . . . . . . . . 10  |-  ( ( n  e.  ( ZZ>= `  K )  /\  (
n  +  1 )  e.  ( K ... N ) )  ->  n  e.  ( K ... N ) )
2726adantl 453 . . . . . . . . 9  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( K ... N ) )
2827expr 599 . . . . . . . 8  |-  ( (
ph  /\  n  e.  ( ZZ>= `  K )
)  ->  ( (
n  +  1 )  e.  ( K ... N )  ->  n  e.  ( K ... N
) ) )
2928imim1d 71 . . . . . . 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 6055 . . . . . . . . . 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 10475 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( ZZ>= `  K
)  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
3231ad2antrl 709 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) ) )
33 elfzuz3 11020 . . . . . . . . . . . . . . . 16  |-  ( ( n  +  1 )  e.  ( K ... N )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
3433ad2antll 710 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  N  e.  ( ZZ>= `  ( n  +  1 ) ) )
35 elfzuzb 11017 . . . . . . . . . . . . . . 15  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  <->  ( (
n  +  1 )  e.  ( ZZ>= `  ( K  +  1 ) )  /\  N  e.  ( ZZ>= `  ( n  +  1 ) ) ) )
3632, 34, 35sylanbrc 646 . . . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. x  e.  ( ( K  +  1 ) ... N ) ( F `  x
)  =  Z )
3938adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  A. x  e.  ( ( K  + 
1 ) ... N
) ( F `  x )  =  Z )
40 fveq2 5695 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
4140eqeq1d 2420 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
4241rspcv 3016 . . . . . . . . . . . . . 14  |-  ( ( n  +  1 )  e.  ( ( K  +  1 ) ... N )  ->  ( A. x  e.  (
( K  +  1 ) ... N ) ( F `  x
)  =  Z  -> 
( F `  (
n  +  1 ) )  =  Z ) )
4336, 39, 42sylc 58 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( F `  ( n  +  1 ) )  =  Z )
4443oveq2d 6064 . . . . . . . . . . . 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 2757 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 oveq1 6055 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( x  .+  Z
)  =  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )
)
49 id 20 . . . . . . . . . . . . . . . 16  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  x  =  (  seq 
M (  .+  ,  F ) `  K
) )
5048, 49eqeq12d 2426 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) ) )
5150rspcv 3016 . . . . . . . . . . . . . 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 58 . . . . . . . . . . . . 13  |-  ( ph  ->  ( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) )
5352adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  ( (  seq  M (  .+  ,  F ) `  K
)  .+  Z )  =  (  seq  M ( 
.+  ,  F ) `
 K ) )
5444, 53eqtr2d 2445 . . . . . . . . . . 11  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  (  seq  M (  .+  ,  F
) `  K )  =  ( (  seq 
M (  .+  ,  F ) `  K
)  .+  ( F `  ( n  +  1 ) ) ) )
55 simprl 733 . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  K  e.  ( ZZ>= `  M )
)
58 uztrn 10466 . . . . . . . . . . . . 13  |-  ( ( n  e.  ( ZZ>= `  K )  /\  K  e.  ( ZZ>= `  M )
)  ->  n  e.  ( ZZ>= `  M )
)
5955, 57, 58syl2anc 643 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( n  e.  ( ZZ>= `  K )  /\  ( n  +  1 )  e.  ( K ... N ) ) )  ->  n  e.  ( ZZ>= `  M )
)
60 seqp1 11301 . . . . . . . . . . . 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 2426 . . . . . . . . . 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 213 . . . . . . . . 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 599 . . . . . . . 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 24 . . . . . . 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 42 . . . . . 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 425 . . . . 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 24 . . . 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 10498 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  ( ph  ->  ( N  e.  ( K ... N )  ->  (  seq  M
(  .+  ,  F
) `  K )  =  (  seq  M ( 
.+  ,  F ) `
 N ) ) ) )
701, 69mpcom 34 . 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 359    = wceq 1649    e. wcel 1721   A.wral 2674   ` cfv 5421  (class class class)co 6048   1c1 8955    + caddc 8957   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007    seq cseq 11286
This theorem is referenced by:  seqcoll  11675  seqcoll2  11676  fsumcvg  12469  ovolicc1  19373  lgsdilem2  21076  fprodcvg  25217
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-seq 11287
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