MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqid2 Unicode version

Theorem seqid2 11324
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 11021 . . 3  |-  ( N  e.  ( ZZ>= `  K
)  ->  N  e.  ( K ... N ) )
31, 2syl 16 . 2  |-  ( ph  ->  N  e.  ( K ... N ) )
4 eleq1 2464 . . . . . 6  |-  ( x  =  K  ->  (
x  e.  ( K ... N )  <->  K  e.  ( K ... N ) ) )
5 fveq2 5687 . . . . . . 7  |-  ( x  =  K  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  K
) )
65eqeq2d 2415 . . . . . 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 2464 . . . . . 6  |-  ( x  =  n  ->  (
x  e.  ( K ... N )  <->  n  e.  ( K ... N ) ) )
10 fveq2 5687 . . . . . . 7  |-  ( x  =  n  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  n
) )
1110eqeq2d 2415 . . . . . 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 2464 . . . . . 6  |-  ( x  =  ( n  + 
1 )  ->  (
x  e.  ( K ... N )  <->  ( n  +  1 )  e.  ( K ... N
) ) )
15 fveq2 5687 . . . . . . 7  |-  ( x  =  ( n  + 
1 )  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  (
n  +  1 ) ) )
1615eqeq2d 2415 . . . . . 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 2464 . . . . . 6  |-  ( x  =  N  ->  (
x  e.  ( K ... N )  <->  N  e.  ( K ... N ) ) )
20 fveq2 5687 . . . . . . 7  |-  ( x  =  N  ->  (  seq  M (  .+  ,  F ) `  x
)  =  (  seq 
M (  .+  ,  F ) `  N
) )
2120eqeq2d 2415 . . . . . 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 2405 . . . . 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 11025 . . . . . . . . . 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 6047 . . . . . . . . . 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 10467 . . . . . . . . . . . . . . . 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 11012 . . . . . . . . . . . . . . . 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 11009 . . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . 16  |-  ( x  =  ( n  + 
1 )  ->  ( F `  x )  =  ( F `  ( n  +  1
) ) )
4140eqeq1d 2412 . . . . . . . . . . . . . . 15  |-  ( x  =  ( n  + 
1 )  ->  (
( F `  x
)  =  Z  <->  ( F `  ( n  +  1 ) )  =  Z ) )
4241rspcv 3008 . . . . . . . . . . . . . 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 6056 . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. x  e.  S  ( x  .+  Z )  =  x )
48 oveq1 6047 . . . . . . . . . . . . . . . 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 2418 . . . . . . . . . . . . . . 15  |-  ( x  =  (  seq  M
(  .+  ,  F
) `  K )  ->  ( ( x  .+  Z )  =  x  <-> 
( (  seq  M
(  .+  ,  F
) `  K )  .+  Z )  =  (  seq  M (  .+  ,  F ) `  K
) ) )
5150rspcv 3008 . . . . . . . . . . . . . 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 2437 . . . . . . . . . . 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 10458 . . . . . . . . . . . . 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 11293 . . . . . . . . . . . 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 2418 . . . . . . . . . 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 10490 . . 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 2666   ` cfv 5413  (class class class)co 6040   1c1 8947    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999    seq cseq 11278
This theorem is referenced by:  seqcoll  11667  seqcoll2  11668  fsumcvg  12461  ovolicc1  19365  lgsdilem2  21068  fprodcvg  25209
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279
  Copyright terms: Public domain W3C validator