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Theorem expcnv 11527
Description: A sequence of powers of a complex number  A with absolute value smaller than 1 converges to zero. (Contributed by NM, 8-May-2006.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
Hypotheses
Ref Expression
expcnv.1  |-  ( ph  ->  A  e.  CC )
expcnv.2  |-  ( ph  ->  ( abs `  A
)  <  1 )
Assertion
Ref Expression
expcnv  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Distinct variable group:    A, n
Allowed substitution hint:    ph( n)

Proof of Theorem expcnv
StepHypRef Expression
1 nnuz 9651 . . 3  |-  NN  =  ( ZZ>= `  1 )
2 1z 9441 . . . 4  |-  1  e.  ZZ
32a1i 10 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  1  e.  ZZ )
4 nn0ex 9359 . . . . 5  |-  NN0  e.  _V
54mptex 5205 . . . 4  |-  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V
65a1i 10 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  (
n  e.  NN0  |->  ( A ^ n ) )  e.  _V )
7 0cn 8261 . . . 4  |-  0  e.  CC
87a1i 10 . . 3  |-  ( (
ph  /\  A  = 
0 )  ->  0  e.  CC )
9 nnnn0 9360 . . . . . 6  |-  ( k  e.  NN  ->  k  e.  NN0 )
10 oveq2 5378 . . . . . . 7  |-  ( n  =  k  ->  ( A ^ n )  =  ( A ^ k
) )
11 eqid 2069 . . . . . . 7  |-  ( n  e.  NN0  |->  ( A ^ n ) )  =  ( n  e. 
NN0  |->  ( A ^
n ) )
12 ovex 5395 . . . . . . 7  |-  ( A ^ k )  e. 
_V
1310, 11, 12fvmpt 5131 . . . . . 6  |-  ( k  e.  NN0  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
149, 13syl 15 . . . . 5  |-  ( k  e.  NN  ->  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k )  =  ( A ^
k ) )
15 simpr 441 . . . . . 6  |-  ( (
ph  /\  A  = 
0 )  ->  A  =  0 )
1615oveq1d 5385 . . . . 5  |-  ( (
ph  /\  A  = 
0 )  ->  ( A ^ k )  =  ( 0 ^ k
) )
1714, 16sylan9eqr 2123 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  k  e.  NN )  ->  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  ( 0 ^ k ) )
18 0exp 10511 . . . . 5  |-  ( k  e.  NN  ->  (
0 ^ k )  =  0 )
1918adantl 446 . . . 4  |-  ( ( ( ph  /\  A  =  0 )  /\  k  e.  NN )  ->  ( 0 ^ k
)  =  0 )
2017, 19eqtrd 2101 . . 3  |-  ( ( ( ph  /\  A  =  0 )  /\  k  e.  NN )  ->  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k )  =  0 )
211, 3, 6, 8, 20climconst 11228 . 2  |-  ( (
ph  /\  A  = 
0 )  ->  (
n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
222a1i 10 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  1  e.  ZZ )
23 expcnv.2 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  A
)  <  1 )
2423adantr 445 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  ( abs `  A )  <  1 )
25 expcnv.1 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
26 absrpcl 10987 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( abs `  A
)  e.  RR+ )
2725, 26sylan 451 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  ( abs `  A )  e.  RR+ )
2827reclt1d 9791 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( abs `  A
)  <  1  <->  1  <  ( 1  /  ( abs `  A ) ) ) )
2924, 28mpbid 199 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  0 )  ->  1  <  ( 1  /  ( abs `  A ) ) )
30 1re 8267 . . . . . . . . 9  |-  1  e.  RR
3127rpreccld 9788 . . . . . . . . . 10  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  /  ( abs `  A ) )  e.  RR+ )
3231rpred 9778 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  /  ( abs `  A ) )  e.  RR )
33 difrp 9775 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( 1  /  ( abs `  A ) )  e.  RR )  -> 
( 1  <  (
1  /  ( abs `  A ) )  <->  ( (
1  /  ( abs `  A ) )  - 
1 )  e.  RR+ ) )
3430, 32, 33sylancr 637 . . . . . . . 8  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  <  ( 1  /  ( abs `  A
) )  <->  ( (
1  /  ( abs `  A ) )  - 
1 )  e.  RR+ ) )
3529, 34mpbid 199 . . . . . . 7  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( 1  /  ( abs `  A ) )  -  1 )  e.  RR+ )
3635rpreccld 9788 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  e.  RR+ )
3736rpcnd 9780 . . . . 5  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  e.  CC )
38 divcnv 11517 . . . . 5  |-  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  e.  CC  ->  ( n  e.  NN  |->  ( ( 1  /  ( ( 1  /  ( abs `  A
) )  -  1 ) )  /  n
) )  ~~>  0 )
3937, 38syl 15 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  NN  |->  ( ( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  /  n ) )  ~~>  0 )
40 nnex 9143 . . . . . 6  |-  NN  e.  _V
4140mptex 5205 . . . . 5  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  e.  _V
4241a1i 10 . . . 4  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  e.  _V )
43 oveq2 5378 . . . . . . 7  |-  ( n  =  k  ->  (
( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  /  n )  =  ( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  k
) )
44 eqid 2069 . . . . . . 7  |-  ( n  e.  NN  |->  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  /  n ) )  =  ( n  e.  NN  |->  ( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  n
) )
45 ovex 5395 . . . . . . 7  |-  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  / 
k )  e.  _V
4643, 44, 45fvmpt 5131 . . . . . 6  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  n
) ) `  k
)  =  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  / 
k ) )
4746adantl 446 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  n
) ) `  k
)  =  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  / 
k ) )
4836rpred 9778 . . . . . 6  |-  ( (
ph  /\  A  =/=  0 )  ->  (
1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  e.  RR )
49 nndivre 9172 . . . . . 6  |-  ( ( ( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  e.  RR  /\  k  e.  NN )  ->  (
( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  /  k )  e.  RR )
5048, 49sylan 451 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  /  k )  e.  RR )
5147, 50eqeltrd 2143 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  n
) ) `  k
)  e.  RR )
52 oveq2 5378 . . . . . . . 8  |-  ( n  =  k  ->  (
( abs `  A
) ^ n )  =  ( ( abs `  A ) ^ k
) )
53 eqid 2069 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( abs `  A ) ^ n ) )  =  ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) )
54 ovex 5395 . . . . . . . 8  |-  ( ( abs `  A ) ^ k )  e. 
_V
5552, 53, 54fvmpt 5131 . . . . . . 7  |-  ( k  e.  NN  ->  (
( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
5655adantl 446 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
57 nnz 9433 . . . . . . 7  |-  ( k  e.  NN  ->  k  e.  ZZ )
58 rpexpcl 10496 . . . . . . 7  |-  ( ( ( abs `  A
)  e.  RR+  /\  k  e.  ZZ )  ->  (
( abs `  A
) ^ k )  e.  RR+ )
5927, 57, 58syl2an 457 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( abs `  A
) ^ k )  e.  RR+ )
6056, 59eqeltrd 2143 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  e.  RR+ )
6160rpred 9778 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  e.  RR )
62 nnrp 9751 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  RR+ )
63 rpmulcl 9763 . . . . . . . 8  |-  ( ( ( ( 1  / 
( abs `  A
) )  -  1 )  e.  RR+  /\  k  e.  RR+ )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  e.  RR+ )
6435, 62, 63syl2an 457 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  e.  RR+ )
6564rpred 9778 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  e.  RR )
66 peano2re 8414 . . . . . . . . . 10  |-  ( ( ( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  e.  RR  ->  (
( ( ( 1  /  ( abs `  A
) )  -  1 )  x.  k )  +  1 )  e.  RR )
6765, 66syl 15 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( ( 1  /  ( abs `  A
) )  -  1 )  x.  k )  +  1 )  e.  RR )
68 rpexpcl 10496 . . . . . . . . . . 11  |-  ( ( ( 1  /  ( abs `  A ) )  e.  RR+  /\  k  e.  ZZ )  ->  (
( 1  /  ( abs `  A ) ) ^ k )  e.  RR+ )
6931, 57, 68syl2an 457 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( 1  /  ( abs `  A ) ) ^ k )  e.  RR+ )
7069rpred 9778 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( 1  /  ( abs `  A ) ) ^ k )  e.  RR )
7165lep1d 9080 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  <_  ( ( ( ( 1  /  ( abs `  A ) )  -  1 )  x.  k )  +  1 ) )
7232adantr 445 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
1  /  ( abs `  A ) )  e.  RR )
739adantl 446 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  k  e.  NN0 )
7431rpge0d 9782 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =/=  0 )  ->  0  <_  ( 1  /  ( abs `  A ) ) )
7574adantr 445 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  0  <_  ( 1  /  ( abs `  A ) ) )
76 bernneq2 10601 . . . . . . . . . 10  |-  ( ( ( 1  /  ( abs `  A ) )  e.  RR  /\  k  e.  NN0  /\  0  <_ 
( 1  /  ( abs `  A ) ) )  ->  ( (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  +  1 )  <_ 
( ( 1  / 
( abs `  A
) ) ^ k
) )
7772, 73, 75, 76syl3anc 1140 . . . . . . . . 9  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( ( 1  /  ( abs `  A
) )  -  1 )  x.  k )  +  1 )  <_ 
( ( 1  / 
( abs `  A
) ) ^ k
) )
7865, 67, 70, 71, 77letrd 8402 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  <_  ( ( 1  /  ( abs `  A
) ) ^ k
) )
7927rpcnne0d 9787 . . . . . . . . 9  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( abs `  A
)  e.  CC  /\  ( abs `  A )  =/=  0 ) )
80 exprec 10517 . . . . . . . . . 10  |-  ( ( ( abs `  A
)  e.  CC  /\  ( abs `  A )  =/=  0  /\  k  e.  ZZ )  ->  (
( 1  /  ( abs `  A ) ) ^ k )  =  ( 1  /  (
( abs `  A
) ^ k ) ) )
81803expa 1110 . . . . . . . . 9  |-  ( ( ( ( abs `  A
)  e.  CC  /\  ( abs `  A )  =/=  0 )  /\  k  e.  ZZ )  ->  ( ( 1  / 
( abs `  A
) ) ^ k
)  =  ( 1  /  ( ( abs `  A ) ^ k
) ) )
8279, 57, 81syl2an 457 . . . . . . . 8  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( 1  /  ( abs `  A ) ) ^ k )  =  ( 1  /  (
( abs `  A
) ^ k ) ) )
8378, 82breqtrd 3620 . . . . . . 7  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k )  <_  ( 1  / 
( ( abs `  A
) ^ k ) ) )
8464, 59, 83lerec2d 9799 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( abs `  A
) ^ k )  <_  ( 1  / 
( ( ( 1  /  ( abs `  A
) )  -  1 )  x.  k ) ) )
8535rpcnne0d 9787 . . . . . . 7  |-  ( (
ph  /\  A  =/=  0 )  ->  (
( ( 1  / 
( abs `  A
) )  -  1 )  e.  CC  /\  ( ( 1  / 
( abs `  A
) )  -  1 )  =/=  0 ) )
86 nncn 9145 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  CC )
87 nnne0 9169 . . . . . . . 8  |-  ( k  e.  NN  ->  k  =/=  0 )
8886, 87jca 512 . . . . . . 7  |-  ( k  e.  NN  ->  (
k  e.  CC  /\  k  =/=  0 ) )
89 recdiv2 8878 . . . . . . 7  |-  ( ( ( ( ( 1  /  ( abs `  A
) )  -  1 )  e.  CC  /\  ( ( 1  / 
( abs `  A
) )  -  1 )  =/=  0 )  /\  ( k  e.  CC  /\  k  =/=  0 ) )  -> 
( ( 1  / 
( ( 1  / 
( abs `  A
) )  -  1 ) )  /  k
)  =  ( 1  /  ( ( ( 1  /  ( abs `  A ) )  - 
1 )  x.  k
) ) )
9085, 88, 89syl2an 457 . . . . . 6  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( 1  /  (
( 1  /  ( abs `  A ) )  -  1 ) )  /  k )  =  ( 1  /  (
( ( 1  / 
( abs `  A
) )  -  1 )  x.  k ) ) )
9184, 90breqtrrd 3622 . . . . 5  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( abs `  A
) ^ k )  <_  ( ( 1  /  ( ( 1  /  ( abs `  A
) )  -  1 ) )  /  k
) )
9291, 56, 473brtr4d 3626 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  (
( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  <_  ( ( n  e.  NN  |->  ( ( 1  /  ( ( 1  /  ( abs `  A ) )  - 
1 ) )  /  n ) ) `  k ) )
9360rpge0d 9782 . . . 4  |-  ( ( ( ph  /\  A  =/=  0 )  /\  k  e.  NN )  ->  0  <_  ( ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) ) `  k
) )
941, 22, 39, 42, 51, 61, 92, 93climsqz2 11326 . . 3  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 )
952a1i 10 . . . . 5  |-  ( ph  ->  1  e.  ZZ )
965a1i 10 . . . . 5  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  e.  _V )
9741a1i 10 . . . . 5  |-  ( ph  ->  ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  e.  _V )
989adantl 446 . . . . . . 7  |-  ( (
ph  /\  k  e.  NN )  ->  k  e. 
NN0 )
9998, 13syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  =  ( A ^ k
) )
100 expcl 10495 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( A ^ k
)  e.  CC )
10125, 9, 100syl2an 457 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( A ^ k )  e.  CC )
10299, 101eqeltrd 2143 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN0  |->  ( A ^ n ) ) `
 k )  e.  CC )
103 absexp 11001 . . . . . . 7  |-  ( ( A  e.  CC  /\  k  e.  NN0 )  -> 
( abs `  ( A ^ k ) )  =  ( ( abs `  A ) ^ k
) )
10425, 9, 103syl2an 457 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( A ^ k
) )  =  ( ( abs `  A
) ^ k ) )
10599fveq2d 5058 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( abs `  ( ( n  e. 
NN0  |->  ( A ^
n ) ) `  k ) )  =  ( abs `  ( A ^ k ) ) )
10655adantl 446 . . . . . 6  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( ( abs `  A ) ^ k
) )
107104, 105, 1063eqtr4rd 2112 . . . . 5  |-  ( (
ph  /\  k  e.  NN )  ->  ( ( n  e.  NN  |->  ( ( abs `  A
) ^ n ) ) `  k )  =  ( abs `  (
( n  e.  NN0  |->  ( A ^ n ) ) `  k ) ) )
1081, 95, 96, 97, 102, 107climabs0 11270 . . . 4  |-  ( ph  ->  ( ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0  <->  (
n  e.  NN  |->  ( ( abs `  A
) ^ n ) )  ~~>  0 ) )
109108biimpar 465 . . 3  |-  ( (
ph  /\  ( n  e.  NN  |->  ( ( abs `  A ) ^ n
) )  ~~>  0 )  ->  ( n  e. 
NN0  |->  ( A ^
n ) )  ~~>  0 )
11094, 109syldan 450 . 2  |-  ( (
ph  /\  A  =/=  0 )  ->  (
n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
11121, 110pm2.61dane 2262 1  |-  ( ph  ->  ( n  e.  NN0  |->  ( A ^ n ) )  ~~>  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 174    /\ wa 356    = wceq 1524    e. wcel 1526    =/= wne 2187   _Vcvv 2480   class class class wbr 3596    e. cmpt 3650   ` cfv 4278  (class class class)co 5370   CCcc 8166   RRcr 8167   0cc0 8168   1c1 8169    + caddc 8171    x. cmul 8173    <_ cle 8294    < clt 8298    - cmin 8462    / cdiv 8828   NNcn 9137   NN0cn0 9353   ZZcz 9412   RR+crp 9742   ^cexp 10478   abscabs 10933    ~~> cli 11169
This theorem is referenced by:  explecnv  11528  geolim  11531  geo2lim  11535  iscmet3lem3  17417  mbfi1fseqlem6  17776  geomcau  24305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1528  ax-11 1529  ax-13 1530  ax-14 1531  ax-17 1533  ax-12o 1567  ax-10 1581  ax-9 1587  ax-4 1594  ax-16 1780  ax-ext 2051  ax-rep 3701  ax-sep 3711  ax-nul 3719  ax-pow 3755  ax-pr 3779  ax-un 4071  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-icn 8226  ax-addcl 8227  ax-addrcl 8228  ax-mulcl 8229  ax-mulrcl 8230  ax-mulcom 8231  ax-addass 8232  ax-mulass 8233  ax-distr 8234  ax-i2m1 8235  ax-1ne0 8236  ax-1rid 8237  ax-rnegex 8238  ax-rrecex 8239  ax-cnre 8240  ax-pre-lttri 8241  ax-pre-lttrn 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-sup 8245
This theorem depends on definitions:  df-bi 175  df-or 357  df-an 358  df-3or 897  df-3an 898  df-tru 1259  df-ex 1451  df-sb 1741  df-eu 1963  df-mo 1964  df-clab 2057  df-cleq 2062  df-clel 2065  df-ne 2189  df-nel 2190  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2286  df-v 2482  df-sbc 2656  df-csb 2738  df-dif 2801  df-un 2803  df-in 2805  df-ss 2809  df-pss 2811  df-nul 3078  df-if 3187  df-pw 3248  df-sn 3266  df-pr 3267  df-tp 3268  df-op 3269  df-uni 3435  df-iun 3512  df-br 3597  df-opab 3651  df-mpt 3652  df-tr 3684  df-eprel 3866  df-id 3870  df-po 3875  df-so 3876  df-fr 3913  df-we 3915  df-ord 3956  df-on 3957  df-lim 3958  df-suc 3959  df-om 4234  df-xp 4280  df-rel 4281  df-cnv 4282  df-co 4283  df-dm 4284  df-rn 4285  df-res 4286  df-ima 4287  df-fun 4288  df-fn 4289  df-f 4290  df-f1 4291  df-fo 4292  df-f1o 4293  df-fv 4294  df-ov 5373  df-oprab 5374  df-mpt2 5375  df-2nd 5625  df-iota 5780  df-recs 5853  df-rdg 5888  df-er 6125  df-pm 6230  df-en 6312  df-dom 6313  df-sdom 6314  df-riota 6478  df-sup 6686  df-pnf 8299  df-mnf 8300  df-xr 8301  df-ltxr 8302  df-le 8303  df-sub 8464  df-neg 8465  df-div 8829  df-n 9138  df-2 9195  df-3 9196  df-n0 9354  df-z 9413  df-uz 9619  df-rp 9743  df-fl 10304  df-seq 10421  df-exp 10479  df-cj 10798  df-re 10799  df-im 10800  df-sqr 10934  df-abs 10935  df-clim 11173  df-rlim 11174
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