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Theorem expnbnd 10429
Description: Exponentiation with a mantissa greater than 1 has no upper bound. (Contributed by NM, 20-Oct-2007.)
Assertion
Ref Expression
expnbnd  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  A  <  ( B ^ k ) )
Distinct variable groups:    A, k    B, k

Proof of Theorem expnbnd
StepHypRef Expression
1 simp1 981 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
21adantr 274 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  A  e.  RR )
3 simp2 982 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR )
43adantr 274 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  B  e.  RR )
5 simpr 109 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  A )
6 simp3 983 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  B )
76adantr 274 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  B )
8 1red 7795 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
91, 8resubcld 8157 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( A  -  1 )  e.  RR )
103, 8resubcld 8157 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 )  e.  RR )
118, 3posdifd 8308 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  B  <->  0  <  ( B  -  1 ) ) )
126, 11mpbid 146 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  ( B  -  1 ) )
1310, 12gt0ap0d 8405 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 ) #  0 )
149, 10, 13redivclapd 8608 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( A  -  1 )  /  ( B  -  1 ) )  e.  RR )
15 arch 8988 . . . . . . 7  |-  ( ( ( A  -  1 )  /  ( B  -  1 ) )  e.  RR  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
1614, 15syl 14 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
17163expa 1181 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  1  <  B
)  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
1817adantrl 469 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
19 simplll 522 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  A  e.  RR )
2019adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  e.  RR )
21 simpllr 523 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  B  e.  RR )
22 1red 7795 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  e.  RR )
2321, 22resubcld 8157 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B  - 
1 )  e.  RR )
24 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN )
2524nnred 8747 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  RR )
2623, 25remulcld 7810 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( B  -  1 )  x.  k )  e.  RR )
2726, 22readdcld 7809 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( B  -  1 )  x.  k )  +  1 )  e.  RR )
2827adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( (
( B  -  1 )  x.  k )  +  1 )  e.  RR )
2924nnnn0d 9044 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN0 )
30 reexpcl 10324 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
3121, 29, 30syl2anc 408 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
3231adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( B ^ k )  e.  RR )
33 simpr 109 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( ( A  -  1 )  /  ( B  - 
1 ) )  < 
k )
34 1red 7795 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  e.  RR )
3520, 34resubcld 8157 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( A  -  1 )  e.  RR )
36 simplr 519 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN )
3736nnred 8747 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  RR )
3821adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  B  e.  RR )
3938, 34resubcld 8157 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( B  -  1 )  e.  RR )
40 simplrr 525 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  <  B
)
4140adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  <  B )
4234, 38posdifd 8308 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( 1  <  B  <->  0  <  ( B  -  1 ) ) )
4341, 42mpbid 146 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <  ( B  -  1 ) )
44 ltdivmul 8648 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR  /\  k  e.  RR  /\  (
( B  -  1 )  e.  RR  /\  0  <  ( B  - 
1 ) ) )  ->  ( ( ( A  -  1 )  /  ( B  - 
1 ) )  < 
k  <->  ( A  - 
1 )  <  (
( B  -  1 )  x.  k ) ) )
4535, 37, 39, 43, 44syl112anc 1220 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( (
( A  -  1 )  /  ( B  -  1 ) )  <  k  <->  ( A  -  1 )  < 
( ( B  - 
1 )  x.  k
) ) )
4633, 45mpbid 146 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( A  -  1 )  < 
( ( B  - 
1 )  x.  k
) )
4739, 37remulcld 7810 . . . . . . . . 9  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( ( B  -  1 )  x.  k )  e.  RR )
4820, 34, 47ltsubaddd 8317 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( ( A  -  1 )  <  ( ( B  -  1 )  x.  k )  <->  A  <  ( ( ( B  - 
1 )  x.  k
)  +  1 ) ) )
4946, 48mpbid 146 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  <  ( ( ( B  - 
1 )  x.  k
)  +  1 ) )
5036nnnn0d 9044 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN0 )
51 0red 7781 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  e.  RR )
52 0lt1 7903 . . . . . . . . . . . 12  |-  0  <  1
53 0re 7780 . . . . . . . . . . . . 13  |-  0  e.  RR
54 1re 7779 . . . . . . . . . . . . 13  |-  1  e.  RR
55 lttr 7852 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5653, 54, 55mp3an12 1305 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5752, 56mpani 426 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
5821, 40, 57sylc 62 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <  B
)
5951, 21, 58ltled 7895 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <_  B
)
6059adantr 274 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <_  B )
61 bernneq2 10427 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0  /\  0  <_  B )  ->  (
( ( B  - 
1 )  x.  k
)  +  1 )  <_  ( B ^
k ) )
6238, 50, 60, 61syl3anc 1216 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  ( (
( B  -  1 )  x.  k )  +  1 )  <_ 
( B ^ k
) )
6320, 28, 32, 49, 62ltletrd 8199 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  <  ( B ^ k ) )
6463ex 114 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( A  -  1 )  /  ( B  - 
1 ) )  < 
k  ->  A  <  ( B ^ k ) ) )
6564reximdva 2534 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  ( E. k  e.  NN  ( ( A  - 
1 )  /  ( B  -  1 ) )  <  k  ->  E. k  e.  NN  A  <  ( B ^
k ) ) )
6618, 65mpd 13 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  E. k  e.  NN  A  <  ( B ^ k ) )
672, 4, 5, 7, 66syl22anc 1217 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
68 1nn 8745 . . 3  |-  1  e.  NN
69 simpr 109 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  B )
70 simpl2 985 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  RR )
7170recnd 7808 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  CC )
72 exp1 10313 . . . . 5  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
7371, 72syl 14 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  -> 
( B ^ 1 )  =  B )
7469, 73breqtrrd 3956 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  ( B ^
1 ) )
75 oveq2 5782 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
7675breq2d 3941 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
7776rspcev 2789 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
7868, 74, 77sylancr 410 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
79 axltwlin 7846 . . . . 5  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8054, 79mp3an1 1302 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8180ancoms 266 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
82813impia 1178 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  A  \/  A  <  B ) )
8367, 78, 82mpjaodan 787 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  A  <  ( B ^ k ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   CCcc 7632   RRcr 7633   0cc0 7634   1c1 7635    + caddc 7637    x. cmul 7639    < clt 7814    <_ cle 7815    - cmin 7947    / cdiv 8446   NNcn 8734   NN0cn0 8991   ^cexp 10306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7725  ax-resscn 7726  ax-1cn 7727  ax-1re 7728  ax-icn 7729  ax-addcl 7730  ax-addrcl 7731  ax-mulcl 7732  ax-mulrcl 7733  ax-addcom 7734  ax-mulcom 7735  ax-addass 7736  ax-mulass 7737  ax-distr 7738  ax-i2m1 7739  ax-0lt1 7740  ax-1rid 7741  ax-0id 7742  ax-rnegex 7743  ax-precex 7744  ax-cnre 7745  ax-pre-ltirr 7746  ax-pre-ltwlin 7747  ax-pre-lttrn 7748  ax-pre-apti 7749  ax-pre-ltadd 7750  ax-pre-mulgt0 7751  ax-pre-mulext 7752  ax-arch 7753
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7816  df-mnf 7817  df-xr 7818  df-ltxr 7819  df-le 7820  df-sub 7949  df-neg 7950  df-reap 8351  df-ap 8358  df-div 8447  df-inn 8735  df-n0 8992  df-z 9069  df-uz 9341  df-seqfrec 10233  df-exp 10307
This theorem is referenced by:  expnlbnd  10430
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