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Theorem expnbnd 10658
Description: Exponentiation with a base 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 998 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
21adantr 276 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  A  e.  RR )
3 simp2 999 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR )
43adantr 276 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  B  e.  RR )
5 simpr 110 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  A )
6 simp3 1000 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  B )
76adantr 276 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  B )
8 1red 7986 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
91, 8resubcld 8352 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( A  -  1 )  e.  RR )
103, 8resubcld 8352 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 )  e.  RR )
118, 3posdifd 8503 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  B  <->  0  <  ( B  -  1 ) ) )
126, 11mpbid 147 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  ( B  -  1 ) )
1310, 12gt0ap0d 8600 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 ) #  0 )
149, 10, 13redivclapd 8806 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( A  -  1 )  /  ( B  -  1 ) )  e.  RR )
15 arch 9187 . . . . . . 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 1204 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  1  <  B
)  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
1817adantrl 478 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
19 simplll 533 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  A  e.  RR )
2019adantr 276 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  e.  RR )
21 simpllr 534 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  B  e.  RR )
22 1red 7986 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  e.  RR )
2321, 22resubcld 8352 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B  - 
1 )  e.  RR )
24 simpr 110 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN )
2524nnred 8946 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  RR )
2623, 25remulcld 8002 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( B  -  1 )  x.  k )  e.  RR )
2726, 22readdcld 8001 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( B  -  1 )  x.  k )  +  1 )  e.  RR )
2827adantr 276 . . . . . . 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 9243 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN0 )
30 reexpcl 10551 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
3121, 29, 30syl2anc 411 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
3231adantr 276 . . . . . . 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 110 . . . . . . . . 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 7986 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  e.  RR )
3520, 34resubcld 8352 . . . . . . . . . 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 528 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN )
3736nnred 8946 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  RR )
3821adantr 276 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  B  e.  RR )
3938, 34resubcld 8352 . . . . . . . . . 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 536 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  <  B
)
4140adantr 276 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  <  B )
4234, 38posdifd 8503 . . . . . . . . . . 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 147 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <  ( B  -  1 ) )
44 ltdivmul 8847 . . . . . . . . . 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 1252 . . . . . . . . 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 147 . . . . . . . 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 8002 . . . . . . . . 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 8512 . . . . . . . 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 147 . . . . . . 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 9243 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN0 )
51 0red 7972 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  e.  RR )
52 0lt1 8098 . . . . . . . . . . . 12  |-  0  <  1
53 0re 7971 . . . . . . . . . . . . 13  |-  0  e.  RR
54 1re 7970 . . . . . . . . . . . . 13  |-  1  e.  RR
55 lttr 8045 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5653, 54, 55mp3an12 1337 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5752, 56mpani 430 . . . . . . . . . . 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 8090 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <_  B
)
6059adantr 276 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <_  B )
61 bernneq2 10656 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0  /\  0  <_  B )  ->  (
( ( B  - 
1 )  x.  k
)  +  1 )  <_  ( B ^
k ) )
6238, 50, 60, 61syl3anc 1248 . . . . . . 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 8394 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  <  ( B ^ k ) )
6463ex 115 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( A  -  1 )  /  ( B  - 
1 ) )  < 
k  ->  A  <  ( B ^ k ) ) )
6564reximdva 2589 . . . 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 1249 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
68 1nn 8944 . . 3  |-  1  e.  NN
69 simpr 110 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  B )
70 simpl2 1002 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  RR )
7170recnd 8000 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  CC )
72 exp1 10540 . . . . 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 4043 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  ( B ^
1 ) )
75 oveq2 5896 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
7675breq2d 4027 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
7776rspcev 2853 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
7868, 74, 77sylancr 414 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
79 axltwlin 8039 . . . . 5  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8054, 79mp3an1 1334 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8180ancoms 268 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
82813impia 1201 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  A  \/  A  <  B ) )
8367, 78, 82mpjaodan 799 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 104    <-> wb 105    \/ wo 709    /\ w3a 979    = wceq 1363    e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   RRcr 7824   0cc0 7825   1c1 7826    + caddc 7828    x. cmul 7830    < clt 8006    <_ cle 8007    - cmin 8142    / cdiv 8643   NNcn 8933   NN0cn0 9190   ^cexp 10533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-seqfrec 10460  df-exp 10534
This theorem is referenced by:  expnlbnd  10659  pclemub  12301
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