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Theorem expnbnd 9540
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 915 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
21adantr 265 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  A  e.  RR )
3 simp2 916 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR )
43adantr 265 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  B  e.  RR )
5 simpr 107 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  A )
6 simp3 917 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  B )
76adantr 265 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  B )
8 1red 7100 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
91, 8resubcld 7451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( A  -  1 )  e.  RR )
103, 8resubcld 7451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 )  e.  RR )
118, 3posdifd 7597 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  B  <->  0  <  ( B  -  1 ) ) )
126, 11mpbid 139 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  0  <  ( B  -  1 ) )
1310, 12gt0ap0d 7693 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 ) #  0 )
149, 10, 13redivclapd 7883 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( A  -  1 )  /  ( B  -  1 ) )  e.  RR )
15 arch 8236 . . . . . . 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 1115 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  1  <  B
)  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
1817adantrl 455 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
19 simplll 493 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  A  e.  RR )
2019adantr 265 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  e.  RR )
21 simpllr 494 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  B  e.  RR )
22 1red 7100 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  e.  RR )
2321, 22resubcld 7451 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B  - 
1 )  e.  RR )
24 simpr 107 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN )
2524nnred 8003 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  RR )
2623, 25remulcld 7115 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( B  -  1 )  x.  k )  e.  RR )
2726, 22readdcld 7114 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( B  -  1 )  x.  k )  +  1 )  e.  RR )
2827adantr 265 . . . . . . 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 8292 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN0 )
30 reexpcl 9437 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
3121, 29, 30syl2anc 397 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
3231adantr 265 . . . . . . 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 107 . . . . . . . . 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 7100 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  e.  RR )
3520, 34resubcld 7451 . . . . . . . . . 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 490 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN )
3736nnred 8003 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  RR )
3821adantr 265 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  B  e.  RR )
3938, 34resubcld 7451 . . . . . . . . . 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 496 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  <  B
)
4140adantr 265 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  <  B )
4234, 38posdifd 7597 . . . . . . . . . . 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 139 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <  ( B  -  1 ) )
44 ltdivmul 7917 . . . . . . . . . 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 1150 . . . . . . . . 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 139 . . . . . . . 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 7115 . . . . . . . . 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 7606 . . . . . . . 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 139 . . . . . . 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 8292 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN0 )
51 0red 7086 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  e.  RR )
52 0lt1 7202 . . . . . . . . . . . 12  |-  0  <  1
53 0re 7085 . . . . . . . . . . . . 13  |-  0  e.  RR
54 1re 7084 . . . . . . . . . . . . 13  |-  1  e.  RR
55 lttr 7151 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5653, 54, 55mp3an12 1233 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5752, 56mpani 414 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
5821, 40, 57sylc 60 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <  B
)
5951, 21, 58ltled 7194 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <_  B
)
6059adantr 265 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <_  B )
61 bernneq2 9538 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0  /\  0  <_  B )  ->  (
( ( B  - 
1 )  x.  k
)  +  1 )  <_  ( B ^
k ) )
6238, 50, 60, 61syl3anc 1146 . . . . . . 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 7492 . . . . . 6  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  <  ( B ^ k ) )
6463ex 112 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( A  -  1 )  /  ( B  - 
1 ) )  < 
k  ->  A  <  ( B ^ k ) ) )
6564reximdva 2438 . . . 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 1147 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
68 1nn 8001 . . 3  |-  1  e.  NN
69 simpr 107 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  B )
70 simpl2 919 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  RR )
7170recnd 7113 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  CC )
72 exp1 9426 . . . . 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 3818 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  ( B ^
1 ) )
75 oveq2 5548 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
7675breq2d 3804 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
7776rspcev 2673 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
7868, 74, 77sylancr 399 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
79 axltwlin 7146 . . . . 5  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8054, 79mp3an1 1230 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8180ancoms 259 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
82813impia 1112 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  A  \/  A  <  B ) )
8367, 78, 82mpjaodan 722 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    + caddc 6950    x. cmul 6952    < clt 7119    <_ cle 7120    - cmin 7245    / cdiv 7725   NNcn 7990   NN0cn0 8239   ^cexp 9419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060  ax-arch 7061
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-if 3360  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-frec 6009  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647  df-div 7726  df-inn 7991  df-n0 8240  df-z 8303  df-uz 8570  df-iseq 9376  df-iexp 9420
This theorem is referenced by:  expnlbnd  9541
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