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Theorem expnbnd 10192
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 946 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
21adantr 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  A  e.  RR )
3 simp2 947 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  B  e.  RR )
43adantr 271 . . 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 948 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  <  B )
76adantr 271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  -> 
1  <  B )
8 1red 7600 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
91, 8resubcld 7956 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( A  -  1 )  e.  RR )
103, 8resubcld 7956 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 )  e.  RR )
118, 3posdifd 8106 . . . . . . . . . 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 8202 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B  -  1 ) #  0 )
149, 10, 13redivclapd 8398 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
( A  -  1 )  /  ( B  -  1 ) )  e.  RR )
15 arch 8768 . . . . . . 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 1146 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  1  <  B
)  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
1817adantrl 463 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  E. k  e.  NN  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)
19 simplll 501 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  A  e.  RR )
2019adantr 271 . . . . . . 7  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  A  e.  RR )
21 simpllr 502 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  B  e.  RR )
22 1red 7600 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  e.  RR )
2321, 22resubcld 7956 . . . . . . . . . 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 8533 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  RR )
2623, 25remulcld 7615 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( B  -  1 )  x.  k )  e.  RR )
2726, 22readdcld 7614 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( ( ( B  -  1 )  x.  k )  +  1 )  e.  RR )
2827adantr 271 . . . . . . 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 8824 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  k  e.  NN0 )
30 reexpcl 10087 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  k  e.  NN0 )  -> 
( B ^ k
)  e.  RR )
3121, 29, 30syl2anc 404 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  ( B ^
k )  e.  RR )
3231adantr 271 . . . . . . 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 7600 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  e.  RR )
3520, 34resubcld 7956 . . . . . . . . . 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 498 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN )
3736nnred 8533 . . . . . . . . . 10  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  RR )
3821adantr 271 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  B  e.  RR )
3938, 34resubcld 7956 . . . . . . . . . 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 504 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  1  <  B
)
4140adantr 271 . . . . . . . . . . 11  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  1  <  B )
4234, 38posdifd 8106 . . . . . . . . . . 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 8434 . . . . . . . . . 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 1185 . . . . . . . . 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 7615 . . . . . . . . 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 8115 . . . . . . . 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 8824 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  k  e.  NN0 )
51 0red 7586 . . . . . . . . . 10  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  e.  RR )
52 0lt1 7707 . . . . . . . . . . . 12  |-  0  <  1
53 0re 7585 . . . . . . . . . . . . 13  |-  0  e.  RR
54 1re 7584 . . . . . . . . . . . . 13  |-  1  e.  RR
55 lttr 7656 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5653, 54, 55mp3an12 1270 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
5752, 56mpani 422 . . . . . . . . . . 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 7699 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  ->  0  <_  B
)
6059adantr 271 . . . . . . . 8  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
1  <  A  /\  1  <  B ) )  /\  k  e.  NN )  /\  ( ( A  -  1 )  / 
( B  -  1 ) )  <  k
)  ->  0  <_  B )
61 bernneq2 10190 . . . . . . . 8  |-  ( ( B  e.  RR  /\  k  e.  NN0  /\  0  <_  B )  ->  (
( ( B  - 
1 )  x.  k
)  +  1 )  <_  ( B ^
k ) )
6238, 50, 60, 61syl3anc 1181 . . . . . . 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 7998 . . . . . 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 2487 . . . 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 1182 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
68 1nn 8531 . . 3  |-  1  e.  NN
69 simpr 109 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  B )
70 simpl2 950 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  RR )
7170recnd 7613 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  B  e.  CC )
72 exp1 10076 . . . . 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 3893 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  A  <  ( B ^
1 ) )
75 oveq2 5698 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
7675breq2d 3879 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
7776rspcev 2736 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
7868, 74, 77sylancr 406 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  B )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
79 axltwlin 7651 . . . . 5  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  A  e.  RR )  ->  (
1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8054, 79mp3an1 1267 . . . 4  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
8180ancoms 265 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  ->  ( 1  <  A  \/  A  <  B ) ) )
82813impia 1143 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  (
1  <  A  \/  A  <  B ) )
8367, 78, 82mpjaodan 750 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 667    /\ w3a 927    = wceq 1296    e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448    + caddc 7450    x. cmul 7452    < clt 7619    <_ cle 7620    - cmin 7750    / cdiv 8236   NNcn 8520   NN0cn0 8771   ^cexp 10069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-ilim 4220  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-frec 6194  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-seqfrec 10001  df-exp 10070
This theorem is referenced by:  expnlbnd  10193
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