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Theorem expnbnd 11977
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 1nn 10321 . . 3  |-  1  e.  NN
2 1re 9373 . . . . . . . 8  |-  1  e.  RR
3 lttr 9439 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( A  <  1  /\  1  <  B )  ->  A  <  B
) )
42, 3mp3an2 1295 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <  1  /\  1  < 
B )  ->  A  <  B ) )
54exp4b 602 . . . . . 6  |-  ( A  e.  RR  ->  ( B  e.  RR  ->  ( A  <  1  -> 
( 1  <  B  ->  A  <  B ) ) ) )
65com34 83 . . . . 5  |-  ( A  e.  RR  ->  ( B  e.  RR  ->  ( 1  <  B  -> 
( A  <  1  ->  A  <  B ) ) ) )
763imp1 1193 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  A  <  B )
8 recn 9360 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
9 exp1 11855 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
108, 9syl 16 . . . . . 6  |-  ( B  e.  RR  ->  ( B ^ 1 )  =  B )
11103ad2ant2 1003 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B ^ 1 )  =  B )
1211adantr 462 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  -> 
( B ^ 1 )  =  B )
137, 12breqtrrd 4306 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  A  <  ( B ^
1 ) )
14 oveq2 6088 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
1514breq2d 4292 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
1615rspcev 3062 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
171, 13, 16sylancr 656 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
18 peano2rem 9663 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
1918adantr 462 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( A  -  1 )  e.  RR )
20 peano2rem 9663 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
2120adantr 462 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( B  -  1 )  e.  RR )
2221adantl 463 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( B  -  1 )  e.  RR )
23 posdif 9820 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  <->  0  <  ( B  - 
1 ) ) )
242, 23mpan 663 . . . . . . . . . . . . 13  |-  ( B  e.  RR  ->  (
1  <  B  <->  0  <  ( B  -  1 ) ) )
2524biimpa 481 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  ( B  -  1 ) )
2625gt0ne0d 9892 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( B  -  1 )  =/=  0 )
2726adantl 463 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( B  -  1 )  =/=  0 )
2819, 22, 27redivcld 10147 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( ( A  -  1 )  /  ( B  - 
1 ) )  e.  RR )
2928adantll 706 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( A  - 
1 )  /  ( B  -  1 ) )  e.  RR )
3018adantl 463 . . . . . . . . . 10  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  ( A  -  1 )  e.  RR )
31 subge0 9840 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( 0  <_  ( A  -  1 )  <->  1  <_  A )
)
322, 31mpan2 664 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
0  <_  ( A  -  1 )  <->  1  <_  A ) )
3332biimparc 484 . . . . . . . . . 10  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  0  <_  ( A  -  1 ) )
3430, 33jca 529 . . . . . . . . 9  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  ( ( A  - 
1 )  e.  RR  /\  0  <_  ( A  -  1 ) ) )
3521, 25jca 529 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( ( B  - 
1 )  e.  RR  /\  0  <  ( B  -  1 ) ) )
36 divge0 10186 . . . . . . . . 9  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  0  <_  ( A  -  1 ) )  /\  ( ( B  -  1 )  e.  RR  /\  0  < 
( B  -  1 ) ) )  -> 
0  <_  ( ( A  -  1 )  /  ( B  - 
1 ) ) )
3734, 35, 36syl2an 474 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <_  ( ( A  -  1 )  /  ( B  - 
1 ) ) )
38 flge0nn0 11650 . . . . . . . 8  |-  ( ( ( ( A  - 
1 )  /  ( B  -  1 ) )  e.  RR  /\  0  <_  ( ( A  -  1 )  / 
( B  -  1 ) ) )  -> 
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  e.  NN0 )
3929, 37, 38syl2anc 654 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  e.  NN0 )
40 nn0p1nn 10607 . . . . . . 7  |-  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  e.  NN0  ->  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 )  e.  NN )
4139, 40syl 16 . . . . . 6  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN )
42 simplr 747 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  A  e.  RR )
4321adantl 463 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( B  -  1 )  e.  RR )
44 peano2nn0 10608 . . . . . . . . . . 11  |-  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  e.  NN0  ->  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 )  e. 
NN0 )
4539, 44syl 16 . . . . . . . . . 10  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN0 )
4645nn0red 10625 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  RR )
4743, 46remulcld 9402 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
48 peano2re 9530 . . . . . . . 8  |-  ( ( ( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  e.  RR  ->  (
( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  +  1 )  e.  RR )
4947, 48syl 16 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( ( B  -  1 )  x.  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) )  +  1 )  e.  RR )
50 simprl 748 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  B  e.  RR )
51 reexpcl 11866 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN0 )  ->  ( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
5250, 45, 51syl2anc 654 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
53 flltp1 11634 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  /  ( B  -  1 ) )  e.  RR  ->  (
( A  -  1 )  /  ( B  -  1 ) )  <  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) )
5429, 53syl 16 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( A  - 
1 )  /  ( B  -  1 ) )  <  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )
5530adantr 462 . . . . . . . . . 10  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( A  -  1 )  e.  RR )
5625adantl 463 . . . . . . . . . 10  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <  ( B  -  1 ) )
57 ltdivmul 10192 . . . . . . . . . 10  |-  ( ( ( A  -  1 )  e.  RR  /\  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  RR  /\  ( ( B  - 
1 )  e.  RR  /\  0  <  ( B  -  1 ) ) )  ->  ( (
( A  -  1 )  /  ( B  -  1 ) )  <  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 )  <->  ( A  -  1 )  < 
( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) ) ) )
5855, 46, 43, 56, 57syl112anc 1215 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( ( A  -  1 )  / 
( B  -  1 ) )  <  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 )  <-> 
( A  -  1 )  <  ( ( B  -  1 )  x.  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) ) ) )
5954, 58mpbid 210 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( A  -  1 )  <  ( ( B  -  1 )  x.  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) ) )
60 ltsubadd 9797 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  (
( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  e.  RR )  -> 
( ( A  - 
1 )  <  (
( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  <-> 
A  <  ( (
( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  +  1 ) ) )
612, 60mp3an2 1295 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )  ->  ( ( A  -  1 )  < 
( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  <->  A  <  ( ( ( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  +  1 ) ) )
6242, 47, 61syl2anc 654 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( A  - 
1 )  <  (
( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  <-> 
A  <  ( (
( B  -  1 )  x.  ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) )  +  1 ) ) )
6359, 62mpbid 210 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  A  <  ( ( ( B  -  1 )  x.  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) )  +  1 ) )
64 0lt1 9850 . . . . . . . . . . . 12  |-  0  <  1
65 0re 9374 . . . . . . . . . . . . 13  |-  0  e.  RR
66 lttr 9439 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
6765, 2, 66mp3an12 1297 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
6864, 67mpani 669 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
69 ltle 9451 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  ->  0  <_  B )
)
7065, 69mpan 663 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
0  <  B  ->  0  <_  B ) )
7168, 70syld 44 . . . . . . . . . 10  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <_  B ) )
7271imp 429 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <_  B )
7372adantl 463 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <_  B )
74 bernneq2 11975 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN0  /\  0  <_  B )  -> 
( ( ( B  -  1 )  x.  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) )  +  1 )  <_  ( B ^ ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) ) )
7550, 45, 73, 74syl3anc 1211 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( ( B  -  1 )  x.  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) )  +  1 )  <_  ( B ^ ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 ) ) )
7642, 49, 52, 63, 75ltletrd 9519 . . . . . 6  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  A  <  ( B ^
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) ) )
77 oveq2 6088 . . . . . . . 8  |-  ( k  =  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 )  ->  ( B ^ k )  =  ( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) ) )
7877breq2d 4292 . . . . . . 7  |-  ( k  =  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 )  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) ) ) )
7978rspcev 3062 . . . . . 6  |-  ( ( ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN  /\  A  <  ( B ^
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
8041, 76, 79syl2anc 654 . . . . 5  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
8180exp43 607 . . . 4  |-  ( 1  <_  A  ->  ( A  e.  RR  ->  ( B  e.  RR  ->  ( 1  <  B  ->  E. k  e.  NN  A  <  ( B ^
k ) ) ) ) )
8281com4l 84 . . 3  |-  ( A  e.  RR  ->  ( B  e.  RR  ->  ( 1  <  B  -> 
( 1  <_  A  ->  E. k  e.  NN  A  <  ( B ^
k ) ) ) ) )
83823imp1 1193 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <_  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
84 simp1 981 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
85 1red 9389 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
8617, 83, 84, 85ltlecasei 9470 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  E. k  e.  NN  A  <  ( B ^ k ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   NN0cn0 10567   |_cfl 11624   ^cexp 11849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850  df-fl 11626  df-seq 11791  df-exp 11850
This theorem is referenced by:  expnlbnd  11978  expmulnbnd  11980  bitsfzolem  13613  bitsfi  13616  pclem  13888  aaliou3lem8  21696  ostth2lem1  22752  ostth3  22772
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