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Theorem expnbnd 12257
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 10543 . . 3  |-  1  e.  NN
2 1re 9591 . . . . . . . 8  |-  1  e.  RR
3 lttr 9657 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( A  <  1  /\  1  <  B )  ->  A  <  B
) )
42, 3mp3an2 1312 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  <  1  /\  1  < 
B )  ->  A  <  B ) )
54exp4b 607 . . . . . 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 1209 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  A  <  B )
8 recn 9578 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
9 exp1 12135 . . . . . . 7  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
108, 9syl 16 . . . . . 6  |-  ( B  e.  RR  ->  ( B ^ 1 )  =  B )
11103ad2ant2 1018 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  ( B ^ 1 )  =  B )
1211adantr 465 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  -> 
( B ^ 1 )  =  B )
137, 12breqtrrd 4473 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  A  <  ( B ^
1 ) )
14 oveq2 6290 . . . . 5  |-  ( k  =  1  ->  ( B ^ k )  =  ( B ^ 1 ) )
1514breq2d 4459 . . . 4  |-  ( k  =  1  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ 1 ) ) )
1615rspcev 3214 . . 3  |-  ( ( 1  e.  NN  /\  A  <  ( B ^
1 ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
171, 13, 16sylancr 663 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  A  <  1 )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
18 peano2rem 9882 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
1918adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( A  -  1 )  e.  RR )
20 peano2rem 9882 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
2120adantr 465 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( B  -  1 )  e.  RR )
2221adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( B  -  1 )  e.  RR )
23 posdif 10041 . . . . . . . . . . . . . 14  |-  ( ( 1  e.  RR  /\  B  e.  RR )  ->  ( 1  <  B  <->  0  <  ( B  - 
1 ) ) )
242, 23mpan 670 . . . . . . . . . . . . 13  |-  ( B  e.  RR  ->  (
1  <  B  <->  0  <  ( B  -  1 ) ) )
2524biimpa 484 . . . . . . . . . . . 12  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
0  <  ( B  -  1 ) )
2625gt0ne0d 10113 . . . . . . . . . . 11  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( B  -  1 )  =/=  0 )
2726adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( B  -  1 )  =/=  0 )
2819, 22, 27redivcld 10368 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  1  <  B ) )  ->  ( ( A  -  1 )  /  ( B  - 
1 ) )  e.  RR )
2928adantll 713 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( A  - 
1 )  /  ( B  -  1 ) )  e.  RR )
3018adantl 466 . . . . . . . . . 10  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  ( A  -  1 )  e.  RR )
31 subge0 10061 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( 0  <_  ( A  -  1 )  <->  1  <_  A )
)
322, 31mpan2 671 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  (
0  <_  ( A  -  1 )  <->  1  <_  A ) )
3332biimparc 487 . . . . . . . . . 10  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  0  <_  ( A  -  1 ) )
3430, 33jca 532 . . . . . . . . 9  |-  ( ( 1  <_  A  /\  A  e.  RR )  ->  ( ( A  - 
1 )  e.  RR  /\  0  <_  ( A  -  1 ) ) )
3521, 25jca 532 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  1  <  B )  -> 
( ( B  - 
1 )  e.  RR  /\  0  <  ( B  -  1 ) ) )
36 divge0 10407 . . . . . . . . 9  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  0  <_  ( A  -  1 ) )  /\  ( ( B  -  1 )  e.  RR  /\  0  < 
( B  -  1 ) ) )  -> 
0  <_  ( ( A  -  1 )  /  ( B  - 
1 ) ) )
3734, 35, 36syl2an 477 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <_  ( ( A  -  1 )  /  ( B  - 
1 ) ) )
38 flge0nn0 11917 . . . . . . . 8  |-  ( ( ( ( A  - 
1 )  /  ( B  -  1 ) )  e.  RR  /\  0  <_  ( ( A  -  1 )  / 
( B  -  1 ) ) )  -> 
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  e.  NN0 )
3929, 37, 38syl2anc 661 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  e.  NN0 )
40 nn0p1nn 10831 . . . . . . 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 754 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  A  e.  RR )
4321adantl 466 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( B  -  1 )  e.  RR )
44 peano2nn0 10832 . . . . . . . . . . 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 10849 . . . . . . . . 9  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  RR )
4743, 46remulcld 9620 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( ( B  - 
1 )  x.  (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
48 peano2re 9748 . . . . . . . 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 755 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  B  e.  RR )
51 reexpcl 12146 . . . . . . . 8  |-  ( ( B  e.  RR  /\  ( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 )  e.  NN0 )  ->  ( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
5250, 45, 51syl2anc 661 . . . . . . 7  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) )  e.  RR )
53 flltp1 11901 . . . . . . . . . 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 465 . . . . . . . . . 10  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
( A  -  1 )  e.  RR )
5625adantl 466 . . . . . . . . . 10  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <  ( B  -  1 ) )
57 ltdivmul 10413 . . . . . . . . . 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 1232 . . . . . . . . 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 10018 . . . . . . . . . 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 1312 . . . . . . . . 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 661 . . . . . . . 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 10071 . . . . . . . . . . . 12  |-  0  <  1
65 0re 9592 . . . . . . . . . . . . 13  |-  0  e.  RR
66 lttr 9657 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  B  e.  RR )  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
6765, 2, 66mp3an12 1314 . . . . . . . . . . . 12  |-  ( B  e.  RR  ->  (
( 0  <  1  /\  1  <  B )  ->  0  <  B
) )
6864, 67mpani 676 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  (
1  <  B  ->  0  <  B ) )
69 ltle 9669 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  ->  0  <_  B )
)
7065, 69mpan 670 . . . . . . . . . . 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 466 . . . . . . . 8  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  -> 
0  <_  B )
74 bernneq2 12255 . . . . . . . 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 1228 . . . . . . 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 9737 . . . . . 6  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  A  <  ( B ^
( ( |_ `  ( ( A  - 
1 )  /  ( B  -  1 ) ) )  +  1 ) ) )
77 oveq2 6290 . . . . . . . 8  |-  ( k  =  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 )  ->  ( B ^ k )  =  ( B ^ (
( |_ `  (
( A  -  1 )  /  ( B  -  1 ) ) )  +  1 ) ) )
7877breq2d 4459 . . . . . . 7  |-  ( k  =  ( ( |_
`  ( ( A  -  1 )  / 
( B  -  1 ) ) )  +  1 )  ->  ( A  <  ( B ^
k )  <->  A  <  ( B ^ ( ( |_ `  ( ( A  -  1 )  /  ( B  - 
1 ) ) )  +  1 ) ) ) )
7978rspcev 3214 . . . . . 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 661 . . . . 5  |-  ( ( ( 1  <_  A  /\  A  e.  RR )  /\  ( B  e.  RR  /\  1  < 
B ) )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
8180exp43 612 . . . 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 1209 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  /\  1  <_  A )  ->  E. k  e.  NN  A  <  ( B ^
k ) )
84 simp1 996 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  A  e.  RR )
85 1red 9607 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  1  <  B )  ->  1  e.  RR )
8617, 83, 84, 85ltlecasei 9688 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   NNcn 10532   NN0cn0 10791   |_cfl 11891   ^cexp 12129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fl 11893  df-seq 12071  df-exp 12130
This theorem is referenced by:  expnlbnd  12258  expmulnbnd  12260  bitsfzolem  13936  bitsfi  13939  pclem  14214  aaliou3lem8  22472  ostth2lem1  23528  ostth3  23548
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