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Theorem pntrmax 22697
Description: There is a bound on the residual valid for all  x. (Contributed by Mario Carneiro, 9-Apr-2016.)
Hypothesis
Ref Expression
pntrval.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
Assertion
Ref Expression
pntrmax  |-  E. c  e.  RR+  A. x  e.  RR+  ( abs `  (
( R `  x
)  /  x ) )  <_  c
Distinct variable groups:    x, a    x, c, R
Allowed substitution hint:    R( a)

Proof of Theorem pntrmax
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rpssre 10988 . . . 4  |-  RR+  C_  RR
21a1i 11 . . 3  |-  ( T. 
->  RR+  C_  RR )
3 1red 9388 . . 3  |-  ( T. 
->  1  e.  RR )
4 pntrval.r . . . . . . . 8  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
54pntrval 22695 . . . . . . 7  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
6 rpre 10984 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
7 chpcl 22346 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
86, 7syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
98, 6resubcld 9763 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (ψ `  x )  -  x
)  e.  RR )
105, 9eqeltrd 2507 . . . . . 6  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
11 rerpdivcl 11005 . . . . . 6  |-  ( ( ( R `  x
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( R `  x )  /  x
)  e.  RR )
1210, 11mpancom 662 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  RR )
1312recnd 9399 . . . 4  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  CC )
1413adantl 463 . . 3  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( R `  x )  /  x )  e.  CC )
155oveq1d 6095 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  -  x
)  /  x ) )
168recnd 9399 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
17 rpcn 10986 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
18 rpne0 10993 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
1916, 17, 17, 18divsubdird 10133 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  /  x
)  =  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) ) )
2017, 18dividd 10092 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  /  x )  =  1 )
2120oveq2d 6096 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) )  =  ( ( (ψ `  x )  /  x )  -  1 ) )
2215, 19, 213eqtrd 2469 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
2322mpteq2ia 4362 . . . 4  |-  ( x  e.  RR+  |->  ( ( R `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )
24 rerpdivcl 11005 . . . . . . 7  |-  ( ( (ψ `  x )  e.  RR  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
258, 24mpancom 662 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  /  x
)  e.  RR )
2625adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
27 1red 9388 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  1  e.  RR )
28 chpo1ub 22613 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
2928a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  e.  O(1) )
30 ax-1cn 9327 . . . . . . 7  |-  1  e.  CC
31 o1const 13080 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  1  e.  CC )  ->  (
x  e.  RR+  |->  1 )  e.  O(1) )
321, 30, 31mp2an 665 . . . . . 6  |-  ( x  e.  RR+  |->  1 )  e.  O(1)
3332a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  1 )  e.  O(1) )
3426, 27, 29, 33o1sub2 13086 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )  e.  O(1) )
3523, 34syl5eqel 2517 . . 3  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( R `  x )  /  x
) )  e.  O(1) )
36 chpcl 22346 . . . . 5  |-  ( y  e.  RR  ->  (ψ `  y )  e.  RR )
37 peano2re 9529 . . . . 5  |-  ( (ψ `  y )  e.  RR  ->  ( (ψ `  y
)  +  1 )  e.  RR )
3836, 37syl 16 . . . 4  |-  ( y  e.  RR  ->  (
(ψ `  y )  +  1 )  e.  RR )
3938ad2antrl 720 . . 3  |-  ( ( T.  /\  ( y  e.  RR  /\  1  <_  y ) )  -> 
( (ψ `  y
)  +  1 )  e.  RR )
40223ad2ant1 1002 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( R `  x
)  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
4140fveq2d 5683 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  =  ( abs `  (
( (ψ `  x
)  /  x )  -  1 ) ) )
42 1re 9372 . . . . . . . . . 10  |-  1  e.  RR
43383ad2ant2 1003 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  1 )  e.  RR )
44 resubcl 9660 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
4542, 43, 44sylancr 656 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
46 0red 9374 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  e.  RR )
47253ad2ant1 1002 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  e.  RR )
48 chpge0 22348 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  0  <_  (ψ `  y )
)
49483ad2ant2 1003 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  y )
)
50363ad2ant2 1003 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  RR )
51 addge02 9837 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  (ψ `  y )  e.  RR )  ->  (
0  <_  (ψ `  y
)  <->  1  <_  (
(ψ `  y )  +  1 ) ) )
5242, 50, 51sylancr 656 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  <_  (ψ `  y
)  <->  1  <_  (
(ψ `  y )  +  1 ) ) )
5349, 52mpbid 210 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  <_  ( (ψ `  y
)  +  1 ) )
54 suble0 9840 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
( 1  -  (
(ψ `  y )  +  1 ) )  <_  0  <->  1  <_  ( (ψ `  y )  +  1 ) ) )
5542, 43, 54sylancr 656 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( 1  -  (
(ψ `  y )  +  1 ) )  <_  0  <->  1  <_  ( (ψ `  y )  +  1 ) ) )
5653, 55mpbird 232 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  0
)
5783ad2ant1 1002 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  e.  RR )
5863ad2ant1 1002 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR )
59 chpge0 22348 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  0  <_  (ψ `  x )
)
6058, 59syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  x )
)
61 rpregt0 10991 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
62613ad2ant1 1002 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  RR  /\  0  <  x ) )
63 divge0 10185 . . . . . . . . . 10  |-  ( ( ( (ψ `  x
)  e.  RR  /\  0  <_  (ψ `  x
) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
(ψ `  x )  /  x ) )
6457, 60, 62, 63syl21anc 1210 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  ( (ψ `  x
)  /  x ) )
6545, 46, 47, 56, 64letrd 9515 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  (
(ψ `  x )  /  x ) )
66 2re 10378 . . . . . . . . . . 11  |-  2  e.  RR
67 readdcl 9352 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e.  RR )  ->  (
(ψ `  y )  +  2 )  e.  RR )
6850, 66, 67sylancl 655 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  e.  RR )
69 1red 9388 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  e.  RR )
7058adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  e.  RR )
71 1red 9388 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  e.  RR )
7266a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
2  e.  RR )
73 simpr 458 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <_  1 )
74 1lt2 10475 . . . . . . . . . . . . . . . 16  |-  1  <  2
7574a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  <  2 )
7670, 71, 72, 73, 75lelttrd 9516 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <  2 )
77 chpeq0 22431 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  (
(ψ `  x )  =  0  <->  x  <  2 ) )
7870, 77syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  =  0  <->  x  <  2 ) )
7976, 78mpbird 232 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
(ψ `  x )  =  0 )
8079oveq1d 6095 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  =  ( 0  /  x ) )
81 simp1 981 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR+ )
8281rpcnne0d 11023 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
83 div0 10009 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 0  /  x
)  =  0 )
8482, 83syl 16 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  /  x )  =  0 )
8584, 49eqbrtrd 4300 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  /  x )  <_  (ψ `  y
) )
8685adantr 462 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( 0  /  x
)  <_  (ψ `  y
) )
8780, 86eqbrtrd 4300 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
8847adantr 462 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  e.  RR )
8957adantr 462 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  RR )
9050adantr 462 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  y )  e.  RR )
91 0lt1 9849 . . . . . . . . . . . . . . . 16  |-  0  <  1
9291a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <  1 )
93 lediv2a 10213 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( 1  e.  RR  /\  0  <  1 )  /\  (
x  e.  RR  /\  0  <  x )  /\  ( (ψ `  x )  e.  RR  /\  0  <_ 
(ψ `  x )
) )  /\  1  <_  x )  ->  (
(ψ `  x )  /  x )  <_  (
(ψ `  x )  /  1 ) )
9493ex 434 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  e.  RR  /\  0  <  1 )  /\  ( x  e.  RR  /\  0  < 
x )  /\  (
(ψ `  x )  e.  RR  /\  0  <_ 
(ψ `  x )
) )  ->  (
1  <_  x  ->  ( (ψ `  x )  /  x )  <_  (
(ψ `  x )  /  1 ) ) )
9569, 92, 62, 57, 60, 94syl212anc 1221 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  <_  x  ->  ( (ψ `  x )  /  x )  <_  (
(ψ `  x )  /  1 ) ) )
9695imp 429 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  ( (ψ `  x )  /  1
) )
9789recnd 9399 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  CC )
9897div1d 10086 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  1 )  =  (ψ `  x
) )
9996, 98breqtrd 4304 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  x
) )
100 simp2 982 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  y  e.  RR )
101 ltle 9450 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  x  <_  y )
)
1026, 101sylan 468 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR )  ->  (
x  <  y  ->  x  <_  y ) )
1031023impia 1177 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  <_  y )
104 chpwordi 22379 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <_  y )  ->  (ψ `  x )  <_  (ψ `  y ) )
10558, 100, 103, 104syl3anc 1211 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  <_  (ψ `  y ) )
106105adantr 462 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  <_  (ψ `  y )
)
10788, 89, 90, 99, 106letrd 9515 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
10858, 69, 87, 107lecasei 9467 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (ψ `  y ) )
109 2nn0 10583 . . . . . . . . . . 11  |-  2  e.  NN0
110 nn0addge1 10613 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e. 
NN0 )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11150, 109, 110sylancl 655 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11247, 50, 68, 108, 111letrd 9515 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
(ψ `  y )  +  2 ) )
113 df-2 10367 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
114113oveq2i 6091 . . . . . . . . . 10  |-  ( (ψ `  y )  +  2 )  =  ( (ψ `  y )  +  ( 1  +  1 ) )
11550recnd 9399 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  CC )
11630a1i 11 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  e.  CC )
117115, 116, 116add12d 9578 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  ( 1  +  1 ) )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
118114, 117syl5eq 2477 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
119112, 118breqtrd 4304 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
1  +  ( (ψ `  y )  +  1 ) ) )
12047, 69, 43absdifled 12904 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( abs `  (
( (ψ `  x
)  /  x )  -  1 ) )  <_  ( (ψ `  y )  +  1 )  <->  ( ( 1  -  ( (ψ `  y )  +  1 ) )  <_  (
(ψ `  x )  /  x )  /\  (
(ψ `  x )  /  x )  <_  (
1  +  ( (ψ `  y )  +  1 ) ) ) ) )
12165, 119, 120mpbir2and 906 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( (ψ `  x )  /  x
)  -  1 ) )  <_  ( (ψ `  y )  +  1 ) )
12241, 121eqbrtrd 4300 . . . . . 6  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
( (ψ `  y
)  +  1 ) )
1231223expb 1181 . . . . 5  |-  ( ( x  e.  RR+  /\  (
y  e.  RR  /\  x  <  y ) )  ->  ( abs `  (
( R `  x
)  /  x ) )  <_  ( (ψ `  y )  +  1 ) )
124123adantrlr 715 . . . 4  |-  ( ( x  e.  RR+  /\  (
( y  e.  RR  /\  1  <_  y )  /\  x  <  y ) )  ->  ( abs `  ( ( R `  x )  /  x
) )  <_  (
(ψ `  y )  +  1 ) )
125124adantll 706 . . 3  |-  ( ( ( T.  /\  x  e.  RR+ )  /\  (
( y  e.  RR  /\  1  <_  y )  /\  x  <  y ) )  ->  ( abs `  ( ( R `  x )  /  x
) )  <_  (
(ψ `  y )  +  1 ) )
1262, 3, 14, 35, 39, 125o1bddrp 13003 . 2  |-  ( T. 
->  E. c  e.  RR+  A. x  e.  RR+  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
c )
127126trud 1371 1  |-  E. c  e.  RR+  A. x  e.  RR+  ( abs `  (
( R `  x
)  /  x ) )  <_  c
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 958    = wceq 1362   T. wtru 1363    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    < clt 9405    <_ cle 9406    - cmin 9582    / cdiv 9980   2c2 10358   NN0cn0 10566   RR+crp 10978   abscabs 12706   O(1)co1 12947  ψcchp 22314
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-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  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-int 4117  df-iun 4161  df-iin 4162  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-se 4667  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-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-o1 12951  df-lo1 12952  df-sum 13147  df-ef 13335  df-e 13336  df-sin 13337  df-cos 13338  df-pi 13340  df-dvds 13518  df-gcd 13673  df-prm 13746  df-pc 13886  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892  df-cxp 21893  df-cht 22318  df-vma 22319  df-chp 22320  df-ppi 22321
This theorem is referenced by:  pntrlog2bnd  22717  pntibnd  22726  pnt3  22745
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