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Theorem pntrmax 22941
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 11107 . . . 4  |-  RR+  C_  RR
21a1i 11 . . 3  |-  ( T. 
->  RR+  C_  RR )
3 1red 9507 . . 3  |-  ( T. 
->  1  e.  RR )
4 pntrval.r . . . . . . . 8  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
54pntrval 22939 . . . . . . 7  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
6 rpre 11103 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
7 chpcl 22590 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
86, 7syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
98, 6resubcld 9882 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (ψ `  x )  -  x
)  e.  RR )
105, 9eqeltrd 2540 . . . . . 6  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
11 rerpdivcl 11124 . . . . . 6  |-  ( ( ( R `  x
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( R `  x )  /  x
)  e.  RR )
1210, 11mpancom 669 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  RR )
1312recnd 9518 . . . 4  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  CC )
1413adantl 466 . . 3  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( R `  x )  /  x )  e.  CC )
155oveq1d 6210 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  -  x
)  /  x ) )
168recnd 9518 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
17 rpcn 11105 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
18 rpne0 11112 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
1916, 17, 17, 18divsubdird 10252 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  /  x
)  =  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) ) )
2017, 18dividd 10211 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  /  x )  =  1 )
2120oveq2d 6211 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) )  =  ( ( (ψ `  x )  /  x )  -  1 ) )
2215, 19, 213eqtrd 2497 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
2322mpteq2ia 4477 . . . 4  |-  ( x  e.  RR+  |->  ( ( R `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )
24 rerpdivcl 11124 . . . . . . 7  |-  ( ( (ψ `  x )  e.  RR  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
258, 24mpancom 669 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  /  x
)  e.  RR )
2625adantl 466 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
27 1red 9507 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  1  e.  RR )
28 chpo1ub 22857 . . . . . 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 9446 . . . . . . 7  |-  1  e.  CC
31 o1const 13210 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  1  e.  CC )  ->  (
x  e.  RR+  |->  1 )  e.  O(1) )
321, 30, 31mp2an 672 . . . . . 6  |-  ( x  e.  RR+  |->  1 )  e.  O(1)
3332a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  1 )  e.  O(1) )
3426, 27, 29, 33o1sub2 13216 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )  e.  O(1) )
3523, 34syl5eqel 2544 . . 3  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( R `  x )  /  x
) )  e.  O(1) )
36 chpcl 22590 . . . . 5  |-  ( y  e.  RR  ->  (ψ `  y )  e.  RR )
37 peano2re 9648 . . . . 5  |-  ( (ψ `  y )  e.  RR  ->  ( (ψ `  y
)  +  1 )  e.  RR )
3836, 37syl 16 . . . 4  |-  ( y  e.  RR  ->  (
(ψ `  y )  +  1 )  e.  RR )
3938ad2antrl 727 . . 3  |-  ( ( T.  /\  ( y  e.  RR  /\  1  <_  y ) )  -> 
( (ψ `  y
)  +  1 )  e.  RR )
40223ad2ant1 1009 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( R `  x
)  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
4140fveq2d 5798 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  =  ( abs `  (
( (ψ `  x
)  /  x )  -  1 ) ) )
42 1re 9491 . . . . . . . . . 10  |-  1  e.  RR
43383ad2ant2 1010 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  1 )  e.  RR )
44 resubcl 9779 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
4542, 43, 44sylancr 663 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
46 0red 9493 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  e.  RR )
47253ad2ant1 1009 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  e.  RR )
48 chpge0 22592 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  0  <_  (ψ `  y )
)
49483ad2ant2 1010 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  y )
)
50363ad2ant2 1010 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  RR )
51 addge02 9956 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  (ψ `  y )  e.  RR )  ->  (
0  <_  (ψ `  y
)  <->  1  <_  (
(ψ `  y )  +  1 ) ) )
5242, 50, 51sylancr 663 . . . . . . . . . . 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 9959 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
( 1  -  (
(ψ `  y )  +  1 ) )  <_  0  <->  1  <_  ( (ψ `  y )  +  1 ) ) )
5542, 43, 54sylancr 663 . . . . . . . . . 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 1009 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  e.  RR )
5863ad2ant1 1009 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR )
59 chpge0 22592 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  0  <_  (ψ `  x )
)
6058, 59syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  x )
)
61 rpregt0 11110 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
62613ad2ant1 1009 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  RR  /\  0  <  x ) )
63 divge0 10304 . . . . . . . . . 10  |-  ( ( ( (ψ `  x
)  e.  RR  /\  0  <_  (ψ `  x
) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
(ψ `  x )  /  x ) )
6457, 60, 62, 63syl21anc 1218 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  ( (ψ `  x
)  /  x ) )
6545, 46, 47, 56, 64letrd 9634 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  (
(ψ `  x )  /  x ) )
66 2re 10497 . . . . . . . . . . 11  |-  2  e.  RR
67 readdcl 9471 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e.  RR )  ->  (
(ψ `  y )  +  2 )  e.  RR )
6850, 66, 67sylancl 662 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  e.  RR )
69 1red 9507 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  e.  RR )
7058adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  e.  RR )
71 1red 9507 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <_  1 )
74 1lt2 10594 . . . . . . . . . . . . . . . 16  |-  1  <  2
7574a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  <  2 )
7670, 71, 72, 73, 75lelttrd 9635 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <  2 )
77 chpeq0 22675 . . . . . . . . . . . . . . 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 6210 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  =  ( 0  /  x ) )
81 simp1 988 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR+ )
8281rpcnne0d 11142 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
83 div0 10128 . . . . . . . . . . . . . . 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 4415 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  /  x )  <_  (ψ `  y
) )
8685adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( 0  /  x
)  <_  (ψ `  y
) )
8780, 86eqbrtrd 4415 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
8847adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  e.  RR )
8957adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  RR )
9050adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  y )  e.  RR )
91 0lt1 9968 . . . . . . . . . . . . . . . 16  |-  0  <  1
9291a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <  1 )
93 lediv2a 10332 . . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . 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 9518 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  CC )
9897div1d 10205 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  1 )  =  (ψ `  x
) )
9996, 98breqtrd 4419 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  x
) )
100 simp2 989 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  y  e.  RR )
101 ltle 9569 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  x  <_  y )
)
1026, 101sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR )  ->  (
x  <  y  ->  x  <_  y ) )
1031023impia 1185 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  <_  y )
104 chpwordi 22623 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <_  y )  ->  (ψ `  x )  <_  (ψ `  y ) )
10558, 100, 103, 104syl3anc 1219 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  <_  (ψ `  y ) )
106105adantr 465 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  <_  (ψ `  y )
)
10788, 89, 90, 99, 106letrd 9634 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
10858, 69, 87, 107lecasei 9586 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (ψ `  y ) )
109 2nn0 10702 . . . . . . . . . . 11  |-  2  e.  NN0
110 nn0addge1 10732 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e. 
NN0 )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11150, 109, 110sylancl 662 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11247, 50, 68, 108, 111letrd 9634 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
(ψ `  y )  +  2 ) )
113 df-2 10486 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
114113oveq2i 6206 . . . . . . . . . 10  |-  ( (ψ `  y )  +  2 )  =  ( (ψ `  y )  +  ( 1  +  1 ) )
11550recnd 9518 . . . . . . . . . . 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 9697 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  ( 1  +  1 ) )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
118114, 117syl5eq 2505 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
119112, 118breqtrd 4419 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
1  +  ( (ψ `  y )  +  1 ) ) )
12047, 69, 43absdifled 13034 . . . . . . . 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 913 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( (ψ `  x )  /  x
)  -  1 ) )  <_  ( (ψ `  y )  +  1 ) )
12241, 121eqbrtrd 4415 . . . . . 6  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
( (ψ `  y
)  +  1 ) )
1231223expb 1189 . . . . 5  |-  ( ( x  e.  RR+  /\  (
y  e.  RR  /\  x  <  y ) )  ->  ( abs `  (
( R `  x
)  /  x ) )  <_  ( (ψ `  y )  +  1 ) )
124123adantrlr 722 . . . 4  |-  ( ( x  e.  RR+  /\  (
( y  e.  RR  /\  1  <_  y )  /\  x  <  y ) )  ->  ( abs `  ( ( R `  x )  /  x
) )  <_  (
(ψ `  y )  +  1 ) )
125124adantll 713 . . 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 13133 . 2  |-  ( T. 
->  E. c  e.  RR+  A. x  e.  RR+  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
c )
127126trud 1379 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 965    = wceq 1370   T. wtru 1371    e. wcel 1758    =/= wne 2645   A.wral 2796   E.wrex 2797    C_ wss 3431   class class class wbr 4395    |-> cmpt 4453   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389    + caddc 9391    < clt 9524    <_ cle 9525    - cmin 9701    / cdiv 10099   2c2 10477   NN0cn0 10685   RR+crp 11097   abscabs 12836   O(1)co1 13077  ψcchp 22558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-ioo 11410  df-ioc 11411  df-ico 11412  df-icc 11413  df-fz 11550  df-fzo 11661  df-fl 11754  df-mod 11821  df-seq 11919  df-exp 11978  df-fac 12164  df-bc 12191  df-hash 12216  df-shft 12669  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-limsup 13062  df-clim 13079  df-rlim 13080  df-o1 13081  df-lo1 13082  df-sum 13277  df-ef 13466  df-e 13467  df-sin 13468  df-cos 13469  df-pi 13471  df-dvds 13649  df-gcd 13804  df-prm 13877  df-pc 14017  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-mulg 15662  df-cntz 15949  df-cmn 16395  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-lp 18867  df-perf 18868  df-cn 18958  df-cnp 18959  df-haus 19046  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-xms 20022  df-ms 20023  df-tms 20024  df-cncf 20581  df-limc 21469  df-dv 21470  df-log 22136  df-cxp 22137  df-cht 22562  df-vma 22563  df-chp 22564  df-ppi 22565
This theorem is referenced by:  pntrlog2bnd  22961  pntibnd  22970  pnt3  22989
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