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Theorem pntrmax 23474
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 11226 . . . 4  |-  RR+  C_  RR
21a1i 11 . . 3  |-  ( T. 
->  RR+  C_  RR )
3 1red 9607 . . 3  |-  ( T. 
->  1  e.  RR )
4 pntrval.r . . . . . . . 8  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
54pntrval 23472 . . . . . . 7  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
6 rpre 11222 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
7 chpcl 23123 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
86, 7syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
98, 6resubcld 9983 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (ψ `  x )  -  x
)  e.  RR )
105, 9eqeltrd 2555 . . . . . 6  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
11 rerpdivcl 11243 . . . . . 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 9618 . . . 4  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  CC )
1413adantl 466 . . 3  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( R `  x )  /  x )  e.  CC )
155oveq1d 6297 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  -  x
)  /  x ) )
168recnd 9618 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
17 rpcn 11224 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
18 rpne0 11231 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
1916, 17, 17, 18divsubdird 10355 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  /  x
)  =  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) ) )
2017, 18dividd 10314 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  /  x )  =  1 )
2120oveq2d 6298 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) )  =  ( ( (ψ `  x )  /  x )  -  1 ) )
2215, 19, 213eqtrd 2512 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
2322mpteq2ia 4529 . . . 4  |-  ( x  e.  RR+  |->  ( ( R `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )
24 rerpdivcl 11243 . . . . . . 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 9607 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  1  e.  RR )
28 chpo1ub 23390 . . . . . 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 9546 . . . . . . 7  |-  1  e.  CC
31 o1const 13398 . . . . . . 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 13404 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )  e.  O(1) )
3523, 34syl5eqel 2559 . . 3  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( R `  x )  /  x
) )  e.  O(1) )
36 chpcl 23123 . . . . 5  |-  ( y  e.  RR  ->  (ψ `  y )  e.  RR )
37 peano2re 9748 . . . . 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 1017 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( R `  x
)  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
4140fveq2d 5868 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  =  ( abs `  (
( (ψ `  x
)  /  x )  -  1 ) ) )
42 1re 9591 . . . . . . . . . 10  |-  1  e.  RR
43383ad2ant2 1018 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  1 )  e.  RR )
44 resubcl 9879 . . . . . . . . . 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 9593 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  e.  RR )
47253ad2ant1 1017 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  e.  RR )
48 chpge0 23125 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  0  <_  (ψ `  y )
)
49483ad2ant2 1018 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  y )
)
50363ad2ant2 1018 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  RR )
51 addge02 10059 . . . . . . . . . . . 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 10062 . . . . . . . . . . 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 1017 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  e.  RR )
5863ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR )
59 chpge0 23125 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  0  <_  (ψ `  x )
)
6058, 59syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  x )
)
61 rpregt0 11229 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
62613ad2ant1 1017 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  RR  /\  0  <  x ) )
63 divge0 10407 . . . . . . . . . 10  |-  ( ( ( (ψ `  x
)  e.  RR  /\  0  <_  (ψ `  x
) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
(ψ `  x )  /  x ) )
6457, 60, 62, 63syl21anc 1227 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  ( (ψ `  x
)  /  x ) )
6545, 46, 47, 56, 64letrd 9734 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  (
(ψ `  x )  /  x ) )
66 2re 10601 . . . . . . . . . . 11  |-  2  e.  RR
67 readdcl 9571 . . . . . . . . . . 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 9607 . . . . . . . . . . 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 9607 . . . . . . . . . . . . . . 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 10698 . . . . . . . . . . . . . . . 16  |-  1  <  2
7574a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  <  2 )
7670, 71, 72, 73, 75lelttrd 9735 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <  2 )
77 chpeq0 23208 . . . . . . . . . . . . . . 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 6297 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  =  ( 0  /  x ) )
81 simp1 996 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR+ )
8281rpcnne0d 11261 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
83 div0 10231 . . . . . . . . . . . . . . 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 4467 . . . . . . . . . . . . 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 4467 . . . . . . . . . . 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 10071 . . . . . . . . . . . . . . . 16  |-  0  <  1
9291a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <  1 )
93 lediv2a 10435 . . . . . . . . . . . . . . . 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 1238 . . . . . . . . . . . . . 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 9618 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  CC )
9897div1d 10308 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  1 )  =  (ψ `  x
) )
9996, 98breqtrd 4471 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  x
) )
100 simp2 997 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  y  e.  RR )
101 ltle 9669 . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  <_  y )
104 chpwordi 23156 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <_  y )  ->  (ψ `  x )  <_  (ψ `  y ) )
10558, 100, 103, 104syl3anc 1228 . . . . . . . . . . . . 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 9734 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
10858, 69, 87, 107lecasei 9686 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (ψ `  y ) )
109 2nn0 10808 . . . . . . . . . . 11  |-  2  e.  NN0
110 nn0addge1 10838 . . . . . . . . . . 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 9734 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
(ψ `  y )  +  2 ) )
113 df-2 10590 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
114113oveq2i 6293 . . . . . . . . . 10  |-  ( (ψ `  y )  +  2 )  =  ( (ψ `  y )  +  ( 1  +  1 ) )
11550recnd 9618 . . . . . . . . . . 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 9797 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  ( 1  +  1 ) )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
118114, 117syl5eq 2520 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
119112, 118breqtrd 4471 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
1  +  ( (ψ `  y )  +  1 ) ) )
12047, 69, 43absdifled 13222 . . . . . . . 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 920 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( (ψ `  x )  /  x
)  -  1 ) )  <_  ( (ψ `  y )  +  1 ) )
12241, 121eqbrtrd 4467 . . . . . 6  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
( (ψ `  y
)  +  1 ) )
1231223expb 1197 . . . . 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 13321 . 2  |-  ( T. 
->  E. c  e.  RR+  A. x  e.  RR+  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
c )
127126trud 1388 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 973    = wceq 1379   T. wtru 1380    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    < clt 9624    <_ cle 9625    - cmin 9801    / cdiv 10202   2c2 10581   NN0cn0 10791   RR+crp 11216   abscabs 13024   O(1)co1 13265  ψcchp 23091
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  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  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  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-int 4283  df-iun 4327  df-iin 4328  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-se 4839  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-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  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-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-mod 11960  df-seq 12071  df-exp 12130  df-fac 12316  df-bc 12343  df-hash 12368  df-shft 12857  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-o1 13269  df-lo1 13270  df-sum 13465  df-ef 13658  df-e 13659  df-sin 13660  df-cos 13661  df-pi 13663  df-dvds 13841  df-gcd 13997  df-prm 14070  df-pc 14213  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-fbas 18184  df-fg 18185  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-ntr 19284  df-cls 19285  df-nei 19362  df-lp 19400  df-perf 19401  df-cn 19491  df-cnp 19492  df-haus 19579  df-tx 19795  df-hmeo 19988  df-fil 20079  df-fm 20171  df-flim 20172  df-flf 20173  df-xms 20555  df-ms 20556  df-tms 20557  df-cncf 21114  df-limc 22002  df-dv 22003  df-log 22669  df-cxp 22670  df-cht 23095  df-vma 23096  df-chp 23097  df-ppi 23098
This theorem is referenced by:  pntrlog2bnd  23494  pntibnd  23503  pnt3  23522
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