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Theorem pntrmax 20713
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 10364 . . . 4  |-  RR+  C_  RR
21a1i 10 . . 3  |-  (  T. 
->  RR+  C_  RR )
3 1re 8837 . . . 4  |-  1  e.  RR
43a1i 10 . . 3  |-  (  T. 
->  1  e.  RR )
5 pntrval.r . . . . . . . 8  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
65pntrval 20711 . . . . . . 7  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
7 rpre 10360 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
8 chpcl 20362 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
97, 8syl 15 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
109, 7resubcld 9211 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (ψ `  x )  -  x
)  e.  RR )
116, 10eqeltrd 2357 . . . . . 6  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
12 rerpdivcl 10381 . . . . . 6  |-  ( ( ( R `  x
)  e.  RR  /\  x  e.  RR+ )  -> 
( ( R `  x )  /  x
)  e.  RR )
1311, 12mpancom 650 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  RR )
1413recnd 8861 . . . 4  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  e.  CC )
1514adantl 452 . . 3  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( R `  x )  /  x )  e.  CC )
166oveq1d 5873 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  -  x
)  /  x ) )
179recnd 8861 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
18 rpcn 10362 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
19 rpne0 10369 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
2017, 18, 18, 19divsubdird 9575 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  /  x
)  =  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) ) )
2118, 19dividd 9534 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  /  x )  =  1 )
2221oveq2d 5874 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  /  x )  -  (
x  /  x ) )  =  ( ( (ψ `  x )  /  x )  -  1 ) )
2316, 20, 223eqtrd 2319 . . . . 5  |-  ( x  e.  RR+  ->  ( ( R `  x )  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
2423mpteq2ia 4102 . . . 4  |-  ( x  e.  RR+  |->  ( ( R `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )
25 rerpdivcl 10381 . . . . . . 7  |-  ( ( (ψ `  x )  e.  RR  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
269, 25mpancom 650 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  /  x
)  e.  RR )
2726adantl 452 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
283a1i 10 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  1  e.  RR )
29 chpo1ub 20629 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O
( 1 )
3029a1i 10 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  e.  O ( 1 ) )
31 ax-1cn 8795 . . . . . . 7  |-  1  e.  CC
32 o1const 12093 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  1  e.  CC )  ->  (
x  e.  RR+  |->  1 )  e.  O ( 1 ) )
331, 31, 32mp2an 653 . . . . . 6  |-  ( x  e.  RR+  |->  1 )  e.  O ( 1 )
3433a1i 10 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  1 )  e.  O
( 1 ) )
3527, 28, 30, 34o1sub2 12099 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  -  1 ) )  e.  O ( 1 ) )
3624, 35syl5eqel 2367 . . 3  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( R `  x )  /  x
) )  e.  O
( 1 ) )
37 chpcl 20362 . . . . 5  |-  ( y  e.  RR  ->  (ψ `  y )  e.  RR )
38 peano2re 8985 . . . . 5  |-  ( (ψ `  y )  e.  RR  ->  ( (ψ `  y
)  +  1 )  e.  RR )
3937, 38syl 15 . . . 4  |-  ( y  e.  RR  ->  (
(ψ `  y )  +  1 )  e.  RR )
4039ad2antrl 708 . . 3  |-  ( (  T.  /\  ( y  e.  RR  /\  1  <_  y ) )  -> 
( (ψ `  y
)  +  1 )  e.  RR )
41233ad2ant1 976 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( R `  x
)  /  x )  =  ( ( (ψ `  x )  /  x
)  -  1 ) )
4241fveq2d 5529 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  =  ( abs `  (
( (ψ `  x
)  /  x )  -  1 ) ) )
43393ad2ant2 977 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  1 )  e.  RR )
44 resubcl 9111 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
453, 43, 44sylancr 644 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  e.  RR )
46 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
4746a1i 10 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  e.  RR )
48263ad2ant1 976 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  e.  RR )
49 chpge0 20364 . . . . . . . . . . . 12  |-  ( y  e.  RR  ->  0  <_  (ψ `  y )
)
50493ad2ant2 977 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  y )
)
51373ad2ant2 977 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  RR )
52 addge02 9285 . . . . . . . . . . . 12  |-  ( ( 1  e.  RR  /\  (ψ `  y )  e.  RR )  ->  (
0  <_  (ψ `  y
)  <->  1  <_  (
(ψ `  y )  +  1 ) ) )
533, 51, 52sylancr 644 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  <_  (ψ `  y
)  <->  1  <_  (
(ψ `  y )  +  1 ) ) )
5450, 53mpbid 201 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  <_  ( (ψ `  y
)  +  1 ) )
55 suble0 9288 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  ( (ψ `  y )  +  1 )  e.  RR )  ->  (
( 1  -  (
(ψ `  y )  +  1 ) )  <_  0  <->  1  <_  ( (ψ `  y )  +  1 ) ) )
563, 43, 55sylancr 644 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
( 1  -  (
(ψ `  y )  +  1 ) )  <_  0  <->  1  <_  ( (ψ `  y )  +  1 ) ) )
5754, 56mpbird 223 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  0
)
5893ad2ant1 976 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  e.  RR )
5973ad2ant1 976 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR )
60 chpge0 20364 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  0  <_  (ψ `  x )
)
6159, 60syl 15 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  (ψ `  x )
)
62 rpregt0 10367 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
63623ad2ant1 976 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  RR  /\  0  <  x ) )
64 divge0 9625 . . . . . . . . . 10  |-  ( ( ( (ψ `  x
)  e.  RR  /\  0  <_  (ψ `  x
) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
(ψ `  x )  /  x ) )
6558, 61, 63, 64syl21anc 1181 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <_  ( (ψ `  x
)  /  x ) )
6645, 47, 48, 57, 65letrd 8973 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  -  ( (ψ `  y )  +  1 ) )  <_  (
(ψ `  x )  /  x ) )
67 2re 9815 . . . . . . . . . . 11  |-  2  e.  RR
68 readdcl 8820 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e.  RR )  ->  (
(ψ `  y )  +  2 )  e.  RR )
6951, 67, 68sylancl 643 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  e.  RR )
703a1i 10 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  e.  RR )
7159adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  e.  RR )
723a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  e.  RR )
7367a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
2  e.  RR )
74 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <_  1 )
75 1lt2 9886 . . . . . . . . . . . . . . . 16  |-  1  <  2
7675a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
1  <  2 )
7771, 72, 73, 74, 76lelttrd 8974 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  ->  x  <  2 )
78 chpeq0 20447 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR  ->  (
(ψ `  x )  =  0  <->  x  <  2 ) )
7971, 78syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  =  0  <->  x  <  2 ) )
8077, 79mpbird 223 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
(ψ `  x )  =  0 )
8180oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  =  ( 0  /  x ) )
82 simp1 955 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  e.  RR+ )
8382rpcnne0d 10399 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
x  e.  CC  /\  x  =/=  0 ) )
84 div0 9452 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 0  /  x
)  =  0 )
8583, 84syl 15 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  /  x )  =  0 )
8685, 50eqbrtrd 4043 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
0  /  x )  <_  (ψ `  y
) )
8786adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( 0  /  x
)  <_  (ψ `  y
) )
8881, 87eqbrtrd 4043 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  x  <_  1 )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
8948adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  e.  RR )
9058adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  RR )
9151adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  y )  e.  RR )
92 0lt1 9296 . . . . . . . . . . . . . . . 16  |-  0  <  1
9392a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  0  <  1 )
94 lediv2a 9650 . . . . . . . . . . . . . . . 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 ) )
9594ex 423 . . . . . . . . . . . . . . 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 ) ) )
9670, 93, 63, 58, 61, 95syl212anc 1192 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
1  <_  x  ->  ( (ψ `  x )  /  x )  <_  (
(ψ `  x )  /  1 ) ) )
9796imp 418 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  ( (ψ `  x )  /  1
) )
9890recnd 8861 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  e.  CC )
9998div1d 9528 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  1 )  =  (ψ `  x
) )
10097, 99breqtrd 4047 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  x
) )
101 simp2 956 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  y  e.  RR )
102 ltle 8910 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  ->  x  <_  y )
)
1037, 102sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  y  e.  RR )  ->  (
x  <  y  ->  x  <_  y ) )
1041033impia 1148 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  x  <_  y )
105 chpwordi 20395 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <_  y )  ->  (ψ `  x )  <_  (ψ `  y ) )
10659, 101, 104, 105syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  x )  <_  (ψ `  y ) )
107106adantr 451 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
(ψ `  x )  <_  (ψ `  y )
)
10889, 90, 91, 100, 107letrd 8973 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR+  /\  y  e.  RR  /\  x  <  y )  /\  1  <_  x )  -> 
( (ψ `  x
)  /  x )  <_  (ψ `  y
) )
10959, 70, 88, 108lecasei 8926 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (ψ `  y ) )
110 2nn0 9982 . . . . . . . . . . 11  |-  2  e.  NN0
111 nn0addge1 10010 . . . . . . . . . . 11  |-  ( ( (ψ `  y )  e.  RR  /\  2  e. 
NN0 )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11251, 110, 111sylancl 643 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  <_  (
(ψ `  y )  +  2 ) )
11348, 51, 69, 109, 112letrd 8973 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
(ψ `  y )  +  2 ) )
114 df-2 9804 . . . . . . . . . . 11  |-  2  =  ( 1  +  1 )
115114oveq2i 5869 . . . . . . . . . 10  |-  ( (ψ `  y )  +  2 )  =  ( (ψ `  y )  +  ( 1  +  1 ) )
11651recnd 8861 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (ψ `  y )  e.  CC )
11731a1i 10 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  1  e.  CC )
118116, 117, 117add12d 9033 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  ( 1  +  1 ) )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
119115, 118syl5eq 2327 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  y )  +  2 )  =  ( 1  +  ( (ψ `  y )  +  1 ) ) )
120113, 119breqtrd 4047 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  (
(ψ `  x )  /  x )  <_  (
1  +  ( (ψ `  y )  +  1 ) ) )
12148, 70, 43absdifled 11917 . . . . . . . 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 ) ) ) ) )
12266, 120, 121mpbir2and 888 . . . . . . 7  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( (ψ `  x )  /  x
)  -  1 ) )  <_  ( (ψ `  y )  +  1 ) )
12342, 122eqbrtrd 4043 . . . . . 6  |-  ( ( x  e.  RR+  /\  y  e.  RR  /\  x  < 
y )  ->  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
( (ψ `  y
)  +  1 ) )
1241233expb 1152 . . . . 5  |-  ( ( x  e.  RR+  /\  (
y  e.  RR  /\  x  <  y ) )  ->  ( abs `  (
( R `  x
)  /  x ) )  <_  ( (ψ `  y )  +  1 ) )
125124adantrlr 703 . . . 4  |-  ( ( x  e.  RR+  /\  (
( y  e.  RR  /\  1  <_  y )  /\  x  <  y ) )  ->  ( abs `  ( ( R `  x )  /  x
) )  <_  (
(ψ `  y )  +  1 ) )
126125adantll 694 . . 3  |-  ( ( (  T.  /\  x  e.  RR+ )  /\  (
( y  e.  RR  /\  1  <_  y )  /\  x  <  y ) )  ->  ( abs `  ( ( R `  x )  /  x
) )  <_  (
(ψ `  y )  +  1 ) )
1272, 4, 15, 36, 40, 126o1bddrp 12016 . 2  |-  (  T. 
->  E. c  e.  RR+  A. x  e.  RR+  ( abs `  ( ( R `
 x )  /  x ) )  <_ 
c )
128127trud 1314 1  |-  E. c  e.  RR+  A. x  e.  RR+  ( abs `  (
( R `  x
)  /  x ) )  <_  c
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   NN0cn0 9965   RR+crp 10354   abscabs 11719   O ( 1 )co1 11960  ψcchp 20330
This theorem is referenced by:  pntrlog2bnd  20733  pntibnd  20742  pnt3  20761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-o1 11964  df-lo1 11965  df-sum 12159  df-ef 12349  df-e 12350  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-cht 20334  df-vma 20335  df-chp 20336  df-ppi 20337
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