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Theorem pntlemg 20674
Description: Lemma for pnt 20690. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434,  M is j^* and  N is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
Hypotheses
Ref Expression
pntlem1.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
pntlem1.a  |-  ( ph  ->  A  e.  RR+ )
pntlem1.b  |-  ( ph  ->  B  e.  RR+ )
pntlem1.l  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
pntlem1.d  |-  D  =  ( A  +  1 )
pntlem1.f  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
pntlem1.u  |-  ( ph  ->  U  e.  RR+ )
pntlem1.u2  |-  ( ph  ->  U  <_  A )
pntlem1.e  |-  E  =  ( U  /  D
)
pntlem1.k  |-  K  =  ( exp `  ( B  /  E ) )
pntlem1.y  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
pntlem1.x  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
pntlem1.c  |-  ( ph  ->  C  e.  RR+ )
pntlem1.w  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
pntlem1.z  |-  ( ph  ->  Z  e.  ( W [,)  +oo ) )
pntlem1.m  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
pntlem1.n  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
Assertion
Ref Expression
pntlemg  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
Distinct variable group:    E, a
Allowed substitution hints:    ph( a)    A( a)    B( a)    C( a)    D( a)    R( a)    U( a)    F( a)    K( a)    L( a)    M( a)    N( a)    W( a)    X( a)    Y( a)    Z( a)

Proof of Theorem pntlemg
StepHypRef Expression
1 pntlem1.m . . 3  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
2 pntlem1.x . . . . . . . . 9  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
32simpld 447 . . . . . . . 8  |-  ( ph  ->  X  e.  RR+ )
43rpred 10322 . . . . . . 7  |-  ( ph  ->  X  e.  RR )
5 1re 8770 . . . . . . . . 9  |-  1  e.  RR
65a1i 12 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
7 pntlem1.y . . . . . . . . . 10  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
87simpld 447 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR+ )
98rpred 10322 . . . . . . . 8  |-  ( ph  ->  Y  e.  RR )
107simprd 451 . . . . . . . 8  |-  ( ph  ->  1  <_  Y )
112simprd 451 . . . . . . . 8  |-  ( ph  ->  Y  <  X )
126, 9, 4, 10, 11lelttrd 8907 . . . . . . 7  |-  ( ph  ->  1  <  X )
134, 12rplogcld 19907 . . . . . 6  |-  ( ph  ->  ( log `  X
)  e.  RR+ )
14 pntlem1.r . . . . . . . . . 10  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
15 pntlem1.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR+ )
16 pntlem1.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
17 pntlem1.l . . . . . . . . . 10  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
18 pntlem1.d . . . . . . . . . 10  |-  D  =  ( A  +  1 )
19 pntlem1.f . . . . . . . . . 10  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
20 pntlem1.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  RR+ )
21 pntlem1.u2 . . . . . . . . . 10  |-  ( ph  ->  U  <_  A )
22 pntlem1.e . . . . . . . . . 10  |-  E  =  ( U  /  D
)
23 pntlem1.k . . . . . . . . . 10  |-  K  =  ( exp `  ( B  /  E ) )
2414, 15, 16, 17, 18, 19, 20, 21, 22, 23pntlemc 20671 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2524simp2d 973 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
2625rpred 10322 . . . . . . 7  |-  ( ph  ->  K  e.  RR )
2724simp3d 974 . . . . . . . 8  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
2827simp2d 973 . . . . . . 7  |-  ( ph  ->  1  <  K )
2926, 28rplogcld 19907 . . . . . 6  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
3013, 29rpdivcld 10339 . . . . 5  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR+ )
3130rprege0d 10329 . . . 4  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  e.  RR  /\  0  <_  ( ( log `  X
)  /  ( log `  K ) ) ) )
32 flge0nn0 10879 . . . 4  |-  ( ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  /\  0  <_ 
( ( log `  X
)  /  ( log `  K ) ) )  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  e.  NN0 )
33 nn0p1nn 9935 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  e. 
NN0  ->  ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  e.  NN )
3431, 32, 333syl 20 . . 3  |-  ( ph  ->  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )  e.  NN )
351, 34syl5eqel 2340 . 2  |-  ( ph  ->  M  e.  NN )
3635nnzd 10048 . . 3  |-  ( ph  ->  M  e.  ZZ )
37 pntlem1.n . . . 4  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
38 pntlem1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
39 pntlem1.w . . . . . . . . . 10  |-  W  =  ( ( ( Y  +  ( 4  / 
( L  x.  E
) ) ) ^
2 )  +  ( ( ( X  x.  ( K ^ 2 ) ) ^ 4 )  +  ( exp `  (
( (; 3 2  x.  B
)  /  ( ( U  -  E )  x.  ( L  x.  ( E ^ 2 ) ) ) )  x.  ( ( U  x.  3 )  +  C
) ) ) ) )
40 pntlem1.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( W [,)  +oo ) )
4114, 15, 16, 17, 18, 19, 20, 21, 22, 23, 7, 2, 38, 39, 40pntlemb 20673 . . . . . . . . 9  |-  ( ph  ->  ( Z  e.  RR+  /\  ( 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_ 
( Z  /  Y
) )  /\  (
( 4  /  ( L  x.  E )
)  <_  ( sqr `  Z )  /\  (
( ( log `  X
)  /  ( log `  K ) )  +  2 )  <_  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  /\  (
( U  x.  3 )  +  C )  <_  ( ( ( U  -  E )  x.  ( ( L  x.  ( E ^
2 ) )  / 
(; 3 2  x.  B
) ) )  x.  ( log `  Z
) ) ) ) )
4241simp1d 972 . . . . . . . 8  |-  ( ph  ->  Z  e.  RR+ )
4342relogcld 19901 . . . . . . 7  |-  ( ph  ->  ( log `  Z
)  e.  RR )
4443, 29rerpdivcld 10349 . . . . . 6  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR )
4544rehalfcld 9890 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  RR )
4645flcld 10861 . . . 4  |-  ( ph  ->  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )  e.  ZZ )
4737, 46syl5eqel 2340 . . 3  |-  ( ph  ->  N  e.  ZZ )
48 0re 8771 . . . . . 6  |-  0  e.  RR
4948a1i 12 . . . . 5  |-  ( ph  ->  0  e.  RR )
50 4nn 9811 . . . . . 6  |-  4  e.  NN
51 nndivre 9714 . . . . . 6  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  e.  RR  /\  4  e.  NN )  ->  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  e.  RR )
5244, 50, 51sylancl 646 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  RR )
5347zred 10049 . . . . . 6  |-  ( ph  ->  N  e.  RR )
5435nnred 9694 . . . . . 6  |-  ( ph  ->  M  e.  RR )
5553, 54resubcld 9144 . . . . 5  |-  ( ph  ->  ( N  -  M
)  e.  RR )
5642rpred 10322 . . . . . . . . 9  |-  ( ph  ->  Z  e.  RR )
5741simp2d 973 . . . . . . . . . 10  |-  ( ph  ->  ( 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_ 
( Z  /  Y
) ) )
5857simp1d 972 . . . . . . . . 9  |-  ( ph  ->  1  <  Z )
5956, 58rplogcld 19907 . . . . . . . 8  |-  ( ph  ->  ( log `  Z
)  e.  RR+ )
6059, 29rpdivcld 10339 . . . . . . 7  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR+ )
61 4re 9752 . . . . . . . 8  |-  4  e.  RR
62 4pos 9765 . . . . . . . 8  |-  0  <  4
6361, 62elrpii 10289 . . . . . . 7  |-  4  e.  RR+
64 rpdivcl 10308 . . . . . . 7  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  e.  RR+  /\  4  e.  RR+ )  ->  ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  e.  RR+ )
6560, 63, 64sylancl 646 . . . . . 6  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  RR+ )
6665rpge0d 10326 . . . . 5  |-  ( ph  ->  0  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
6752recnd 8794 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  CC )
6835nncnd 9695 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
69 ax-1cn 8728 . . . . . . . . . 10  |-  1  e.  CC
7069a1i 12 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
7167, 68, 70addassd 8790 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  =  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( M  +  1 ) ) )
7254, 6readdcld 8795 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  RR )
7352, 72readdcld 8795 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  e.  RR )
74 peano2re 8918 . . . . . . . . . 10  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
7553, 74syl 17 . . . . . . . . 9  |-  ( ph  ->  ( N  +  1 )  e.  RR )
7630rpred 10322 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR )
77 2re 9748 . . . . . . . . . . . . . 14  |-  2  e.  RR
7877a1i 12 . . . . . . . . . . . . 13  |-  ( ph  ->  2  e.  RR )
7976, 78readdcld 8795 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  e.  RR )
80 reflcl 10859 . . . . . . . . . . . . . . . . 17  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  e.  RR )
8176, 80syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  e.  RR )
8281recnd 8794 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  e.  CC )
8382, 70, 70addassd 8790 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  ( 1  +  1 ) ) )
841oveq1i 5767 . . . . . . . . . . . . . 14  |-  ( M  +  1 )  =  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )
85 df-2 9737 . . . . . . . . . . . . . . 15  |-  2  =  ( 1  +  1 )
8685oveq2i 5768 . . . . . . . . . . . . . 14  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  2 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  ( 1  +  1 ) )
8783, 84, 863eqtr4g 2313 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  1 )  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  2 ) )
88 flle 10862 . . . . . . . . . . . . . . 15  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  <_  ( ( log `  X )  /  ( log `  K ) ) )
8976, 88syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  <_  ( ( log `  X )  /  ( log `  K ) ) )
9081, 76, 78, 89leadd1dd 9319 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  2 )  <_ 
( ( ( log `  X )  /  ( log `  K ) )  +  2 ) )
9187, 90eqbrtrd 3983 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  1 )  <_  ( (
( log `  X
)  /  ( log `  K ) )  +  2 ) )
9241simp3d 974 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  / 
( L  x.  E
) )  <_  ( sqr `  Z )  /\  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  <_ 
( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  /\  ( ( U  x.  3 )  +  C
)  <_  ( (
( U  -  E
)  x.  ( ( L  x.  ( E ^ 2 ) )  /  (; 3 2  x.  B
) ) )  x.  ( log `  Z
) ) ) )
9392simp2d 973 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  <_ 
( ( ( log `  Z )  /  ( log `  K ) )  /  4 ) )
9472, 79, 52, 91, 93letrd 8906 . . . . . . . . . . 11  |-  ( ph  ->  ( M  +  1 )  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
9572, 52, 52, 94leadd2dd 9320 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
9644recnd 8794 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  CC )
97 2cn 9749 . . . . . . . . . . . . . . 15  |-  2  e.  CC
9897a1i 12 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  e.  CC )
99 2ne0 9762 . . . . . . . . . . . . . . 15  |-  2  =/=  0
10099a1i 12 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  =/=  0 )
10196, 98, 98, 100, 100divdiv1d 9500 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  /  2 )  =  ( ( ( log `  Z )  /  ( log `  K
) )  /  (
2  x.  2 ) ) )
102 2t2e4 9803 . . . . . . . . . . . . . 14  |-  ( 2  x.  2 )  =  4
103102oveq2i 5768 . . . . . . . . . . . . 13  |-  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
( 2  x.  2 ) )  =  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )
104101, 103syl6eq 2304 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  /  2 )  =  ( ( ( log `  Z )  /  ( log `  K
) )  /  4
) )
105104oveq2d 5773 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( 2  x.  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
10645recnd 8794 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  CC )
107106, 98, 100divcan2d 9471 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )
108672timesd 9886 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) )  =  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
109105, 107, 1083eqtr3d 2296 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  =  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
11095, 109breqtrrd 3989 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
) )
111 fllep1 10864 . . . . . . . . . . 11  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 ) )
11245, 111syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 ) )
11337oveq1i 5767 . . . . . . . . . 10  |-  ( N  +  1 )  =  ( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 )
114112, 113syl6breqr 4003 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( N  +  1 ) )
11573, 45, 75, 110, 114letrd 8906 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( N  + 
1 ) )
11671, 115eqbrtrd 3983 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  <_  ( N  + 
1 ) )
11752, 54readdcld 8795 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  e.  RR )
118117, 53, 6leadd1d 9299 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  <_  N  <->  ( (
( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  +  M )  +  1 )  <_  ( N  +  1 ) ) )
119116, 118mpbird 225 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  <_  N )
120 leaddsub 9183 . . . . . . 7  |-  ( ( ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  RR  /\  M  e.  RR  /\  N  e.  RR )  ->  (
( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  <_  N  <->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
12152, 54, 53, 120syl3anc 1187 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  <_  N  <->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
122119, 121mpbid 203 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  <_ 
( N  -  M
) )
12349, 52, 55, 66, 122letrd 8906 . . . 4  |-  ( ph  ->  0  <_  ( N  -  M ) )
12453, 54subge0d 9295 . . . 4  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
125123, 124mpbid 203 . . 3  |-  ( ph  ->  M  <_  N )
126 eluz2 10168 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
12736, 47, 125, 126syl3anbrc 1141 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
12835, 127, 1223jca 1137 1  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963    e. cmpt 4017   ` cfv 4638  (class class class)co 5757   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    +oocpnf 8797    < clt 8800    <_ cle 8801    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   3c3 9729   4c4 9730   NN0cn0 9897   ZZcz 9956  ;cdc 10056   ZZ>=cuz 10162   RR+crp 10286   (,)cioo 10587   [,)cico 10589   |_cfl 10855   ^cexp 11035   sqrcsqr 11648   expce 12270   _eceu 12271   logclog 19839  ψcchp 20257
This theorem is referenced by:  pntlemh  20675  pntlemq  20677  pntlemr  20678  pntlemj  20679  pntlemf  20681
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-e 12277  df-sin 12278  df-cos 12279  df-pi 12281  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841
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