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Theorem pntlemg 23761
Description: Lemma for pnt 23777. 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 459 . . . . . . . 8  |-  ( ph  ->  X  e.  RR+ )
43rpred 11267 . . . . . . 7  |-  ( ph  ->  X  e.  RR )
5 1red 9614 . . . . . . . 8  |-  ( ph  ->  1  e.  RR )
6 pntlem1.y . . . . . . . . . 10  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
76simpld 459 . . . . . . . . 9  |-  ( ph  ->  Y  e.  RR+ )
87rpred 11267 . . . . . . . 8  |-  ( ph  ->  Y  e.  RR )
96simprd 463 . . . . . . . 8  |-  ( ph  ->  1  <_  Y )
102simprd 463 . . . . . . . 8  |-  ( ph  ->  Y  <  X )
115, 8, 4, 9, 10lelttrd 9743 . . . . . . 7  |-  ( ph  ->  1  <  X )
124, 11rplogcld 22992 . . . . . 6  |-  ( ph  ->  ( log `  X
)  e.  RR+ )
13 pntlem1.r . . . . . . . . . 10  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
14 pntlem1.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR+ )
15 pntlem1.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
16 pntlem1.l . . . . . . . . . 10  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
17 pntlem1.d . . . . . . . . . 10  |-  D  =  ( A  +  1 )
18 pntlem1.f . . . . . . . . . 10  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
19 pntlem1.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  RR+ )
20 pntlem1.u2 . . . . . . . . . 10  |-  ( ph  ->  U  <_  A )
21 pntlem1.e . . . . . . . . . 10  |-  E  =  ( U  /  D
)
22 pntlem1.k . . . . . . . . . 10  |-  K  =  ( exp `  ( B  /  E ) )
2313, 14, 15, 16, 17, 18, 19, 20, 21, 22pntlemc 23758 . . . . . . . . 9  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
2423simp2d 1010 . . . . . . . 8  |-  ( ph  ->  K  e.  RR+ )
2524rpred 11267 . . . . . . 7  |-  ( ph  ->  K  e.  RR )
2623simp3d 1011 . . . . . . . 8  |-  ( ph  ->  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E
)  e.  RR+ )
)
2726simp2d 1010 . . . . . . 7  |-  ( ph  ->  1  <  K )
2825, 27rplogcld 22992 . . . . . 6  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
2912, 28rpdivcld 11284 . . . . 5  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR+ )
3029rprege0d 11274 . . . 4  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  e.  RR  /\  0  <_  ( ( log `  X
)  /  ( log `  K ) ) ) )
31 flge0nn0 11936 . . . 4  |-  ( ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  /\  0  <_ 
( ( log `  X
)  /  ( log `  K ) ) )  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  e.  NN0 )
32 nn0p1nn 10842 . . . 4  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  e. 
NN0  ->  ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  e.  NN )
3330, 31, 323syl 20 . . 3  |-  ( ph  ->  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )  e.  NN )
341, 33syl5eqel 2535 . 2  |-  ( ph  ->  M  e.  NN )
3534nnzd 10975 . . 3  |-  ( ph  ->  M  e.  ZZ )
36 pntlem1.n . . . 4  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
37 pntlem1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
38 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
) ) ) ) )
39 pntlem1.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( W [,) +oo ) )
4013, 14, 15, 16, 17, 18, 19, 20, 21, 22, 6, 2, 37, 38, 39pntlemb 23760 . . . . . . . . 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
) ) ) ) )
4140simp1d 1009 . . . . . . . 8  |-  ( ph  ->  Z  e.  RR+ )
4241relogcld 22986 . . . . . . 7  |-  ( ph  ->  ( log `  Z
)  e.  RR )
4342, 28rerpdivcld 11294 . . . . . 6  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR )
4443rehalfcld 10792 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  RR )
4544flcld 11917 . . . 4  |-  ( ph  ->  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )  e.  ZZ )
4636, 45syl5eqel 2535 . . 3  |-  ( ph  ->  N  e.  ZZ )
47 0red 9600 . . . . 5  |-  ( ph  ->  0  e.  RR )
48 4nn 10702 . . . . . 6  |-  4  e.  NN
49 nndivre 10578 . . . . . 6  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  e.  RR  /\  4  e.  NN )  ->  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  e.  RR )
5043, 48, 49sylancl 662 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  RR )
5146zred 10976 . . . . . 6  |-  ( ph  ->  N  e.  RR )
5234nnred 10558 . . . . . 6  |-  ( ph  ->  M  e.  RR )
5351, 52resubcld 9994 . . . . 5  |-  ( ph  ->  ( N  -  M
)  e.  RR )
5441rpred 11267 . . . . . . . . 9  |-  ( ph  ->  Z  e.  RR )
5540simp2d 1010 . . . . . . . . . 10  |-  ( ph  ->  ( 1  <  Z  /\  _e  <_  ( sqr `  Z )  /\  ( sqr `  Z )  <_ 
( Z  /  Y
) ) )
5655simp1d 1009 . . . . . . . . 9  |-  ( ph  ->  1  <  Z )
5754, 56rplogcld 22992 . . . . . . . 8  |-  ( ph  ->  ( log `  Z
)  e.  RR+ )
5857, 28rpdivcld 11284 . . . . . . 7  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR+ )
59 4re 10619 . . . . . . . 8  |-  4  e.  RR
60 4pos 10638 . . . . . . . 8  |-  0  <  4
6159, 60elrpii 11234 . . . . . . 7  |-  4  e.  RR+
62 rpdivcl 11253 . . . . . . 7  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  e.  RR+  /\  4  e.  RR+ )  ->  ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  e.  RR+ )
6358, 61, 62sylancl 662 . . . . . 6  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  RR+ )
6463rpge0d 11271 . . . . 5  |-  ( ph  ->  0  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
6550recnd 9625 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  CC )
6634nncnd 10559 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
67 1cnd 9615 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
6865, 66, 67addassd 9621 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  =  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( M  +  1 ) ) )
6952, 5readdcld 9626 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  RR )
7050, 69readdcld 9626 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  e.  RR )
71 peano2re 9756 . . . . . . . . . 10  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
7251, 71syl 16 . . . . . . . . 9  |-  ( ph  ->  ( N  +  1 )  e.  RR )
7329rpred 11267 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR )
74 2re 10612 . . . . . . . . . . . . . 14  |-  2  e.  RR
7574a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  2  e.  RR )
7673, 75readdcld 9626 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  e.  RR )
77 reflcl 11915 . . . . . . . . . . . . . . . . 17  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  e.  RR )
7873, 77syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  e.  RR )
7978recnd 9625 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  e.  CC )
8079, 67, 67addassd 9621 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  ( 1  +  1 ) ) )
811oveq1i 6291 . . . . . . . . . . . . . 14  |-  ( M  +  1 )  =  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )
82 df-2 10601 . . . . . . . . . . . . . . 15  |-  2  =  ( 1  +  1 )
8382oveq2i 6292 . . . . . . . . . . . . . 14  |-  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  2 )  =  ( ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  +  ( 1  +  1 ) )
8480, 81, 833eqtr4g 2509 . . . . . . . . . . . . 13  |-  ( ph  ->  ( M  +  1 )  =  ( ( |_ `  ( ( log `  X )  /  ( log `  K
) ) )  +  2 ) )
85 flle 11918 . . . . . . . . . . . . . . 15  |-  ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  <_  ( ( log `  X )  /  ( log `  K ) ) )
8673, 85syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  <_  ( ( log `  X )  /  ( log `  K ) ) )
8778, 73, 75, 86leadd1dd 10173 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  2 )  <_ 
( ( ( log `  X )  /  ( log `  K ) )  +  2 ) )
8884, 87eqbrtrd 4457 . . . . . . . . . . . 12  |-  ( ph  ->  ( M  +  1 )  <_  ( (
( log `  X
)  /  ( log `  K ) )  +  2 ) )
8940simp3d 1011 . . . . . . . . . . . . 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
) ) ) )
9089simp2d 1010 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  <_ 
( ( ( log `  Z )  /  ( log `  K ) )  /  4 ) )
9169, 76, 50, 88, 90letrd 9742 . . . . . . . . . . 11  |-  ( ph  ->  ( M  +  1 )  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
9269, 50, 50, 91leadd2dd 10174 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
9343recnd 9625 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  CC )
94 2cnd 10615 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  e.  CC )
95 2ne0 10635 . . . . . . . . . . . . . . 15  |-  2  =/=  0
9695a1i 11 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  =/=  0 )
9793, 94, 94, 96, 96divdiv1d 10358 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  /  2 )  =  ( ( ( log `  Z )  /  ( log `  K
) )  /  (
2  x.  2 ) ) )
98 2t2e4 10692 . . . . . . . . . . . . . 14  |-  ( 2  x.  2 )  =  4
9998oveq2i 6292 . . . . . . . . . . . . 13  |-  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
( 2  x.  2 ) )  =  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )
10097, 99syl6eq 2500 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  /  2 )  =  ( ( ( log `  Z )  /  ( log `  K
) )  /  4
) )
101100oveq2d 6297 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( 2  x.  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
10244recnd 9625 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  CC )
103102, 94, 96divcan2d 10329 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )
104652timesd 10788 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) )  =  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
105101, 103, 1043eqtr3d 2492 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  =  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
10692, 105breqtrrd 4463 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
) )
107 fllep1 11920 . . . . . . . . . . 11  |-  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
2 )  e.  RR  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 ) )
10844, 107syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 ) )
10936oveq1i 6291 . . . . . . . . . 10  |-  ( N  +  1 )  =  ( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 )
110108, 109syl6breqr 4477 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( N  +  1 ) )
11170, 44, 72, 106, 110letrd 9742 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( N  + 
1 ) )
11268, 111eqbrtrd 4457 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  <_  ( N  + 
1 ) )
11350, 52readdcld 9626 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  e.  RR )
114113, 51, 5leadd1d 10153 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  <_  N  <->  ( (
( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  +  M )  +  1 )  <_  ( N  +  1 ) ) )
115112, 114mpbird 232 . . . . . 6  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  <_  N )
116 leaddsub 10035 . . . . . . 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 )
) )
11750, 52, 51, 116syl3anc 1229 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  <_  N  <->  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
118115, 117mpbid 210 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  <_ 
( N  -  M
) )
11947, 50, 53, 64, 118letrd 9742 . . . 4  |-  ( ph  ->  0  <_  ( N  -  M ) )
12051, 52subge0d 10149 . . . 4  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
121119, 120mpbid 210 . . 3  |-  ( ph  ->  M  <_  N )
122 eluz2 11098 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  N  e.  ZZ  /\  M  <_  N ) )
12335, 46, 121, 122syl3anbrc 1181 . 2  |-  ( ph  ->  N  e.  ( ZZ>= `  M ) )
12434, 123, 1183jca 1177 1  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437    |-> cmpt 4495   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   +oocpnf 9628    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   NNcn 10543   2c2 10592   3c3 10593   4c4 10594   NN0cn0 10802   ZZcz 10871  ;cdc 10986   ZZ>=cuz 11092   RR+crp 11231   (,)cioo 11540   [,)cico 11542   |_cfl 11909   ^cexp 12148   sqrcsqrt 13048   expce 13779   _eceu 13780   logclog 22920  ψcchp 23344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-e 13786  df-sin 13787  df-cos 13788  df-pi 13790  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-limc 22248  df-dv 22249  df-log 22922
This theorem is referenced by:  pntlemh  23762  pntlemq  23764  pntlemr  23765  pntlemj  23766  pntlemf  23768
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