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Theorem pntlemg 20695
Description: Lemma for pnt 20711. 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 10343 . . . . . . 7  |-  ( ph  ->  X  e.  RR )
5 1re 8791 . . . . . . . . 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 10343 . . . . . . . 8  |-  ( ph  ->  Y  e.  RR )
107simprd 451 . . . . . . . 8  |-  ( ph  ->  1  <_  Y )
112simprd 451 . . . . . . . 8  |-  ( ph  ->  Y  <  X )
126, 9, 4, 10, 11lelttrd 8928 . . . . . . 7  |-  ( ph  ->  1  <  X )
134, 12rplogcld 19928 . . . . . 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 20692 . . . . . . . . 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 10343 . . . . . . 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 19928 . . . . . 6  |-  ( ph  ->  ( log `  K
)  e.  RR+ )
3013, 29rpdivcld 10360 . . . . 5  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR+ )
3130rprege0d 10350 . . . 4  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  e.  RR  /\  0  <_  ( ( log `  X
)  /  ( log `  K ) ) ) )
32 flge0nn0 10900 . . . 4  |-  ( ( ( ( log `  X
)  /  ( log `  K ) )  e.  RR  /\  0  <_ 
( ( log `  X
)  /  ( log `  K ) ) )  ->  ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  e.  NN0 )
33 nn0p1nn 9956 . . . 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 10069 . . 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 20694 . . . . . . . . 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 19922 . . . . . . 7  |-  ( ph  ->  ( log `  Z
)  e.  RR )
4443, 29rerpdivcld 10370 . . . . . 6  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR )
4544rehalfcld 9911 . . . . 5  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  RR )
4645flcld 10882 . . . 4  |-  ( ph  ->  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )  e.  ZZ )
4737, 46syl5eqel 2340 . . 3  |-  ( ph  ->  N  e.  ZZ )
48 0re 8792 . . . . . 6  |-  0  e.  RR
4948a1i 12 . . . . 5  |-  ( ph  ->  0  e.  RR )
50 4nn 9832 . . . . . 6  |-  4  e.  NN
51 nndivre 9735 . . . . . 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 10070 . . . . . 6  |-  ( ph  ->  N  e.  RR )
5435nnred 9715 . . . . . 6  |-  ( ph  ->  M  e.  RR )
5553, 54resubcld 9165 . . . . 5  |-  ( ph  ->  ( N  -  M
)  e.  RR )
5642rpred 10343 . . . . . . . . 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 19928 . . . . . . . 8  |-  ( ph  ->  ( log `  Z
)  e.  RR+ )
6059, 29rpdivcld 10360 . . . . . . 7  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  RR+ )
61 4re 9773 . . . . . . . 8  |-  4  e.  RR
62 4pos 9786 . . . . . . . 8  |-  0  <  4
6361, 62elrpii 10310 . . . . . . 7  |-  4  e.  RR+
64 rpdivcl 10329 . . . . . . 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 10347 . . . . 5  |-  ( ph  ->  0  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
6752recnd 8815 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  4 )  e.  CC )
6835nncnd 9716 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
69 ax-1cn 8749 . . . . . . . . . 10  |-  1  e.  CC
7069a1i 12 . . . . . . . . 9  |-  ( ph  ->  1  e.  CC )
7167, 68, 70addassd 8811 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  =  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( M  +  1 ) ) )
7254, 6readdcld 8816 . . . . . . . . . 10  |-  ( ph  ->  ( M  +  1 )  e.  RR )
7352, 72readdcld 8816 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  e.  RR )
74 peano2re 8939 . . . . . . . . . 10  |-  ( N  e.  RR  ->  ( N  +  1 )  e.  RR )
7553, 74syl 17 . . . . . . . . 9  |-  ( ph  ->  ( N  +  1 )  e.  RR )
7630rpred 10343 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( log `  X
)  /  ( log `  K ) )  e.  RR )
77 2re 9769 . . . . . . . . . . . . . 14  |-  2  e.  RR
7877a1i 12 . . . . . . . . . . . . 13  |-  ( ph  ->  2  e.  RR )
7976, 78readdcld 8816 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  X )  /  ( log `  K ) )  +  2 )  e.  RR )
80 reflcl 10880 . . . . . . . . . . . . . . . . 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 8815 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( |_ `  (
( log `  X
)  /  ( log `  K ) ) )  e.  CC )
8382, 70, 70addassd 8811 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  ( 1  +  1 ) ) )
841oveq1i 5788 . . . . . . . . . . . . . 14  |-  ( M  +  1 )  =  ( ( ( |_
`  ( ( log `  X )  /  ( log `  K ) ) )  +  1 )  +  1 )
85 df-2 9758 . . . . . . . . . . . . . . 15  |-  2  =  ( 1  +  1 )
8685oveq2i 5789 . . . . . . . . . . . . . 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 10883 . . . . . . . . . . . . . . 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 9340 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  2 )  <_ 
( ( ( log `  X )  /  ( log `  K ) )  +  2 ) )
9187, 90eqbrtrd 4003 . . . . . . . . . . . 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 8927 . . . . . . . . . . 11  |-  ( ph  ->  ( M  +  1 )  <_  ( (
( log `  Z
)  /  ( log `  K ) )  / 
4 ) )
9572, 52, 52, 94leadd2dd 9341 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
9644recnd 8815 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( log `  Z
)  /  ( log `  K ) )  e.  CC )
97 2cn 9770 . . . . . . . . . . . . . . 15  |-  2  e.  CC
9897a1i 12 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  e.  CC )
99 2ne0 9783 . . . . . . . . . . . . . . 15  |-  2  =/=  0
10099a1i 12 . . . . . . . . . . . . . 14  |-  ( ph  ->  2  =/=  0 )
10196, 98, 98, 100, 100divdiv1d 9521 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  2
)  /  2 )  =  ( ( ( log `  Z )  /  ( log `  K
) )  /  (
2  x.  2 ) ) )
102 2t2e4 9824 . . . . . . . . . . . . . 14  |-  ( 2  x.  2 )  =  4
103102oveq2i 5789 . . . . . . . . . . . . 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 5794 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( 2  x.  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 ) ) )
10645recnd 8815 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  e.  CC )
107106, 98, 100divcan2d 9492 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  / 
2 ) )  =  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )
108672timesd 9907 . . . . . . . . . . 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 4009 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( ( ( log `  Z )  /  ( log `  K
) )  /  2
) )
111 fllep1 10885 . . . . . . . . . . 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 5788 . . . . . . . . . 10  |-  ( N  +  1 )  =  ( ( |_ `  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 ) )  +  1 )
114112, 113syl6breqr 4023 . . . . . . . . 9  |-  ( ph  ->  ( ( ( log `  Z )  /  ( log `  K ) )  /  2 )  <_ 
( N  +  1 ) )
11573, 45, 75, 110, 114letrd 8927 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  ( M  +  1 ) )  <_  ( N  + 
1 ) )
11671, 115eqbrtrd 4003 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  +  M
)  +  1 )  <_  ( N  + 
1 ) )
11752, 54readdcld 8816 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( log `  Z )  /  ( log `  K
) )  /  4
)  +  M )  e.  RR )
118117, 53, 6leadd1d 9320 . . . . . . 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 9204 . . . . . . 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 8927 . . . 4  |-  ( ph  ->  0  <_  ( N  -  M ) )
12453, 54subge0d 9316 . . . 4  |-  ( ph  ->  ( 0  <_  ( N  -  M )  <->  M  <_  N ) )
125123, 124mpbid 203 . . 3  |-  ( ph  ->  M  <_  N )
126 eluz2 10189 . . 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 3983    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    +oocpnf 8818    < clt 8821    <_ cle 8822    - cmin 8991    / cdiv 9377   NNcn 9700   2c2 9749   3c3 9750   4c4 9751   NN0cn0 9918   ZZcz 9977  ;cdc 10077   ZZ>=cuz 10183   RR+crp 10307   (,)cioo 10608   [,)cico 10610   |_cfl 10876   ^cexp 11056   sqrcsqr 11669   expce 12291   _eceu 12292   logclog 19860  ψcchp 20278
This theorem is referenced by:  pntlemh  20696  pntlemq  20698  pntlemr  20699  pntlemj  20700  pntlemf  20702
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
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 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297  df-e 12298  df-sin 12299  df-cos 12300  df-pi 12302  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862
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