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Theorem pntlemq 23912
Description: Lemma for pntlemj 23914. (Contributed by Mario Carneiro, 7-Jun-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 ) )
pntlem1.U  |-  ( ph  ->  A. z  e.  ( Y [,) +oo )
( abs `  (
( R `  z
)  /  z ) )  <_  U )
pntlem1.K  |-  ( ph  ->  A. y  e.  ( X (,) +oo ) E. z  e.  RR+  (
( y  <  z  /\  ( ( 1  +  ( L  x.  E
) )  x.  z
)  <  ( K  x.  y ) )  /\  A. u  e.  ( z [,] ( ( 1  +  ( L  x.  E ) )  x.  z ) ) ( abs `  ( ( R `  u )  /  u ) )  <_  E ) )
pntlem1.o  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
pntlem1.v  |-  ( ph  ->  V  e.  RR+ )
pntlem1.V  |-  ( ph  ->  ( ( ( K ^ J )  < 
V  /\  ( (
1  +  ( L  x.  E ) )  x.  V )  < 
( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] (
( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  (
( R `  u
)  /  u ) )  <_  E )
)
pntlem1.j  |-  ( ph  ->  J  e.  ( M..^ N ) )
pntlem1.i  |-  I  =  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )
Assertion
Ref Expression
pntlemq  |-  ( ph  ->  I  C_  O )
Distinct variable groups:    z, C    y, z, J    y, u, z, L    y, K, z   
z, M    z, O    z, N    u, R, y, z    u, V    z, U    z, W    y, X, z    z, Y    u, a,
y, z, E    u, Z, z
Allowed substitution hints:    ph( y, z, u, a)    A( y, z, u, a)    B( y, z, u, a)    C( y, u, a)    D( y, z, u, a)    R( a)    U( y, u, a)    F( y, z, u, a)    I( y, z, u, a)    J( u, a)    K( u, a)    L( a)    M( y, u, a)    N( y, u, a)    O( y, u, a)    V( y, z, a)    W( y, u, a)    X( u, a)    Y( y, u, a)    Z( y, a)

Proof of Theorem pntlemq
StepHypRef Expression
1 pntlem1.r . . . . . . . . . 10  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
2 pntlem1.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR+ )
3 pntlem1.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR+ )
4 pntlem1.l . . . . . . . . . 10  |-  ( ph  ->  L  e.  ( 0 (,) 1 ) )
5 pntlem1.d . . . . . . . . . 10  |-  D  =  ( A  +  1 )
6 pntlem1.f . . . . . . . . . 10  |-  F  =  ( ( 1  -  ( 1  /  D
) )  x.  (
( L  /  (; 3 2  x.  B ) )  /  ( D ^
2 ) ) )
7 pntlem1.u . . . . . . . . . 10  |-  ( ph  ->  U  e.  RR+ )
8 pntlem1.u2 . . . . . . . . . 10  |-  ( ph  ->  U  <_  A )
9 pntlem1.e . . . . . . . . . 10  |-  E  =  ( U  /  D
)
10 pntlem1.k . . . . . . . . . 10  |-  K  =  ( exp `  ( B  /  E ) )
11 pntlem1.y . . . . . . . . . 10  |-  ( ph  ->  ( Y  e.  RR+  /\  1  <_  Y )
)
12 pntlem1.x . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  RR+  /\  Y  <  X ) )
13 pntlem1.c . . . . . . . . . 10  |-  ( ph  ->  C  e.  RR+ )
14 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
) ) ) ) )
15 pntlem1.z . . . . . . . . . 10  |-  ( ph  ->  Z  e.  ( W [,) +oo ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15pntlemb 23908 . . . . . . . . 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
) ) ) ) )
1716simp1d 1008 . . . . . . . 8  |-  ( ph  ->  Z  e.  RR+ )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 23906 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1918simp2d 1009 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR+ )
20 pntlem1.j . . . . . . . . . . 11  |-  ( ph  ->  J  e.  ( M..^ N ) )
21 elfzoelz 11826 . . . . . . . . . . 11  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
2220, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  J  e.  ZZ )
2322peano2zd 10993 . . . . . . . . 9  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
2419, 23rpexpcld 12336 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR+ )
2517, 24rpdivcld 11298 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR+ )
2625rpred 11281 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR )
2726flcld 11938 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  e.  ZZ )
28 1rp 11249 . . . . . . . . . 10  |-  1  e.  RR+
291, 2, 3, 4, 5, 6pntlemd 23905 . . . . . . . . . . . 12  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
3029simp1d 1008 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  RR+ )
3118simp1d 1008 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  RR+ )
3230, 31rpmulcld 11297 . . . . . . . . . 10  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
33 rpaddcl 11265 . . . . . . . . . 10  |-  ( ( 1  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
1  +  ( L  x.  E ) )  e.  RR+ )
3428, 32, 33sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  ( L  x.  E ) )  e.  RR+ )
35 pntlem1.v . . . . . . . . 9  |-  ( ph  ->  V  e.  RR+ )
3634, 35rpmulcld 11297 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR+ )
3717, 36rpdivcld 11298 . . . . . . 7  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR+ )
3837rpred 11281 . . . . . 6  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR )
3938flcld 11938 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ )
4036rpred 11281 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR )
4124rpred 11281 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR )
42 pntlem1.V . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( K ^ J )  < 
V  /\  ( (
1  +  ( L  x.  E ) )  x.  V )  < 
( K  x.  ( K ^ J ) ) )  /\  A. u  e.  ( V [,] (
( 1  +  ( L  x.  E ) )  x.  V ) ) ( abs `  (
( R `  u
)  /  u ) )  <_  E )
)
4342simpld 459 . . . . . . . . . 10  |-  ( ph  ->  ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) ) )
4443simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) )
4519rpcnd 11283 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  CC )
4619, 22rpexpcld 12336 . . . . . . . . . . . 12  |-  ( ph  ->  ( K ^ J
)  e.  RR+ )
4746rpcnd 11283 . . . . . . . . . . 11  |-  ( ph  ->  ( K ^ J
)  e.  CC )
4845, 47mulcomd 9634 . . . . . . . . . 10  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( ( K ^ J )  x.  K ) )
49 pntlem1.m . . . . . . . . . . . . . . 15  |-  M  =  ( ( |_ `  ( ( log `  X
)  /  ( log `  K ) ) )  +  1 )
50 pntlem1.n . . . . . . . . . . . . . . 15  |-  N  =  ( |_ `  (
( ( log `  Z
)  /  ( log `  K ) )  / 
2 ) )
511, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 49, 50pntlemg 23909 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
5251simp1d 1008 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  NN )
53 elfzouz 11830 . . . . . . . . . . . . . 14  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ( ZZ>= `  M )
)
5420, 53syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ( ZZ>= `  M ) )
55 eluznn 11177 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
5652, 54, 55syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  NN )
5756nnnn0d 10873 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  NN0 )
5845, 57expp1d 12314 . . . . . . . . . 10  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  =  ( ( K ^ J )  x.  K ) )
5948, 58eqtr4d 2501 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( K ^
( J  +  1 ) ) )
6044, 59breqtrd 4480 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K ^ ( J  + 
1 ) ) )
6140, 41, 60ltled 9750 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <_  ( K ^ ( J  + 
1 ) ) )
6236, 24, 17lediv2d 11305 . . . . . . 7  |-  ( ph  ->  ( ( ( 1  +  ( L  x.  E ) )  x.  V )  <_  ( K ^ ( J  + 
1 ) )  <->  ( Z  /  ( K ^
( J  +  1 ) ) )  <_ 
( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
6361, 62mpbid 210 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  <_  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )
64 flwordi 11951 . . . . . 6  |-  ( ( ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR  /\  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR  /\  ( Z  /  ( K ^
( J  +  1 ) ) )  <_ 
( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) )  ->  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  <_ 
( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
6526, 38, 63, 64syl3anc 1228 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  <_  ( |_ `  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
66 eluz2 11112 . . . . 5  |-  ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) )  <-> 
( ( |_ `  ( Z  /  ( K ^ ( J  + 
1 ) ) ) )  e.  ZZ  /\  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ  /\  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  <_ 
( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) ) ) )
6727, 39, 65, 66syl3anbrc 1180 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) ) )
68 eluzp1p1 11131 . . . 4  |-  ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) )  ->  ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 )  e.  (
ZZ>= `  ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ) )
69 fzss1 11748 . . . 4  |-  ( ( ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  +  1 )  e.  ( ZZ>= `  ( ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) )  -> 
( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) )
7067, 68, 693syl 20 . . 3  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) )
7117, 35rpdivcld 11298 . . . . . . 7  |-  ( ph  ->  ( Z  /  V
)  e.  RR+ )
7271rpred 11281 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  e.  RR )
7372flcld 11938 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  e.  ZZ )
7417, 46rpdivcld 11298 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR+ )
7574rpred 11281 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR )
7675flcld 11938 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ )
7746rpred 11281 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  e.  RR )
7835rpred 11281 . . . . . . . 8  |-  ( ph  ->  V  e.  RR )
7943simpld 459 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  <  V )
8077, 78, 79ltled 9750 . . . . . . 7  |-  ( ph  ->  ( K ^ J
)  <_  V )
8146, 35, 17lediv2d 11305 . . . . . . 7  |-  ( ph  ->  ( ( K ^ J )  <_  V  <->  ( Z  /  V )  <_  ( Z  / 
( K ^ J
) ) ) )
8280, 81mpbid 210 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  <_  ( Z  /  ( K ^ J ) ) )
83 flwordi 11951 . . . . . 6  |-  ( ( ( Z  /  V
)  e.  RR  /\  ( Z  /  ( K ^ J ) )  e.  RR  /\  ( Z  /  V )  <_ 
( Z  /  ( K ^ J ) ) )  ->  ( |_ `  ( Z  /  V
) )  <_  ( |_ `  ( Z  / 
( K ^ J
) ) ) )
8472, 75, 82, 83syl3anc 1228 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  <_  ( |_ `  ( Z  /  ( K ^ J ) ) ) )
85 eluz2 11112 . . . . 5  |-  ( ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) )  <-> 
( ( |_ `  ( Z  /  V
) )  e.  ZZ  /\  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ  /\  ( |_ `  ( Z  /  V ) )  <_ 
( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
8673, 76, 84, 85syl3anbrc 1180 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) ) )
87 fzss2 11749 . . . 4  |-  ( ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) )  ->  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )  C_  ( (
( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
8886, 87syl 16 . . 3  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
8970, 88sstrd 3509 . 2  |-  ( ph  ->  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) ) 
C_  ( ( ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  /  ( K ^ J ) ) ) ) )
90 pntlem1.i . 2  |-  I  =  ( ( ( |_
`  ( Z  / 
( ( 1  +  ( L  x.  E
) )  x.  V
) ) )  +  1 ) ... ( |_ `  ( Z  /  V ) ) )
91 pntlem1.o . 2  |-  O  =  ( ( ( |_
`  ( Z  / 
( K ^ ( J  +  1 ) ) ) )  +  1 ) ... ( |_ `  ( Z  / 
( K ^ J
) ) ) )
9289, 90, 913sstr4g 3540 1  |-  ( ph  ->  I  C_  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   +oocpnf 9642    < clt 9645    <_ cle 9646    - cmin 9824    / cdiv 10227   NNcn 10556   2c2 10606   3c3 10607   4c4 10608   ZZcz 10885  ;cdc 11000   ZZ>=cuz 11106   RR+crp 11245   (,)cioo 11554   [,)cico 11556   [,]cicc 11557   ...cfz 11697  ..^cfzo 11821   |_cfl 11930   ^cexp 12169   sqrcsqrt 13078   abscabs 13079   expce 13809   _eceu 13810   logclog 23068  ψcchp 23492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ioc 11559  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11822  df-fl 11932  df-mod 12000  df-seq 12111  df-exp 12170  df-fac 12357  df-bc 12384  df-hash 12409  df-shft 12912  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-limsup 13306  df-clim 13323  df-rlim 13324  df-sum 13521  df-ef 13815  df-e 13816  df-sin 13817  df-cos 13818  df-pi 13820  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-rest 14840  df-topn 14841  df-0g 14859  df-gsum 14860  df-topgen 14861  df-pt 14862  df-prds 14865  df-xrs 14919  df-qtop 14924  df-imas 14925  df-xps 14927  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-mulg 16187  df-cntz 16482  df-cmn 16927  df-psmet 18538  df-xmet 18539  df-met 18540  df-bl 18541  df-mopn 18542  df-fbas 18543  df-fg 18544  df-cnfld 18548  df-top 19526  df-bases 19528  df-topon 19529  df-topsp 19530  df-cld 19647  df-ntr 19648  df-cls 19649  df-nei 19726  df-lp 19764  df-perf 19765  df-cn 19855  df-cnp 19856  df-haus 19943  df-tx 20189  df-hmeo 20382  df-fil 20473  df-fm 20565  df-flim 20566  df-flf 20567  df-xms 20949  df-ms 20950  df-tms 20951  df-cncf 21508  df-limc 22396  df-dv 22397  df-log 23070
This theorem is referenced by:  pntlemj  23914
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