MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntlemq Structured version   Unicode version

Theorem pntlemq 22734
Description: Lemma for pntlemj 22736. (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 22730 . . . . . . . . 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 993 . . . . . . . 8  |-  ( ph  ->  Z  e.  RR+ )
181, 2, 3, 4, 5, 6, 7, 8, 9, 10pntlemc 22728 . . . . . . . . . 10  |-  ( ph  ->  ( E  e.  RR+  /\  K  e.  RR+  /\  ( E  e.  ( 0 (,) 1 )  /\  1  <  K  /\  ( U  -  E )  e.  RR+ ) ) )
1918simp2d 994 . . . . . . . . 9  |-  ( ph  ->  K  e.  RR+ )
20 pntlem1.j . . . . . . . . . . 11  |-  ( ph  ->  J  e.  ( M..^ N ) )
21 elfzoelz 11536 . . . . . . . . . . 11  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ZZ )
2220, 21syl 16 . . . . . . . . . 10  |-  ( ph  ->  J  e.  ZZ )
2322peano2zd 10737 . . . . . . . . 9  |-  ( ph  ->  ( J  +  1 )  e.  ZZ )
2419, 23rpexpcld 12014 . . . . . . . 8  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  e.  RR+ )
2517, 24rpdivcld 11031 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR+ )
2625rpred 11014 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ ( J  + 
1 ) ) )  e.  RR )
2726flcld 11631 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  e.  ZZ )
28 1rp 10982 . . . . . . . . . 10  |-  1  e.  RR+
291, 2, 3, 4, 5, 6pntlemd 22727 . . . . . . . . . . . 12  |-  ( ph  ->  ( L  e.  RR+  /\  D  e.  RR+  /\  F  e.  RR+ ) )
3029simp1d 993 . . . . . . . . . . 11  |-  ( ph  ->  L  e.  RR+ )
3118simp1d 993 . . . . . . . . . . 11  |-  ( ph  ->  E  e.  RR+ )
3230, 31rpmulcld 11030 . . . . . . . . . 10  |-  ( ph  ->  ( L  x.  E
)  e.  RR+ )
33 rpaddcl 10998 . . . . . . . . . 10  |-  ( ( 1  e.  RR+  /\  ( L  x.  E )  e.  RR+ )  ->  (
1  +  ( L  x.  E ) )  e.  RR+ )
3428, 32, 33sylancr 656 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  ( L  x.  E ) )  e.  RR+ )
35 pntlem1.v . . . . . . . . 9  |-  ( ph  ->  V  e.  RR+ )
3634, 35rpmulcld 11030 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR+ )
3717, 36rpdivcld 11031 . . . . . . 7  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR+ )
3837rpred 11014 . . . . . 6  |-  ( ph  ->  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) )  e.  RR )
3938flcld 11631 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ZZ )
4036rpred 11014 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  e.  RR )
4124rpred 11014 . . . . . . . 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 456 . . . . . . . . . 10  |-  ( ph  ->  ( ( K ^ J )  <  V  /\  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) ) )
4443simprd 460 . . . . . . . . 9  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K  x.  ( K ^ J
) ) )
4519rpcnd 11016 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  CC )
4619, 22rpexpcld 12014 . . . . . . . . . . . 12  |-  ( ph  ->  ( K ^ J
)  e.  RR+ )
4746rpcnd 11016 . . . . . . . . . . 11  |-  ( ph  ->  ( K ^ J
)  e.  CC )
4845, 47mulcomd 9394 . . . . . . . . . 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 22731 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  ( ZZ>= `  M )  /\  (
( ( log `  Z
)  /  ( log `  K ) )  / 
4 )  <_  ( N  -  M )
) )
5251simp1d 993 . . . . . . . . . . . . 13  |-  ( ph  ->  M  e.  NN )
53 elfzouz 11540 . . . . . . . . . . . . . 14  |-  ( J  e.  ( M..^ N
)  ->  J  e.  ( ZZ>= `  M )
)
5420, 53syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  J  e.  ( ZZ>= `  M ) )
55 eluznn 10912 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  J  e.  ( ZZ>= `  M ) )  ->  J  e.  NN )
5652, 54, 55syl2anc 654 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  NN )
5756nnnn0d 10623 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  NN0 )
5845, 57expp1d 11992 . . . . . . . . . 10  |-  ( ph  ->  ( K ^ ( J  +  1 ) )  =  ( ( K ^ J )  x.  K ) )
5948, 58eqtr4d 2468 . . . . . . . . 9  |-  ( ph  ->  ( K  x.  ( K ^ J ) )  =  ( K ^
( J  +  1 ) ) )
6044, 59breqtrd 4304 . . . . . . . 8  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <  ( K ^ ( J  + 
1 ) ) )
6140, 41, 60ltled 9509 . . . . . . 7  |-  ( ph  ->  ( ( 1  +  ( L  x.  E
) )  x.  V
)  <_  ( K ^ ( J  + 
1 ) ) )
6236, 24, 17lediv2d 11038 . . . . . . 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 11643 . . . . . 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 1211 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^
( J  +  1 ) ) ) )  <_  ( |_ `  ( Z  /  (
( 1  +  ( L  x.  E ) )  x.  V ) ) ) )
66 eluz2 10854 . . . . 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 1165 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( ( 1  +  ( L  x.  E ) )  x.  V ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  / 
( K ^ ( J  +  1 ) ) ) ) ) )
68 eluzp1p1 10873 . . . 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 11483 . . . 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 11031 . . . . . . 7  |-  ( ph  ->  ( Z  /  V
)  e.  RR+ )
7271rpred 11014 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  e.  RR )
7372flcld 11631 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  e.  ZZ )
7417, 46rpdivcld 11031 . . . . . . 7  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR+ )
7574rpred 11014 . . . . . 6  |-  ( ph  ->  ( Z  /  ( K ^ J ) )  e.  RR )
7675flcld 11631 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ZZ )
7746rpred 11014 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  e.  RR )
7835rpred 11014 . . . . . . . 8  |-  ( ph  ->  V  e.  RR )
7943simpld 456 . . . . . . . 8  |-  ( ph  ->  ( K ^ J
)  <  V )
8077, 78, 79ltled 9509 . . . . . . 7  |-  ( ph  ->  ( K ^ J
)  <_  V )
8146, 35, 17lediv2d 11038 . . . . . . 7  |-  ( ph  ->  ( ( K ^ J )  <_  V  <->  ( Z  /  V )  <_  ( Z  / 
( K ^ J
) ) ) )
8280, 81mpbid 210 . . . . . 6  |-  ( ph  ->  ( Z  /  V
)  <_  ( Z  /  ( K ^ J ) ) )
83 flwordi 11643 . . . . . 6  |-  ( ( ( Z  /  V
)  e.  RR  /\  ( Z  /  ( K ^ J ) )  e.  RR  /\  ( Z  /  V )  <_ 
( Z  /  ( K ^ J ) ) )  ->  ( |_ `  ( Z  /  V
) )  <_  ( |_ `  ( Z  / 
( K ^ J
) ) ) )
8472, 75, 82, 83syl3anc 1211 . . . . 5  |-  ( ph  ->  ( |_ `  ( Z  /  V ) )  <_  ( |_ `  ( Z  /  ( K ^ J ) ) ) )
85 eluz2 10854 . . . . 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 1165 . . . 4  |-  ( ph  ->  ( |_ `  ( Z  /  ( K ^ J ) ) )  e.  ( ZZ>= `  ( |_ `  ( Z  /  V ) ) ) )
87 fzss2 11484 . . . 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 3354 . 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 3385 1  |-  ( ph  ->  I  C_  O )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   +oocpnf 9402    < clt 9405    <_ cle 9406    - cmin 9582    / cdiv 9980   NNcn 10309   2c2 10358   3c3 10359   4c4 10360   ZZcz 10633  ;cdc 10742   ZZ>=cuz 10848   RR+crp 10978   (,)cioo 11287   [,)cico 11289   [,]cicc 11290   ...cfz 11423  ..^cfzo 11531   |_cfl 11623   ^cexp 11848   sqrcsqr 12705   abscabs 12706   expce 13329   _eceu 13330   logclog 21890  ψcchp 22314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335  df-e 13336  df-sin 13337  df-cos 13338  df-pi 13340  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892
This theorem is referenced by:  pntlemj  22736
  Copyright terms: Public domain W3C validator