MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ef01bndlem Structured version   Unicode version

Theorem ef01bndlem 14205
Description: Lemma for sin01bnd 14206 and cos01bnd 14207. (Contributed by Paul Chapman, 19-Jan-2008.)
Hypothesis
Ref Expression
ef01bnd.1  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
Assertion
Ref Expression
ef01bndlem  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <  ( ( A ^ 4 )  /  6 ) )
Distinct variable groups:    k, n, A    k, F
Allowed substitution hint:    F( n)

Proof of Theorem ef01bndlem
StepHypRef Expression
1 ax-icn 9587 . . . . 5  |-  _i  e.  CC
2 0xr 9676 . . . . . . . 8  |-  0  e.  RR*
3 1re 9631 . . . . . . . 8  |-  1  e.  RR
4 elioc2 11686 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
52, 3, 4mp2an 676 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
65simp1bi 1020 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
76recnd 9658 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
8 mulcl 9612 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
91, 7, 8sylancr 667 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
_i  x.  A )  e.  CC )
10 4nn0 10877 . . . 4  |-  4  e.  NN0
11 ef01bnd.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
1211eftlcl 14128 . . . 4  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
139, 10, 12sylancl 666 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
1413abscld 13465 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  e.  RR )
15 reexpcl 12275 . . . 4  |-  ( ( A  e.  RR  /\  4  e.  NN0 )  -> 
( A ^ 4 )  e.  RR )
166, 10, 15sylancl 666 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR )
17 4re 10675 . . . . 5  |-  4  e.  RR
1817, 3readdcli 9645 . . . 4  |-  ( 4  +  1 )  e.  RR
19 faccl 12455 . . . . . 6  |-  ( 4  e.  NN0  ->  ( ! `
 4 )  e.  NN )
2010, 19ax-mp 5 . . . . 5  |-  ( ! `
 4 )  e.  NN
21 4nn 10758 . . . . 5  |-  4  e.  NN
2220, 21nnmulcli 10622 . . . 4  |-  ( ( ! `  4 )  x.  4 )  e.  NN
23 nndivre 10634 . . . 4  |-  ( ( ( 4  +  1 )  e.  RR  /\  ( ( ! ` 
4 )  x.  4 )  e.  NN )  ->  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) )  e.  RR )
2418, 22, 23mp2an 676 . . 3  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  e.  RR
25 remulcl 9613 . . 3  |-  ( ( ( A ^ 4 )  e.  RR  /\  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )  e.  RR )  ->  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) )  e.  RR )
2616, 24, 25sylancl 666 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  e.  RR )
27 6nn 10760 . . 3  |-  6  e.  NN
28 nndivre 10634 . . 3  |-  ( ( ( A ^ 4 )  e.  RR  /\  6  e.  NN )  ->  ( ( A ^
4 )  /  6
)  e.  RR )
2916, 27, 28sylancl 666 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  e.  RR )
30 eqid 2420 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( abs `  (
_i  x.  A )
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( abs `  ( _i  x.  A ) ) ^ n )  / 
( ! `  n
) ) )
31 eqid 2420 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( ( abs `  (
_i  x.  A )
) ^ 4 )  /  ( ! ` 
4 ) )  x.  ( ( 1  / 
( 4  +  1 ) ) ^ n
) ) )  =  ( n  e.  NN0  |->  ( ( ( ( abs `  ( _i  x.  A ) ) ^ 4 )  / 
( ! `  4
) )  x.  (
( 1  /  (
4  +  1 ) ) ^ n ) ) )
3221a1i 11 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  4  e.  NN )
33 absmul 13325 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
_i  x.  A )
)  =  ( ( abs `  _i )  x.  ( abs `  A
) ) )
341, 7, 33sylancr 667 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  ( ( abs `  _i )  x.  ( abs `  A ) ) )
35 absi 13317 . . . . . . . 8  |-  ( abs `  _i )  =  1
3635oveq1i 6306 . . . . . . 7  |-  ( ( abs `  _i )  x.  ( abs `  A
) )  =  ( 1  x.  ( abs `  A ) )
375simp2bi 1021 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
386, 37elrpd 11327 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR+ )
39 rpre 11297 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
40 rpge0 11303 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  0  <_  A )
4139, 40absidd 13452 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( abs `  A )  =  A )
4238, 41syl 17 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  A )  =  A )
4342oveq2d 6312 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
4436, 43syl5eq 2473 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  _i )  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
457mulid2d 9650 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  A )  =  A )
4634, 44, 453eqtrd 2465 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  A )
475simp3bi 1022 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
4846, 47eqbrtrd 4437 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  <_ 
1 )
4911, 30, 31, 32, 9, 48eftlub 14130 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( (
( abs `  (
_i  x.  A )
) ^ 4 )  x.  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) ) ) )
5046oveq1d 6311 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  (
_i  x.  A )
) ^ 4 )  =  ( A ^
4 ) )
5150oveq1d 6311 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( abs `  (
_i  x.  A )
) ^ 4 )  x.  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) ) )  =  ( ( A ^
4 )  x.  (
( 4  +  1 )  /  ( ( ! `  4 )  x.  4 ) ) ) )
5249, 51breqtrd 4441 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) ) )
53 3pos 10692 . . . . . . . . 9  |-  0  <  3
54 0re 9632 . . . . . . . . . 10  |-  0  e.  RR
55 3re 10672 . . . . . . . . . 10  |-  3  e.  RR
56 5re 10677 . . . . . . . . . 10  |-  5  e.  RR
5754, 55, 56ltadd1i 10157 . . . . . . . . 9  |-  ( 0  <  3  <->  ( 0  +  5 )  < 
( 3  +  5 ) )
5853, 57mpbi 211 . . . . . . . 8  |-  ( 0  +  5 )  < 
( 3  +  5 )
59 5cn 10678 . . . . . . . . 9  |-  5  e.  CC
6059addid2i 9810 . . . . . . . 8  |-  ( 0  +  5 )  =  5
61 cu2 12359 . . . . . . . . 9  |-  ( 2 ^ 3 )  =  8
62 5p3e8 10737 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
63 3nn 10757 . . . . . . . . . . 11  |-  3  e.  NN
6463nncni 10608 . . . . . . . . . 10  |-  3  e.  CC
6559, 64addcomi 9813 . . . . . . . . 9  |-  ( 5  +  3 )  =  ( 3  +  5 )
6661, 62, 653eqtr2ri 2456 . . . . . . . 8  |-  ( 3  +  5 )  =  ( 2 ^ 3 )
6758, 60, 663brtr3i 4444 . . . . . . 7  |-  5  <  ( 2 ^ 3 )
68 2re 10668 . . . . . . . 8  |-  2  e.  RR
69 1le2 10812 . . . . . . . 8  |-  1  <_  2
70 4z 10960 . . . . . . . . 9  |-  4  e.  ZZ
71 3lt4 10768 . . . . . . . . . 10  |-  3  <  4
7255, 17, 71ltleii 9746 . . . . . . . . 9  |-  3  <_  4
7363nnzi 10950 . . . . . . . . . 10  |-  3  e.  ZZ
7473eluz1i 11155 . . . . . . . . 9  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 4  e.  ZZ  /\  3  <_ 
4 ) )
7570, 72, 74mpbir2an 928 . . . . . . . 8  |-  4  e.  ( ZZ>= `  3 )
76 leexp2a 12314 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  1  <_  2  /\  4  e.  ( ZZ>= `  3 )
)  ->  ( 2 ^ 3 )  <_ 
( 2 ^ 4 ) )
7768, 69, 75, 76mp3an 1360 . . . . . . 7  |-  ( 2 ^ 3 )  <_ 
( 2 ^ 4 )
78 8re 10683 . . . . . . . . 9  |-  8  e.  RR
7961, 78eqeltri 2504 . . . . . . . 8  |-  ( 2 ^ 3 )  e.  RR
80 2nn 10756 . . . . . . . . . 10  |-  2  e.  NN
81 nnexpcl 12271 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  4  e.  NN0 )  -> 
( 2 ^ 4 )  e.  NN )
8280, 10, 81mp2an 676 . . . . . . . . 9  |-  ( 2 ^ 4 )  e.  NN
8382nnrei 10607 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  RR
8456, 79, 83ltletri 9751 . . . . . . 7  |-  ( ( 5  <  ( 2 ^ 3 )  /\  ( 2 ^ 3 )  <_  ( 2 ^ 4 ) )  ->  5  <  (
2 ^ 4 ) )
8567, 77, 84mp2an 676 . . . . . 6  |-  5  <  ( 2 ^ 4 )
86 6re 10679 . . . . . . . 8  |-  6  e.  RR
8786, 83remulcli 9646 . . . . . . 7  |-  ( 6  x.  ( 2 ^ 4 ) )  e.  RR
88 6pos 10697 . . . . . . . 8  |-  0  <  6
8982nngt0i 10632 . . . . . . . 8  |-  0  <  ( 2 ^ 4 )
9086, 83, 88, 89mulgt0ii 9757 . . . . . . 7  |-  0  <  ( 6  x.  (
2 ^ 4 ) )
9156, 83, 87, 90ltdiv1ii 10525 . . . . . 6  |-  ( 5  <  ( 2 ^ 4 )  <->  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) ) )
9285, 91mpbi 211 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
93 df-5 10660 . . . . . 6  |-  5  =  ( 4  +  1 )
94 df-4 10659 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
9594fveq2i 5875 . . . . . . . . . 10  |-  ( ! `
 4 )  =  ( ! `  (
3  +  1 ) )
96 3nn0 10876 . . . . . . . . . . 11  |-  3  e.  NN0
97 facp1 12450 . . . . . . . . . . 11  |-  ( 3  e.  NN0  ->  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) ) )
9896, 97ax-mp 5 . . . . . . . . . 10  |-  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) )
99 sq2 12357 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
10099, 94eqtr2i 2450 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  ( 2 ^ 2 )
101100oveq2i 6307 . . . . . . . . . 10  |-  ( ( ! `  3 )  x.  ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
10295, 98, 1013eqtri 2453 . . . . . . . . 9  |-  ( ! `
 4 )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
103102oveq1i 6306 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ( ! `
 3 )  x.  ( 2 ^ 2 ) )  x.  (
2 ^ 2 ) )
10499oveq2i 6307 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
4 )  x.  4 )
105 fac3 12452 . . . . . . . . . 10  |-  ( ! `
 3 )  =  6
106 6cn 10680 . . . . . . . . . 10  |-  6  e.  CC
107105, 106eqeltri 2504 . . . . . . . . 9  |-  ( ! `
 3 )  e.  CC
10817recni 9644 . . . . . . . . . 10  |-  4  e.  CC
10999, 108eqeltri 2504 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
110107, 109, 109mulassi 9641 . . . . . . . 8  |-  ( ( ( ! `  3
)  x.  ( 2 ^ 2 ) )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
111103, 104, 1103eqtr3i 2457 . . . . . . 7  |-  ( ( ! `  4 )  x.  4 )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
112 2p2e4 10716 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
113112oveq2i 6307 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( 2 ^ 4 )
114 2cn 10669 . . . . . . . . . 10  |-  2  e.  CC
115 2nn0 10875 . . . . . . . . . 10  |-  2  e.  NN0
116 expadd 12300 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  2  e.  NN0  /\  2  e.  NN0 )  ->  (
2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
117114, 115, 115, 116mp3an 1360 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
118113, 117eqtr3i 2451 . . . . . . . 8  |-  ( 2 ^ 4 )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
119118oveq2i 6307 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
120105oveq1i 6306 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( 6  x.  (
2 ^ 4 ) )
121111, 119, 1203eqtr2ri 2456 . . . . . 6  |-  ( 6  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
4 )  x.  4 )
12293, 121oveq12i 6308 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )
12382nncni 10608 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  CC
124123mulid2i 9635 . . . . . . 7  |-  ( 1  x.  ( 2 ^ 4 ) )  =  ( 2 ^ 4 )
125124oveq1i 6306 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
12682nnne0i 10633 . . . . . . . . 9  |-  ( 2 ^ 4 )  =/=  0
127123, 126dividi 10329 . . . . . . . 8  |-  ( ( 2 ^ 4 )  /  ( 2 ^ 4 ) )  =  1
128127oveq2i 6307 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  / 
6 )  x.  1 )
129 ax-1cn 9586 . . . . . . . 8  |-  1  e.  CC
13086, 88gt0ne0ii 10139 . . . . . . . 8  |-  6  =/=  0
131129, 106, 123, 123, 130, 126divmuldivi 10356 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  x.  ( 2 ^ 4 ) )  /  (
6  x.  ( 2 ^ 4 ) ) )
13286, 130rereccli 10361 . . . . . . . . 9  |-  ( 1  /  6 )  e.  RR
133132recni 9644 . . . . . . . 8  |-  ( 1  /  6 )  e.  CC
134133mulid1i 9634 . . . . . . 7  |-  ( ( 1  /  6 )  x.  1 )  =  ( 1  /  6
)
135128, 131, 1343eqtr3i 2457 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
136125, 135eqtr3i 2451 . . . . 5  |-  ( ( 2 ^ 4 )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
13792, 122, 1363brtr3i 4444 . . . 4  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  < 
( 1  /  6
)
138 rpexpcl 12277 . . . . . 6  |-  ( ( A  e.  RR+  /\  4  e.  ZZ )  ->  ( A ^ 4 )  e.  RR+ )
13938, 70, 138sylancl 666 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR+ )
140 elrp 11293 . . . . . 6  |-  ( ( A ^ 4 )  e.  RR+  <->  ( ( A ^ 4 )  e.  RR  /\  0  < 
( A ^ 4 ) ) )
141 ltmul2 10445 . . . . . . 7  |-  ( ( ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )  e.  RR  /\  ( 1  /  6
)  e.  RR  /\  ( ( A ^
4 )  e.  RR  /\  0  <  ( A ^ 4 ) ) )  ->  ( (
( 4  +  1 )  /  ( ( ! `  4 )  x.  4 ) )  <  ( 1  / 
6 )  <->  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) )  <  (
( A ^ 4 )  x.  ( 1  /  6 ) ) ) )
14224, 132, 141mp3an12 1350 . . . . . 6  |-  ( ( ( A ^ 4 )  e.  RR  /\  0  <  ( A ^
4 ) )  -> 
( ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) )  <  (
1  /  6 )  <-> 
( ( A ^
4 )  x.  (
( 4  +  1 )  /  ( ( ! `  4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  x.  ( 1  / 
6 ) ) ) )
143140, 142sylbi 198 . . . . 5  |-  ( ( A ^ 4 )  e.  RR+  ->  ( ( ( 4  +  1 )  /  ( ( ! `  4 )  x.  4 ) )  <  ( 1  / 
6 )  <->  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) )  <  (
( A ^ 4 )  x.  ( 1  /  6 ) ) ) )
144139, 143syl 17 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )  <  ( 1  /  6 )  <->  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) )  <  (
( A ^ 4 )  x.  ( 1  /  6 ) ) ) )
145137, 144mpbii 214 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
14616recnd 9658 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  CC )
147 divrec 10275 . . . . 5  |-  ( ( ( A ^ 4 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
148106, 130, 147mp3an23 1352 . . . 4  |-  ( ( A ^ 4 )  e.  CC  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
149146, 148syl 17 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
150145, 149breqtrrd 4443 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  / 
6 ) )
15114, 26, 29, 52, 150lelttrd 9782 1  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <  ( ( A ^ 4 )  /  6 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   class class class wbr 4417    |-> cmpt 4475   ` cfv 5592  (class class class)co 6296   CCcc 9526   RRcr 9527   0cc0 9528   1c1 9529   _ici 9530    + caddc 9531    x. cmul 9533   RR*cxr 9663    < clt 9664    <_ cle 9665    / cdiv 10258   NNcn 10598   2c2 10648   3c3 10649   4c4 10650   5c5 10651   6c6 10652   8c8 10654   NN0cn0 10858   ZZcz 10926   ZZ>=cuz 11148   RR+crp 11291   (,]cioc 11625   ^cexp 12258   !cfa 12445   abscabs 13265   sum_csu 13719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606  ax-addf 9607  ax-mulf 9608
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-pm 7474  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-sup 7953  df-inf 7954  df-oi 8016  df-card 8363  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-n0 10859  df-z 10927  df-uz 11149  df-rp 11292  df-ioc 11629  df-ico 11630  df-fz 11772  df-fzo 11903  df-fl 12014  df-seq 12200  df-exp 12259  df-fac 12446  df-hash 12502  df-shft 13098  df-cj 13130  df-re 13131  df-im 13132  df-sqrt 13266  df-abs 13267  df-limsup 13493  df-clim 13519  df-rlim 13520  df-sum 13720
This theorem is referenced by:  sin01bnd  14206  cos01bnd  14207
  Copyright terms: Public domain W3C validator