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Theorem ef01bndlem 13773
Description: Lemma for sin01bnd 13774 and cos01bnd 13775. (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 9547 . . . . 5  |-  _i  e.  CC
2 0xr 9636 . . . . . . . 8  |-  0  e.  RR*
3 1re 9591 . . . . . . . 8  |-  1  e.  RR
4 elioc2 11583 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
52, 3, 4mp2an 672 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
65simp1bi 1011 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
76recnd 9618 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
8 mulcl 9572 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
91, 7, 8sylancr 663 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
_i  x.  A )  e.  CC )
10 4nn0 10810 . . . 4  |-  4  e.  NN0
11 ef01bnd.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
1211eftlcl 13696 . . . 4  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
139, 10, 12sylancl 662 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
1413abscld 13223 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  e.  RR )
15 reexpcl 12146 . . . 4  |-  ( ( A  e.  RR  /\  4  e.  NN0 )  -> 
( A ^ 4 )  e.  RR )
166, 10, 15sylancl 662 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR )
17 4re 10608 . . . . 5  |-  4  e.  RR
1817, 3readdcli 9605 . . . 4  |-  ( 4  +  1 )  e.  RR
19 faccl 12325 . . . . . 6  |-  ( 4  e.  NN0  ->  ( ! `
 4 )  e.  NN )
2010, 19ax-mp 5 . . . . 5  |-  ( ! `
 4 )  e.  NN
21 4nn 10691 . . . . 5  |-  4  e.  NN
2220, 21nnmulcli 10556 . . . 4  |-  ( ( ! `  4 )  x.  4 )  e.  NN
23 nndivre 10567 . . . 4  |-  ( ( ( 4  +  1 )  e.  RR  /\  ( ( ! ` 
4 )  x.  4 )  e.  NN )  ->  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) )  e.  RR )
2418, 22, 23mp2an 672 . . 3  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  e.  RR
25 remulcl 9573 . . 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 662 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  e.  RR )
27 6nn 10693 . . 3  |-  6  e.  NN
28 nndivre 10567 . . 3  |-  ( ( ( A ^ 4 )  e.  RR  /\  6  e.  NN )  ->  ( ( A ^
4 )  /  6
)  e.  RR )
2916, 27, 28sylancl 662 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  e.  RR )
30 eqid 2467 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( abs `  (
_i  x.  A )
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( abs `  ( _i  x.  A ) ) ^ n )  / 
( ! `  n
) ) )
31 eqid 2467 . . . 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 13084 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
_i  x.  A )
)  =  ( ( abs `  _i )  x.  ( abs `  A
) ) )
341, 7, 33sylancr 663 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  ( ( abs `  _i )  x.  ( abs `  A ) ) )
35 absi 13076 . . . . . . . 8  |-  ( abs `  _i )  =  1
3635oveq1i 6292 . . . . . . 7  |-  ( ( abs `  _i )  x.  ( abs `  A
) )  =  ( 1  x.  ( abs `  A ) )
375simp2bi 1012 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
386, 37elrpd 11250 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR+ )
39 rpre 11222 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
40 rpge0 11228 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  0  <_  A )
4139, 40absidd 13210 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( abs `  A )  =  A )
4238, 41syl 16 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  A )  =  A )
4342oveq2d 6298 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
4436, 43syl5eq 2520 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  _i )  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
457mulid2d 9610 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  A )  =  A )
4634, 44, 453eqtrd 2512 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  A )
475simp3bi 1013 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
4846, 47eqbrtrd 4467 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  <_ 
1 )
4911, 30, 31, 32, 9, 48eftlub 13698 . . 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 6297 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  (
_i  x.  A )
) ^ 4 )  =  ( A ^
4 ) )
5150oveq1d 6297 . . 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 4471 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) ) )
53 3pos 10625 . . . . . . . . 9  |-  0  <  3
54 0re 9592 . . . . . . . . . 10  |-  0  e.  RR
55 3re 10605 . . . . . . . . . 10  |-  3  e.  RR
56 5re 10610 . . . . . . . . . 10  |-  5  e.  RR
5754, 55, 56ltadd1i 10103 . . . . . . . . 9  |-  ( 0  <  3  <->  ( 0  +  5 )  < 
( 3  +  5 ) )
5853, 57mpbi 208 . . . . . . . 8  |-  ( 0  +  5 )  < 
( 3  +  5 )
59 5cn 10611 . . . . . . . . 9  |-  5  e.  CC
6059addid2i 9763 . . . . . . . 8  |-  ( 0  +  5 )  =  5
61 cu2 12228 . . . . . . . . 9  |-  ( 2 ^ 3 )  =  8
62 5p3e8 10670 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
63 3nn 10690 . . . . . . . . . . 11  |-  3  e.  NN
6463nncni 10542 . . . . . . . . . 10  |-  3  e.  CC
6559, 64addcomi 9766 . . . . . . . . 9  |-  ( 5  +  3 )  =  ( 3  +  5 )
6661, 62, 653eqtr2ri 2503 . . . . . . . 8  |-  ( 3  +  5 )  =  ( 2 ^ 3 )
6758, 60, 663brtr3i 4474 . . . . . . 7  |-  5  <  ( 2 ^ 3 )
68 2re 10601 . . . . . . . 8  |-  2  e.  RR
69 1le2 10745 . . . . . . . 8  |-  1  <_  2
7021nnzi 10884 . . . . . . . . 9  |-  4  e.  ZZ
71 3lt4 10701 . . . . . . . . . 10  |-  3  <  4
7255, 17, 71ltleii 9703 . . . . . . . . 9  |-  3  <_  4
7363nnzi 10884 . . . . . . . . . 10  |-  3  e.  ZZ
7473eluz1i 11085 . . . . . . . . 9  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 4  e.  ZZ  /\  3  <_ 
4 ) )
7570, 72, 74mpbir2an 918 . . . . . . . 8  |-  4  e.  ( ZZ>= `  3 )
76 leexp2a 12183 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  1  <_  2  /\  4  e.  ( ZZ>= `  3 )
)  ->  ( 2 ^ 3 )  <_ 
( 2 ^ 4 ) )
7768, 69, 75, 76mp3an 1324 . . . . . . 7  |-  ( 2 ^ 3 )  <_ 
( 2 ^ 4 )
78 8re 10616 . . . . . . . . 9  |-  8  e.  RR
7961, 78eqeltri 2551 . . . . . . . 8  |-  ( 2 ^ 3 )  e.  RR
80 2nn 10689 . . . . . . . . . 10  |-  2  e.  NN
81 nnexpcl 12142 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  4  e.  NN0 )  -> 
( 2 ^ 4 )  e.  NN )
8280, 10, 81mp2an 672 . . . . . . . . 9  |-  ( 2 ^ 4 )  e.  NN
8382nnrei 10541 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  RR
8456, 79, 83ltletri 9708 . . . . . . 7  |-  ( ( 5  <  ( 2 ^ 3 )  /\  ( 2 ^ 3 )  <_  ( 2 ^ 4 ) )  ->  5  <  (
2 ^ 4 ) )
8567, 77, 84mp2an 672 . . . . . 6  |-  5  <  ( 2 ^ 4 )
86 6re 10612 . . . . . . . 8  |-  6  e.  RR
8786, 83remulcli 9606 . . . . . . 7  |-  ( 6  x.  ( 2 ^ 4 ) )  e.  RR
88 6pos 10630 . . . . . . . 8  |-  0  <  6
8982nngt0i 10565 . . . . . . . 8  |-  0  <  ( 2 ^ 4 )
9086, 83, 88, 89mulgt0ii 9713 . . . . . . 7  |-  0  <  ( 6  x.  (
2 ^ 4 ) )
9156, 83, 87, 90ltdiv1ii 10471 . . . . . 6  |-  ( 5  <  ( 2 ^ 4 )  <->  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) ) )
9285, 91mpbi 208 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
93 df-5 10593 . . . . . 6  |-  5  =  ( 4  +  1 )
94 df-4 10592 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
9594fveq2i 5867 . . . . . . . . . 10  |-  ( ! `
 4 )  =  ( ! `  (
3  +  1 ) )
96 3nn0 10809 . . . . . . . . . . 11  |-  3  e.  NN0
97 facp1 12320 . . . . . . . . . . 11  |-  ( 3  e.  NN0  ->  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) ) )
9896, 97ax-mp 5 . . . . . . . . . 10  |-  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) )
99 sq2 12226 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
10099, 94eqtr2i 2497 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  ( 2 ^ 2 )
101100oveq2i 6293 . . . . . . . . . 10  |-  ( ( ! `  3 )  x.  ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
10295, 98, 1013eqtri 2500 . . . . . . . . 9  |-  ( ! `
 4 )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
103102oveq1i 6292 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ( ! `
 3 )  x.  ( 2 ^ 2 ) )  x.  (
2 ^ 2 ) )
10499oveq2i 6293 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
4 )  x.  4 )
105 fac3 12322 . . . . . . . . . 10  |-  ( ! `
 3 )  =  6
106 6cn 10613 . . . . . . . . . 10  |-  6  e.  CC
107105, 106eqeltri 2551 . . . . . . . . 9  |-  ( ! `
 3 )  e.  CC
10817recni 9604 . . . . . . . . . 10  |-  4  e.  CC
10999, 108eqeltri 2551 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
110107, 109, 109mulassi 9601 . . . . . . . 8  |-  ( ( ( ! `  3
)  x.  ( 2 ^ 2 ) )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
111103, 104, 1103eqtr3i 2504 . . . . . . 7  |-  ( ( ! `  4 )  x.  4 )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
112 2p2e4 10649 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
113112oveq2i 6293 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( 2 ^ 4 )
114 2cn 10602 . . . . . . . . . 10  |-  2  e.  CC
115 2nn0 10808 . . . . . . . . . 10  |-  2  e.  NN0
116 expadd 12170 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  2  e.  NN0  /\  2  e.  NN0 )  ->  (
2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
117114, 115, 115, 116mp3an 1324 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
118113, 117eqtr3i 2498 . . . . . . . 8  |-  ( 2 ^ 4 )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
119118oveq2i 6293 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
120105oveq1i 6292 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( 6  x.  (
2 ^ 4 ) )
121111, 119, 1203eqtr2ri 2503 . . . . . 6  |-  ( 6  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
4 )  x.  4 )
12293, 121oveq12i 6294 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )
12382nncni 10542 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  CC
124123mulid2i 9595 . . . . . . 7  |-  ( 1  x.  ( 2 ^ 4 ) )  =  ( 2 ^ 4 )
125124oveq1i 6292 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
12682nnne0i 10566 . . . . . . . . 9  |-  ( 2 ^ 4 )  =/=  0
127123, 126dividi 10273 . . . . . . . 8  |-  ( ( 2 ^ 4 )  /  ( 2 ^ 4 ) )  =  1
128127oveq2i 6293 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  / 
6 )  x.  1 )
129 ax-1cn 9546 . . . . . . . 8  |-  1  e.  CC
13086, 88gt0ne0ii 10085 . . . . . . . 8  |-  6  =/=  0
131129, 106, 123, 123, 130, 126divmuldivi 10300 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  x.  ( 2 ^ 4 ) )  /  (
6  x.  ( 2 ^ 4 ) ) )
13286, 130rereccli 10305 . . . . . . . . 9  |-  ( 1  /  6 )  e.  RR
133132recni 9604 . . . . . . . 8  |-  ( 1  /  6 )  e.  CC
134133mulid1i 9594 . . . . . . 7  |-  ( ( 1  /  6 )  x.  1 )  =  ( 1  /  6
)
135128, 131, 1343eqtr3i 2504 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
136125, 135eqtr3i 2498 . . . . 5  |-  ( ( 2 ^ 4 )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
13792, 122, 1363brtr3i 4474 . . . 4  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  < 
( 1  /  6
)
138 rpexpcl 12148 . . . . . 6  |-  ( ( A  e.  RR+  /\  4  e.  ZZ )  ->  ( A ^ 4 )  e.  RR+ )
13938, 70, 138sylancl 662 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR+ )
140 elrp 11218 . . . . . 6  |-  ( ( A ^ 4 )  e.  RR+  <->  ( ( A ^ 4 )  e.  RR  /\  0  < 
( A ^ 4 ) ) )
141 ltmul2 10389 . . . . . . 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 1314 . . . . . 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 195 . . . . 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 16 . . . 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 211 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
14616recnd 9618 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  CC )
147 divrec 10219 . . . . 5  |-  ( ( ( A ^ 4 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
148106, 130, 147mp3an23 1316 . . . 4  |-  ( ( A ^ 4 )  e.  CC  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
149146, 148syl 16 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
150145, 149breqtrrd 4473 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  / 
6 ) )
15114, 26, 29, 52, 150lelttrd 9735 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489   _ici 9490    + caddc 9491    x. cmul 9493   RR*cxr 9623    < clt 9624    <_ cle 9625    / cdiv 10202   NNcn 10532   2c2 10581   3c3 10582   4c4 10583   5c5 10584   6c6 10585   8c8 10587   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   RR+crp 11216   (,]cioc 11526   ^cexp 12129   !cfa 12315   abscabs 13024   sum_csu 13464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ioc 11530  df-ico 11531  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12071  df-exp 12130  df-fac 12316  df-hash 12368  df-shft 12857  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-limsup 13250  df-clim 13267  df-rlim 13268  df-sum 13465
This theorem is referenced by:  sin01bnd  13774  cos01bnd  13775
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