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Theorem ef01bndlem 13570
Description: Lemma for sin01bnd 13571 and cos01bnd 13572. (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 9442 . . . . 5  |-  _i  e.  CC
2 0xr 9531 . . . . . . . 8  |-  0  e.  RR*
3 1re 9486 . . . . . . . 8  |-  1  e.  RR
4 elioc2 11459 . . . . . . . 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 1003 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
76recnd 9513 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
8 mulcl 9467 . . . . 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 10699 . . . 4  |-  4  e.  NN0
11 ef01bnd.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
1211eftlcl 13493 . . . 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 13024 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  e.  RR )
15 reexpcl 11983 . . . 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 10499 . . . . 5  |-  4  e.  RR
1817, 3readdcli 9500 . . . 4  |-  ( 4  +  1 )  e.  RR
19 faccl 12162 . . . . . 6  |-  ( 4  e.  NN0  ->  ( ! `
 4 )  e.  NN )
2010, 19ax-mp 5 . . . . 5  |-  ( ! `
 4 )  e.  NN
21 4nn 10582 . . . . 5  |-  4  e.  NN
2220, 21nnmulcli 10447 . . . 4  |-  ( ( ! `  4 )  x.  4 )  e.  NN
23 nndivre 10458 . . . 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 9468 . . 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 10584 . . 3  |-  6  e.  NN
28 nndivre 10458 . . 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 2451 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( abs `  (
_i  x.  A )
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( abs `  ( _i  x.  A ) ) ^ n )  / 
( ! `  n
) ) )
31 eqid 2451 . . . 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 12885 . . . . . . 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 12877 . . . . . . . 8  |-  ( abs `  _i )  =  1
3635oveq1i 6200 . . . . . . 7  |-  ( ( abs `  _i )  x.  ( abs `  A
) )  =  ( 1  x.  ( abs `  A ) )
375simp2bi 1004 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
386, 37elrpd 11126 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR+ )
39 rpre 11098 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
40 rpge0 11104 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  0  <_  A )
4139, 40absidd 13011 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( abs `  A )  =  A )
4238, 41syl 16 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  A )  =  A )
4342oveq2d 6206 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
4436, 43syl5eq 2504 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  _i )  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
457mulid2d 9505 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  A )  =  A )
4634, 44, 453eqtrd 2496 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  A )
475simp3bi 1005 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
4846, 47eqbrtrd 4410 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  <_ 
1 )
4911, 30, 31, 32, 9, 48eftlub 13495 . . 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 6205 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  (
_i  x.  A )
) ^ 4 )  =  ( A ^
4 ) )
5150oveq1d 6205 . . 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 4414 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) ) )
53 3pos 10516 . . . . . . . . 9  |-  0  <  3
54 0re 9487 . . . . . . . . . 10  |-  0  e.  RR
55 3re 10496 . . . . . . . . . 10  |-  3  e.  RR
56 5re 10501 . . . . . . . . . 10  |-  5  e.  RR
5754, 55, 56ltadd1i 9995 . . . . . . . . 9  |-  ( 0  <  3  <->  ( 0  +  5 )  < 
( 3  +  5 ) )
5853, 57mpbi 208 . . . . . . . 8  |-  ( 0  +  5 )  < 
( 3  +  5 )
59 5cn 10502 . . . . . . . . 9  |-  5  e.  CC
6059addid2i 9658 . . . . . . . 8  |-  ( 0  +  5 )  =  5
61 cu2 12065 . . . . . . . . 9  |-  ( 2 ^ 3 )  =  8
62 5p3e8 10561 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
63 3nn 10581 . . . . . . . . . . 11  |-  3  e.  NN
6463nncni 10433 . . . . . . . . . 10  |-  3  e.  CC
6559, 64addcomi 9661 . . . . . . . . 9  |-  ( 5  +  3 )  =  ( 3  +  5 )
6661, 62, 653eqtr2ri 2487 . . . . . . . 8  |-  ( 3  +  5 )  =  ( 2 ^ 3 )
6758, 60, 663brtr3i 4417 . . . . . . 7  |-  5  <  ( 2 ^ 3 )
68 2re 10492 . . . . . . . 8  |-  2  e.  RR
69 1le2 10636 . . . . . . . 8  |-  1  <_  2
7021nnzi 10771 . . . . . . . . 9  |-  4  e.  ZZ
71 3lt4 10592 . . . . . . . . . 10  |-  3  <  4
7255, 17, 71ltleii 9598 . . . . . . . . 9  |-  3  <_  4
7363nnzi 10771 . . . . . . . . . 10  |-  3  e.  ZZ
7473eluz1i 10969 . . . . . . . . 9  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 4  e.  ZZ  /\  3  <_ 
4 ) )
7570, 72, 74mpbir2an 911 . . . . . . . 8  |-  4  e.  ( ZZ>= `  3 )
76 leexp2a 12020 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  1  <_  2  /\  4  e.  ( ZZ>= `  3 )
)  ->  ( 2 ^ 3 )  <_ 
( 2 ^ 4 ) )
7768, 69, 75, 76mp3an 1315 . . . . . . 7  |-  ( 2 ^ 3 )  <_ 
( 2 ^ 4 )
78 8re 10507 . . . . . . . . 9  |-  8  e.  RR
7961, 78eqeltri 2535 . . . . . . . 8  |-  ( 2 ^ 3 )  e.  RR
80 2nn 10580 . . . . . . . . . 10  |-  2  e.  NN
81 nnexpcl 11979 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  4  e.  NN0 )  -> 
( 2 ^ 4 )  e.  NN )
8280, 10, 81mp2an 672 . . . . . . . . 9  |-  ( 2 ^ 4 )  e.  NN
8382nnrei 10432 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  RR
8456, 79, 83ltletri 9603 . . . . . . 7  |-  ( ( 5  <  ( 2 ^ 3 )  /\  ( 2 ^ 3 )  <_  ( 2 ^ 4 ) )  ->  5  <  (
2 ^ 4 ) )
8567, 77, 84mp2an 672 . . . . . 6  |-  5  <  ( 2 ^ 4 )
86 6re 10503 . . . . . . . 8  |-  6  e.  RR
8786, 83remulcli 9501 . . . . . . 7  |-  ( 6  x.  ( 2 ^ 4 ) )  e.  RR
88 6pos 10521 . . . . . . . 8  |-  0  <  6
8982nngt0i 10456 . . . . . . . 8  |-  0  <  ( 2 ^ 4 )
9086, 83, 88, 89mulgt0ii 9608 . . . . . . 7  |-  0  <  ( 6  x.  (
2 ^ 4 ) )
9156, 83, 87, 90ltdiv1ii 10363 . . . . . 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 10484 . . . . . 6  |-  5  =  ( 4  +  1 )
94 df-4 10483 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
9594fveq2i 5792 . . . . . . . . . 10  |-  ( ! `
 4 )  =  ( ! `  (
3  +  1 ) )
96 3nn0 10698 . . . . . . . . . . 11  |-  3  e.  NN0
97 facp1 12157 . . . . . . . . . . 11  |-  ( 3  e.  NN0  ->  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) ) )
9896, 97ax-mp 5 . . . . . . . . . 10  |-  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) )
99 sq2 12063 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
10099, 94eqtr2i 2481 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  ( 2 ^ 2 )
101100oveq2i 6201 . . . . . . . . . 10  |-  ( ( ! `  3 )  x.  ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
10295, 98, 1013eqtri 2484 . . . . . . . . 9  |-  ( ! `
 4 )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
103102oveq1i 6200 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ( ! `
 3 )  x.  ( 2 ^ 2 ) )  x.  (
2 ^ 2 ) )
10499oveq2i 6201 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
4 )  x.  4 )
105 fac3 12159 . . . . . . . . . 10  |-  ( ! `
 3 )  =  6
106 6cn 10504 . . . . . . . . . 10  |-  6  e.  CC
107105, 106eqeltri 2535 . . . . . . . . 9  |-  ( ! `
 3 )  e.  CC
10817recni 9499 . . . . . . . . . 10  |-  4  e.  CC
10999, 108eqeltri 2535 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
110107, 109, 109mulassi 9496 . . . . . . . 8  |-  ( ( ( ! `  3
)  x.  ( 2 ^ 2 ) )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
111103, 104, 1103eqtr3i 2488 . . . . . . 7  |-  ( ( ! `  4 )  x.  4 )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
112 2p2e4 10540 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
113112oveq2i 6201 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( 2 ^ 4 )
114 2cn 10493 . . . . . . . . . 10  |-  2  e.  CC
115 2nn0 10697 . . . . . . . . . 10  |-  2  e.  NN0
116 expadd 12007 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  2  e.  NN0  /\  2  e.  NN0 )  ->  (
2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
117114, 115, 115, 116mp3an 1315 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
118113, 117eqtr3i 2482 . . . . . . . 8  |-  ( 2 ^ 4 )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
119118oveq2i 6201 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
120105oveq1i 6200 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( 6  x.  (
2 ^ 4 ) )
121111, 119, 1203eqtr2ri 2487 . . . . . 6  |-  ( 6  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
4 )  x.  4 )
12293, 121oveq12i 6202 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )
12382nncni 10433 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  CC
124123mulid2i 9490 . . . . . . 7  |-  ( 1  x.  ( 2 ^ 4 ) )  =  ( 2 ^ 4 )
125124oveq1i 6200 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
12682nnne0i 10457 . . . . . . . . 9  |-  ( 2 ^ 4 )  =/=  0
127123, 126dividi 10165 . . . . . . . 8  |-  ( ( 2 ^ 4 )  /  ( 2 ^ 4 ) )  =  1
128127oveq2i 6201 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  / 
6 )  x.  1 )
129 ax-1cn 9441 . . . . . . . 8  |-  1  e.  CC
13086, 88gt0ne0ii 9977 . . . . . . . 8  |-  6  =/=  0
131129, 106, 123, 123, 130, 126divmuldivi 10192 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  x.  ( 2 ^ 4 ) )  /  (
6  x.  ( 2 ^ 4 ) ) )
13286, 130rereccli 10197 . . . . . . . . 9  |-  ( 1  /  6 )  e.  RR
133132recni 9499 . . . . . . . 8  |-  ( 1  /  6 )  e.  CC
134133mulid1i 9489 . . . . . . 7  |-  ( ( 1  /  6 )  x.  1 )  =  ( 1  /  6
)
135128, 131, 1343eqtr3i 2488 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
136125, 135eqtr3i 2482 . . . . 5  |-  ( ( 2 ^ 4 )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
13792, 122, 1363brtr3i 4417 . . . 4  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  < 
( 1  /  6
)
138 rpexpcl 11985 . . . . . 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 11094 . . . . . 6  |-  ( ( A ^ 4 )  e.  RR+  <->  ( ( A ^ 4 )  e.  RR  /\  0  < 
( A ^ 4 ) ) )
141 ltmul2 10281 . . . . . . 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 1305 . . . . . 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 9513 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  CC )
147 divrec 10111 . . . . 5  |-  ( ( ( A ^ 4 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
148106, 130, 147mp3an23 1307 . . . 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 4416 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  / 
6 ) )
15114, 26, 29, 52, 150lelttrd 9630 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4390    |-> cmpt 4448   ` cfv 5516  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384   _ici 9385    + caddc 9386    x. cmul 9388   RR*cxr 9518    < clt 9519    <_ cle 9520    / cdiv 10094   NNcn 10423   2c2 10472   3c3 10473   4c4 10474   5c5 10475   6c6 10476   8c8 10478   NN0cn0 10680   ZZcz 10747   ZZ>=cuz 10962   RR+crp 11092   (,]cioc 11402   ^cexp 11966   !cfa 12152   abscabs 12825   sum_csu 13265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-ioc 11406  df-ico 11407  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-fac 12153  df-hash 12205  df-shft 12658  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-limsup 13051  df-clim 13068  df-rlim 13069  df-sum 13266
This theorem is referenced by:  sin01bnd  13571  cos01bnd  13572
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