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Theorem ef01bndlem 12787
Description: Lemma for sin01bnd 12788 and cos01bnd 12789. (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 9051 . . . . 5  |-  _i  e.  CC
2 0xr 9133 . . . . . . . 8  |-  0  e.  RR*
3 1re 9092 . . . . . . . 8  |-  1  e.  RR
4 elioc2 10975 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
52, 3, 4mp2an 655 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
65simp1bi 973 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
76recnd 9116 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
8 mulcl 9076 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
91, 7, 8sylancr 646 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
_i  x.  A )  e.  CC )
10 4nn0 10242 . . . 4  |-  4  e.  NN0
11 ef01bnd.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
1211eftlcl 12710 . . . 4  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
139, 10, 12sylancl 645 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
1413abscld 12240 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  e.  RR )
15 reexpcl 11400 . . . 4  |-  ( ( A  e.  RR  /\  4  e.  NN0 )  -> 
( A ^ 4 )  e.  RR )
166, 10, 15sylancl 645 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR )
17 4re 10075 . . . . 5  |-  4  e.  RR
1817, 3readdcli 9105 . . . 4  |-  ( 4  +  1 )  e.  RR
19 faccl 11578 . . . . . 6  |-  ( 4  e.  NN0  ->  ( ! `
 4 )  e.  NN )
2010, 19ax-mp 8 . . . . 5  |-  ( ! `
 4 )  e.  NN
21 4nn 10137 . . . . 5  |-  4  e.  NN
2220, 21nnmulcli 10026 . . . 4  |-  ( ( ! `  4 )  x.  4 )  e.  NN
23 nndivre 10037 . . . 4  |-  ( ( ( 4  +  1 )  e.  RR  /\  ( ( ! ` 
4 )  x.  4 )  e.  NN )  ->  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) )  e.  RR )
2418, 22, 23mp2an 655 . . 3  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  e.  RR
25 remulcl 9077 . . 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 645 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  e.  RR )
27 6nn 10139 . . 3  |-  6  e.  NN
28 nndivre 10037 . . 3  |-  ( ( ( A ^ 4 )  e.  RR  /\  6  e.  NN )  ->  ( ( A ^
4 )  /  6
)  e.  RR )
2916, 27, 28sylancl 645 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  e.  RR )
30 eqid 2438 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( abs `  (
_i  x.  A )
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( abs `  ( _i  x.  A ) ) ^ n )  / 
( ! `  n
) ) )
31 eqid 2438 . . . 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 12101 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
_i  x.  A )
)  =  ( ( abs `  _i )  x.  ( abs `  A
) ) )
341, 7, 33sylancr 646 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  ( ( abs `  _i )  x.  ( abs `  A ) ) )
35 absi 12093 . . . . . . . 8  |-  ( abs `  _i )  =  1
3635oveq1i 6093 . . . . . . 7  |-  ( ( abs `  _i )  x.  ( abs `  A
) )  =  ( 1  x.  ( abs `  A ) )
375simp2bi 974 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
386, 37elrpd 10648 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR+ )
39 rpre 10620 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
40 rpge0 10626 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  0  <_  A )
4139, 40absidd 12227 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( abs `  A )  =  A )
4238, 41syl 16 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  A )  =  A )
4342oveq2d 6099 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
4436, 43syl5eq 2482 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  _i )  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
457mulid2d 9108 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  A )  =  A )
4634, 44, 453eqtrd 2474 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  A )
475simp3bi 975 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
4846, 47eqbrtrd 4234 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  <_ 
1 )
4911, 30, 31, 32, 9, 48eftlub 12712 . . 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 6098 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  (
_i  x.  A )
) ^ 4 )  =  ( A ^
4 ) )
5150oveq1d 6098 . . 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 4238 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) ) )
53 3pos 10086 . . . . . . . . 9  |-  0  <  3
54 0re 9093 . . . . . . . . . 10  |-  0  e.  RR
55 3re 10073 . . . . . . . . . 10  |-  3  e.  RR
56 5re 10077 . . . . . . . . . 10  |-  5  e.  RR
5754, 55, 56ltadd1i 9583 . . . . . . . . 9  |-  ( 0  <  3  <->  ( 0  +  5 )  < 
( 3  +  5 ) )
5853, 57mpbi 201 . . . . . . . 8  |-  ( 0  +  5 )  < 
( 3  +  5 )
5956recni 9104 . . . . . . . . 9  |-  5  e.  CC
6059addid2i 9256 . . . . . . . 8  |-  ( 0  +  5 )  =  5
61 cu2 11481 . . . . . . . . 9  |-  ( 2 ^ 3 )  =  8
62 5p3e8 10119 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
63 3nn 10136 . . . . . . . . . . 11  |-  3  e.  NN
6463nncni 10012 . . . . . . . . . 10  |-  3  e.  CC
6559, 64addcomi 9259 . . . . . . . . 9  |-  ( 5  +  3 )  =  ( 3  +  5 )
6661, 62, 653eqtr2ri 2465 . . . . . . . 8  |-  ( 3  +  5 )  =  ( 2 ^ 3 )
6758, 60, 663brtr3i 4241 . . . . . . 7  |-  5  <  ( 2 ^ 3 )
68 2re 10071 . . . . . . . 8  |-  2  e.  RR
69 2nn 10135 . . . . . . . . 9  |-  2  e.  NN
70 nnge1 10028 . . . . . . . . 9  |-  ( 2  e.  NN  ->  1  <_  2 )
7169, 70ax-mp 8 . . . . . . . 8  |-  1  <_  2
7221nnzi 10307 . . . . . . . . 9  |-  4  e.  ZZ
73 3lt4 10147 . . . . . . . . . 10  |-  3  <  4
7455, 17, 73ltleii 9198 . . . . . . . . 9  |-  3  <_  4
7563nnzi 10307 . . . . . . . . . 10  |-  3  e.  ZZ
7675eluz1i 10497 . . . . . . . . 9  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 4  e.  ZZ  /\  3  <_ 
4 ) )
7772, 74, 76mpbir2an 888 . . . . . . . 8  |-  4  e.  ( ZZ>= `  3 )
78 leexp2a 11437 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  1  <_  2  /\  4  e.  ( ZZ>= `  3 )
)  ->  ( 2 ^ 3 )  <_ 
( 2 ^ 4 ) )
7968, 71, 77, 78mp3an 1280 . . . . . . 7  |-  ( 2 ^ 3 )  <_ 
( 2 ^ 4 )
80 8re 10080 . . . . . . . . 9  |-  8  e.  RR
8161, 80eqeltri 2508 . . . . . . . 8  |-  ( 2 ^ 3 )  e.  RR
82 nnexpcl 11396 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  4  e.  NN0 )  -> 
( 2 ^ 4 )  e.  NN )
8369, 10, 82mp2an 655 . . . . . . . . 9  |-  ( 2 ^ 4 )  e.  NN
8483nnrei 10011 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  RR
8556, 81, 84ltletri 9203 . . . . . . 7  |-  ( ( 5  <  ( 2 ^ 3 )  /\  ( 2 ^ 3 )  <_  ( 2 ^ 4 ) )  ->  5  <  (
2 ^ 4 ) )
8667, 79, 85mp2an 655 . . . . . 6  |-  5  <  ( 2 ^ 4 )
87 6re 10078 . . . . . . . 8  |-  6  e.  RR
8887, 84remulcli 9106 . . . . . . 7  |-  ( 6  x.  ( 2 ^ 4 ) )  e.  RR
89 6pos 10090 . . . . . . . 8  |-  0  <  6
9083nngt0i 10035 . . . . . . . 8  |-  0  <  ( 2 ^ 4 )
9187, 84, 89, 90mulgt0ii 9208 . . . . . . 7  |-  0  <  ( 6  x.  (
2 ^ 4 ) )
9256, 84, 88, 91ltdiv1ii 9942 . . . . . 6  |-  ( 5  <  ( 2 ^ 4 )  <->  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) ) )
9386, 92mpbi 201 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
94 df-5 10063 . . . . . 6  |-  5  =  ( 4  +  1 )
95 df-4 10062 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
9695fveq2i 5733 . . . . . . . . . 10  |-  ( ! `
 4 )  =  ( ! `  (
3  +  1 ) )
97 3nn0 10241 . . . . . . . . . . 11  |-  3  e.  NN0
98 facp1 11573 . . . . . . . . . . 11  |-  ( 3  e.  NN0  ->  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) ) )
9997, 98ax-mp 8 . . . . . . . . . 10  |-  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) )
100 sq2 11479 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
101100, 95eqtr2i 2459 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  ( 2 ^ 2 )
102101oveq2i 6094 . . . . . . . . . 10  |-  ( ( ! `  3 )  x.  ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
10396, 99, 1023eqtri 2462 . . . . . . . . 9  |-  ( ! `
 4 )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
104103oveq1i 6093 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ( ! `
 3 )  x.  ( 2 ^ 2 ) )  x.  (
2 ^ 2 ) )
105100oveq2i 6094 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
4 )  x.  4 )
106 fac3 11575 . . . . . . . . . 10  |-  ( ! `
 3 )  =  6
10787recni 9104 . . . . . . . . . 10  |-  6  e.  CC
108106, 107eqeltri 2508 . . . . . . . . 9  |-  ( ! `
 3 )  e.  CC
10917recni 9104 . . . . . . . . . 10  |-  4  e.  CC
110100, 109eqeltri 2508 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
111108, 110, 110mulassi 9101 . . . . . . . 8  |-  ( ( ( ! `  3
)  x.  ( 2 ^ 2 ) )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
112104, 105, 1113eqtr3i 2466 . . . . . . 7  |-  ( ( ! `  4 )  x.  4 )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
113 2p2e4 10100 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
114113oveq2i 6094 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( 2 ^ 4 )
115 2cn 10072 . . . . . . . . . 10  |-  2  e.  CC
116 2nn0 10240 . . . . . . . . . 10  |-  2  e.  NN0
117 expadd 11424 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  2  e.  NN0  /\  2  e.  NN0 )  ->  (
2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
118115, 116, 116, 117mp3an 1280 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
119114, 118eqtr3i 2460 . . . . . . . 8  |-  ( 2 ^ 4 )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
120119oveq2i 6094 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
121106oveq1i 6093 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( 6  x.  (
2 ^ 4 ) )
122112, 120, 1213eqtr2ri 2465 . . . . . 6  |-  ( 6  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
4 )  x.  4 )
12394, 122oveq12i 6095 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )
12483nncni 10012 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  CC
125124mulid2i 9095 . . . . . . 7  |-  ( 1  x.  ( 2 ^ 4 ) )  =  ( 2 ^ 4 )
126125oveq1i 6093 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
12783nnne0i 10036 . . . . . . . . 9  |-  ( 2 ^ 4 )  =/=  0
128124, 127dividi 9749 . . . . . . . 8  |-  ( ( 2 ^ 4 )  /  ( 2 ^ 4 ) )  =  1
129128oveq2i 6094 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  / 
6 )  x.  1 )
130 ax-1cn 9050 . . . . . . . 8  |-  1  e.  CC
13187, 89gt0ne0ii 9565 . . . . . . . 8  |-  6  =/=  0
132130, 107, 124, 124, 131, 127divmuldivi 9776 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  x.  ( 2 ^ 4 ) )  /  (
6  x.  ( 2 ^ 4 ) ) )
13387, 131rereccli 9781 . . . . . . . . 9  |-  ( 1  /  6 )  e.  RR
134133recni 9104 . . . . . . . 8  |-  ( 1  /  6 )  e.  CC
135134mulid1i 9094 . . . . . . 7  |-  ( ( 1  /  6 )  x.  1 )  =  ( 1  /  6
)
136129, 132, 1353eqtr3i 2466 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
137126, 136eqtr3i 2460 . . . . 5  |-  ( ( 2 ^ 4 )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
13893, 123, 1373brtr3i 4241 . . . 4  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  < 
( 1  /  6
)
139 rpexpcl 11402 . . . . . 6  |-  ( ( A  e.  RR+  /\  4  e.  ZZ )  ->  ( A ^ 4 )  e.  RR+ )
14038, 72, 139sylancl 645 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR+ )
141 elrp 10616 . . . . . 6  |-  ( ( A ^ 4 )  e.  RR+  <->  ( ( A ^ 4 )  e.  RR  /\  0  < 
( A ^ 4 ) ) )
142 ltmul2 9863 . . . . . . 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 ) ) ) )
14324, 133, 142mp3an12 1270 . . . . . 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 ) ) ) )
144141, 143sylbi 189 . . . . 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 ) ) ) )
145140, 144syl 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 ) ) ) )
146138, 145mpbii 204 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
14716recnd 9116 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  CC )
148 divrec 9696 . . . . 5  |-  ( ( ( A ^ 4 )  e.  CC  /\  6  e.  CC  /\  6  =/=  0 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
149107, 131, 148mp3an23 1272 . . . 4  |-  ( ( A ^ 4 )  e.  CC  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
150147, 149syl 16 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
151146, 150breqtrrd 4240 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  / 
6 ) )
15214, 26, 29, 52, 151lelttrd 9230 1  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <  ( ( A ^ 4 )  /  6 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214    e. cmpt 4268   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993   _ici 8994    + caddc 8995    x. cmul 8997   RR*cxr 9121    < clt 9122    <_ cle 9123    / cdiv 9679   NNcn 10002   2c2 10051   3c3 10052   4c4 10053   5c5 10054   6c6 10055   8c8 10057   NN0cn0 10223   ZZcz 10284   ZZ>=cuz 10490   RR+crp 10614   (,]cioc 10919   ^cexp 11384   !cfa 11568   abscabs 12041   sum_csu 12481
This theorem is referenced by:  sin01bnd  12788  cos01bnd  12789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-ioc 10923  df-ico 10924  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-fac 11569  df-hash 11621  df-shft 11884  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-limsup 12267  df-clim 12284  df-rlim 12285  df-sum 12482
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