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Theorem ef01bndlem 11452
Description: Lemma for sin01bnd 11453 and cos01bnd 11454. (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 7708 . . . . 5  |-  _i  e.  CC
2 0xr 7805 . . . . . . . 8  |-  0  e.  RR*
3 1re 7758 . . . . . . . 8  |-  1  e.  RR
4 elioc2 9712 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  1  e.  RR )  ->  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) ) )
52, 3, 4mp2an 422 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  <->  ( A  e.  RR  /\  0  < 
A  /\  A  <_  1 ) )
65simp1bi 996 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR )
76recnd 7787 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  CC )
8 mulcl 7740 . . . . 5  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
91, 7, 8sylancr 410 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
_i  x.  A )  e.  CC )
10 4nn0 8989 . . . 4  |-  4  e.  NN0
11 ef01bnd.1 . . . . 5  |-  F  =  ( n  e.  NN0  |->  ( ( ( _i  x.  A ) ^
n )  /  ( ! `  n )
) )
1211eftlcl 11383 . . . 4  |-  ( ( ( _i  x.  A
)  e.  CC  /\  4  e.  NN0 )  ->  sum_ k  e.  ( ZZ>= ` 
4 ) ( F `
 k )  e.  CC )
139, 10, 12sylancl 409 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  sum_ k  e.  ( ZZ>= `  4 )
( F `  k
)  e.  CC )
1413abscld 10946 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  e.  RR )
15 reexpcl 10303 . . . 4  |-  ( ( A  e.  RR  /\  4  e.  NN0 )  -> 
( A ^ 4 )  e.  RR )
166, 10, 15sylancl 409 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR )
17 4re 8790 . . . . 5  |-  4  e.  RR
1817, 3readdcli 7772 . . . 4  |-  ( 4  +  1 )  e.  RR
19 faccl 10474 . . . . . 6  |-  ( 4  e.  NN0  ->  ( ! `
 4 )  e.  NN )
2010, 19ax-mp 5 . . . . 5  |-  ( ! `
 4 )  e.  NN
21 4nn 8876 . . . . 5  |-  4  e.  NN
2220, 21nnmulcli 8735 . . . 4  |-  ( ( ! `  4 )  x.  4 )  e.  NN
23 nndivre 8749 . . . 4  |-  ( ( ( 4  +  1 )  e.  RR  /\  ( ( ! ` 
4 )  x.  4 )  e.  NN )  ->  ( ( 4  +  1 )  / 
( ( ! ` 
4 )  x.  4 ) )  e.  RR )
2418, 22, 23mp2an 422 . . 3  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  e.  RR
25 remulcl 7741 . . 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 409 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  e.  RR )
27 6nn 8878 . . 3  |-  6  e.  NN
28 nndivre 8749 . . 3  |-  ( ( ( A ^ 4 )  e.  RR  /\  6  e.  NN )  ->  ( ( A ^
4 )  /  6
)  e.  RR )
2916, 27, 28sylancl 409 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  e.  RR )
30 eqid 2137 . . . 4  |-  ( n  e.  NN0  |->  ( ( ( abs `  (
_i  x.  A )
) ^ n )  /  ( ! `  n ) ) )  =  ( n  e. 
NN0  |->  ( ( ( abs `  ( _i  x.  A ) ) ^ n )  / 
( ! `  n
) ) )
31 eqid 2137 . . . 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 9 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  4  e.  NN )
33 absmul 10834 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( abs `  (
_i  x.  A )
)  =  ( ( abs `  _i )  x.  ( abs `  A
) ) )
341, 7, 33sylancr 410 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  ( ( abs `  _i )  x.  ( abs `  A ) ) )
35 absi 10824 . . . . . . . 8  |-  ( abs `  _i )  =  1
3635oveq1i 5777 . . . . . . 7  |-  ( ( abs `  _i )  x.  ( abs `  A
) )  =  ( 1  x.  ( abs `  A ) )
375simp2bi 997 . . . . . . . . . 10  |-  ( A  e.  ( 0 (,] 1 )  ->  0  <  A )
386, 37elrpd 9474 . . . . . . . . 9  |-  ( A  e.  ( 0 (,] 1 )  ->  A  e.  RR+ )
39 rpre 9441 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  A  e.  RR )
40 rpge0 9447 . . . . . . . . . 10  |-  ( A  e.  RR+  ->  0  <_  A )
4139, 40absidd 10932 . . . . . . . . 9  |-  ( A  e.  RR+  ->  ( abs `  A )  =  A )
4238, 41syl 14 . . . . . . . 8  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  A )  =  A )
4342oveq2d 5783 . . . . . . 7  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
4436, 43syl5eq 2182 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  _i )  x.  ( abs `  A ) )  =  ( 1  x.  A
) )
457mulid2d 7777 . . . . . 6  |-  ( A  e.  ( 0 (,] 1 )  ->  (
1  x.  A )  =  A )
4634, 44, 453eqtrd 2174 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  =  A )
475simp3bi 998 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  A  <_  1 )
4846, 47eqbrtrd 3945 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  ( _i  x.  A ) )  <_ 
1 )
4911, 30, 31, 32, 9, 48eftlub 11385 . . 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 5782 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( abs `  (
_i  x.  A )
) ^ 4 )  =  ( A ^
4 ) )
5150oveq1d 5782 . . 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 3949 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  ( abs `  sum_ k  e.  (
ZZ>= `  4 ) ( F `  k ) )  <_  ( ( A ^ 4 )  x.  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) ) ) )
53 3pos 8807 . . . . . . . . 9  |-  0  <  3
54 0re 7759 . . . . . . . . . 10  |-  0  e.  RR
55 3re 8787 . . . . . . . . . 10  |-  3  e.  RR
56 5re 8792 . . . . . . . . . 10  |-  5  e.  RR
5754, 55, 56ltadd1i 8257 . . . . . . . . 9  |-  ( 0  <  3  <->  ( 0  +  5 )  < 
( 3  +  5 ) )
5853, 57mpbi 144 . . . . . . . 8  |-  ( 0  +  5 )  < 
( 3  +  5 )
59 5cn 8793 . . . . . . . . 9  |-  5  e.  CC
6059addid2i 7898 . . . . . . . 8  |-  ( 0  +  5 )  =  5
61 cu2 10384 . . . . . . . . 9  |-  ( 2 ^ 3 )  =  8
62 5p3e8 8860 . . . . . . . . 9  |-  ( 5  +  3 )  =  8
63 3cn 8788 . . . . . . . . . 10  |-  3  e.  CC
6459, 63addcomi 7899 . . . . . . . . 9  |-  ( 5  +  3 )  =  ( 3  +  5 )
6561, 62, 643eqtr2ri 2165 . . . . . . . 8  |-  ( 3  +  5 )  =  ( 2 ^ 3 )
6658, 60, 653brtr3i 3952 . . . . . . 7  |-  5  <  ( 2 ^ 3 )
67 2re 8783 . . . . . . . 8  |-  2  e.  RR
68 1le2 8921 . . . . . . . 8  |-  1  <_  2
69 4z 9077 . . . . . . . . 9  |-  4  e.  ZZ
70 3lt4 8885 . . . . . . . . . 10  |-  3  <  4
7155, 17, 70ltleii 7859 . . . . . . . . 9  |-  3  <_  4
72 3z 9076 . . . . . . . . . 10  |-  3  e.  ZZ
7372eluz1i 9326 . . . . . . . . 9  |-  ( 4  e.  ( ZZ>= `  3
)  <->  ( 4  e.  ZZ  /\  3  <_ 
4 ) )
7469, 71, 73mpbir2an 926 . . . . . . . 8  |-  4  e.  ( ZZ>= `  3 )
75 leexp2a 10339 . . . . . . . 8  |-  ( ( 2  e.  RR  /\  1  <_  2  /\  4  e.  ( ZZ>= `  3 )
)  ->  ( 2 ^ 3 )  <_ 
( 2 ^ 4 ) )
7667, 68, 74, 75mp3an 1315 . . . . . . 7  |-  ( 2 ^ 3 )  <_ 
( 2 ^ 4 )
77 8re 8798 . . . . . . . . 9  |-  8  e.  RR
7861, 77eqeltri 2210 . . . . . . . 8  |-  ( 2 ^ 3 )  e.  RR
79 2nn 8874 . . . . . . . . . 10  |-  2  e.  NN
80 nnexpcl 10299 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  4  e.  NN0 )  -> 
( 2 ^ 4 )  e.  NN )
8179, 10, 80mp2an 422 . . . . . . . . 9  |-  ( 2 ^ 4 )  e.  NN
8281nnrei 8722 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  RR
8356, 78, 82ltletri 7863 . . . . . . 7  |-  ( ( 5  <  ( 2 ^ 3 )  /\  ( 2 ^ 3 )  <_  ( 2 ^ 4 ) )  ->  5  <  (
2 ^ 4 ) )
8466, 76, 83mp2an 422 . . . . . 6  |-  5  <  ( 2 ^ 4 )
85 6re 8794 . . . . . . . 8  |-  6  e.  RR
8685, 82remulcli 7773 . . . . . . 7  |-  ( 6  x.  ( 2 ^ 4 ) )  e.  RR
87 6pos 8814 . . . . . . . 8  |-  0  <  6
8881nngt0i 8743 . . . . . . . 8  |-  0  <  ( 2 ^ 4 )
8985, 82, 87, 88mulgt0ii 7867 . . . . . . 7  |-  0  <  ( 6  x.  (
2 ^ 4 ) )
9056, 82, 86, 89ltdiv1ii 8680 . . . . . 6  |-  ( 5  <  ( 2 ^ 4 )  <->  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) ) )
9184, 90mpbi 144 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  < 
( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
92 df-5 8775 . . . . . 6  |-  5  =  ( 4  +  1 )
93 df-4 8774 . . . . . . . . . . 11  |-  4  =  ( 3  +  1 )
9493fveq2i 5417 . . . . . . . . . 10  |-  ( ! `
 4 )  =  ( ! `  (
3  +  1 ) )
95 3nn0 8988 . . . . . . . . . . 11  |-  3  e.  NN0
96 facp1 10469 . . . . . . . . . . 11  |-  ( 3  e.  NN0  ->  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) ) )
9795, 96ax-mp 5 . . . . . . . . . 10  |-  ( ! `
 ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
3  +  1 ) )
98 sq2 10381 . . . . . . . . . . . 12  |-  ( 2 ^ 2 )  =  4
9998, 93eqtr2i 2159 . . . . . . . . . . 11  |-  ( 3  +  1 )  =  ( 2 ^ 2 )
10099oveq2i 5778 . . . . . . . . . 10  |-  ( ( ! `  3 )  x.  ( 3  +  1 ) )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
10194, 97, 1003eqtri 2162 . . . . . . . . 9  |-  ( ! `
 4 )  =  ( ( ! ` 
3 )  x.  (
2 ^ 2 ) )
102101oveq1i 5777 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ( ! `
 3 )  x.  ( 2 ^ 2 ) )  x.  (
2 ^ 2 ) )
10398oveq2i 5778 . . . . . . . 8  |-  ( ( ! `  4 )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
4 )  x.  4 )
104 fac3 10471 . . . . . . . . . 10  |-  ( ! `
 3 )  =  6
105 6cn 8795 . . . . . . . . . 10  |-  6  e.  CC
106104, 105eqeltri 2210 . . . . . . . . 9  |-  ( ! `
 3 )  e.  CC
10717recni 7771 . . . . . . . . . 10  |-  4  e.  CC
10898, 107eqeltri 2210 . . . . . . . . 9  |-  ( 2 ^ 2 )  e.  CC
109106, 108, 108mulassi 7768 . . . . . . . 8  |-  ( ( ( ! `  3
)  x.  ( 2 ^ 2 ) )  x.  ( 2 ^ 2 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
110102, 103, 1093eqtr3i 2166 . . . . . . 7  |-  ( ( ! `  4 )  x.  4 )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
111 2p2e4 8840 . . . . . . . . . 10  |-  ( 2  +  2 )  =  4
112111oveq2i 5778 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( 2 ^ 4 )
113 2cn 8784 . . . . . . . . . 10  |-  2  e.  CC
114 2nn0 8987 . . . . . . . . . 10  |-  2  e.  NN0
115 expadd 10328 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  2  e.  NN0  /\  2  e.  NN0 )  ->  (
2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
116113, 114, 114, 115mp3an 1315 . . . . . . . . 9  |-  ( 2 ^ ( 2  +  2 ) )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
117112, 116eqtr3i 2160 . . . . . . . 8  |-  ( 2 ^ 4 )  =  ( ( 2 ^ 2 )  x.  (
2 ^ 2 ) )
118117oveq2i 5778 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
3 )  x.  (
( 2 ^ 2 )  x.  ( 2 ^ 2 ) ) )
119104oveq1i 5777 . . . . . . 7  |-  ( ( ! `  3 )  x.  ( 2 ^ 4 ) )  =  ( 6  x.  (
2 ^ 4 ) )
120110, 118, 1193eqtr2ri 2165 . . . . . 6  |-  ( 6  x.  ( 2 ^ 4 ) )  =  ( ( ! ` 
4 )  x.  4 )
12192, 120oveq12i 5779 . . . . 5  |-  ( 5  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 4  +  1 )  /  (
( ! `  4
)  x.  4 ) )
12281nncni 8723 . . . . . . . 8  |-  ( 2 ^ 4 )  e.  CC
123122mulid2i 7762 . . . . . . 7  |-  ( 1  x.  ( 2 ^ 4 ) )  =  ( 2 ^ 4 )
124123oveq1i 5777 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( ( 2 ^ 4 )  /  (
6  x.  ( 2 ^ 4 ) ) )
12582, 88gt0ap0ii 8383 . . . . . . . . 9  |-  ( 2 ^ 4 ) #  0
126122, 125dividapi 8498 . . . . . . . 8  |-  ( ( 2 ^ 4 )  /  ( 2 ^ 4 ) )  =  1
127126oveq2i 5778 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  / 
6 )  x.  1 )
128 ax-1cn 7706 . . . . . . . 8  |-  1  e.  CC
12985, 87gt0ap0ii 8383 . . . . . . . 8  |-  6 #  0
130128, 105, 122, 122, 129, 125divmuldivapi 8525 . . . . . . 7  |-  ( ( 1  /  6 )  x.  ( ( 2 ^ 4 )  / 
( 2 ^ 4 ) ) )  =  ( ( 1  x.  ( 2 ^ 4 ) )  /  (
6  x.  ( 2 ^ 4 ) ) )
13185, 129rerecclapi 8530 . . . . . . . . 9  |-  ( 1  /  6 )  e.  RR
132131recni 7771 . . . . . . . 8  |-  ( 1  /  6 )  e.  CC
133132mulid1i 7761 . . . . . . 7  |-  ( ( 1  /  6 )  x.  1 )  =  ( 1  /  6
)
134127, 130, 1333eqtr3i 2166 . . . . . 6  |-  ( ( 1  x.  ( 2 ^ 4 ) )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
135124, 134eqtr3i 2160 . . . . 5  |-  ( ( 2 ^ 4 )  /  ( 6  x.  ( 2 ^ 4 ) ) )  =  ( 1  /  6
)
13691, 121, 1353brtr3i 3952 . . . 4  |-  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) )  < 
( 1  /  6
)
137 rpexpcl 10305 . . . . . 6  |-  ( ( A  e.  RR+  /\  4  e.  ZZ )  ->  ( A ^ 4 )  e.  RR+ )
13838, 69, 137sylancl 409 . . . . 5  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  RR+ )
139 elrp 9436 . . . . . 6  |-  ( ( A ^ 4 )  e.  RR+  <->  ( ( A ^ 4 )  e.  RR  /\  0  < 
( A ^ 4 ) ) )
140 ltmul2 8607 . . . . . . 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 ) ) ) )
14124, 131, 140mp3an12 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 ) ) ) )
142139, 141sylbi 120 . . . . 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 ) ) ) )
143138, 142syl 14 . . . 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 ) ) ) )
144136, 143mpbii 147 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
14516recnd 7787 . . . 4  |-  ( A  e.  ( 0 (,] 1 )  ->  ( A ^ 4 )  e.  CC )
146 divrecap 8441 . . . . 5  |-  ( ( ( A ^ 4 )  e.  CC  /\  6  e.  CC  /\  6 #  0 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
147105, 129, 146mp3an23 1307 . . . 4  |-  ( ( A ^ 4 )  e.  CC  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
148145, 147syl 14 . . 3  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  /  6 )  =  ( ( A ^ 4 )  x.  ( 1  /  6
) ) )
149144, 148breqtrrd 3951 . 2  |-  ( A  e.  ( 0 (,] 1 )  ->  (
( A ^ 4 )  x.  ( ( 4  +  1 )  /  ( ( ! `
 4 )  x.  4 ) ) )  <  ( ( A ^ 4 )  / 
6 ) )
15014, 26, 29, 52, 149lelttrd 7880 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    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924    |-> cmpt 3984   ` cfv 5118  (class class class)co 5767   CCcc 7611   RRcr 7612   0cc0 7613   1c1 7614   _ici 7615    + caddc 7616    x. cmul 7618   RR*cxr 7792    < clt 7793    <_ cle 7794   # cap 8336    / cdiv 8425   NNcn 8713   2c2 8764   3c3 8765   4c4 8766   5c5 8767   6c6 8768   8c8 8770   NN0cn0 8970   ZZcz 9047   ZZ>=cuz 9319   RR+crp 9434   (,]cioc 9665   ^cexp 10285   !cfa 10464   abscabs 10762   sum_csu 11115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-isom 5127  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-5 8775  df-6 8776  df-7 8777  df-8 8778  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-ioc 9669  df-ico 9670  df-fz 9784  df-fzo 9913  df-seqfrec 10212  df-exp 10286  df-fac 10465  df-ihash 10515  df-shft 10580  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-clim 11041  df-sumdc 11116
This theorem is referenced by:  sin01bnd  11453  cos01bnd  11454
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