Theorem ef01bndlem 11101

Description: Lemma for sin01bnd 11102 and cos01bnd 11103. (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

Step	Hyp	Ref	Expression
1		ax-icn 7494	. . . . 5 |- _i e. CC
2		0xr 7588	. . . . . . . 8 |- 0 e. RR*
3		1re 7541	. . . . . . . 8 |- 1 e. RR
4		elioc2 9408	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
5	2, 3, 4	mp2an 418	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
6	5	simp1bi 959	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
7	6	recnd 7570	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
8		mulcl 7523	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
9	1, 7, 8	sylancr 406	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( _i x. A ) e. CC )
10		4nn0 8746	. . . 4 |- 4 e. NN0
11		ef01bnd.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
12	11	eftlcl 11032	. . . 4 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
13	9, 10, 12	sylancl 405	. . 3 |- ( A e. ( 0 (,] 1 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
14	13	abscld 10668	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) e. RR )
15		reexpcl 10026	. . . 4 |- ( ( A e. RR /\ 4 e. NN0 ) -> ( A ^ 4 ) e. RR )
16	6, 10, 15	sylancl 405	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR )
17		4re 8553	. . . . 5 |- 4 e. RR
18	17, 3	readdcli 7555	. . . 4 |- ( 4 + 1 ) e. RR
19		faccl 10197	. . . . . 6 |- ( 4 e. NN0 -> ( ! ` 4 ) e. NN )
20	10, 19	ax-mp 7	. . . . 5 |- ( ! ` 4 ) e. NN
21		4nn 8633	. . . . 5 |- 4 e. NN
22	20, 21	nnmulcli 8498	. . . 4 |- ( ( ! ` 4 ) x. 4 ) e. NN
23		nndivre 8512	. . . 4 |- ( ( ( 4 + 1 ) e. RR /\ ( ( ! ` 4 ) x. 4 ) e. NN ) -> ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR )
24	18, 22, 23	mp2an 418	. . 3 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR
25		remulcl 7524	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
26	16, 24, 25	sylancl 405	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
27		6nn 8635	. . 3 |- 6 e. NN
28		nndivre 8512	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ 6 e. NN ) -> ( ( A ^ 4 ) / 6 ) e. RR )
29	16, 27, 28	sylancl 405	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) e. RR )
30		eqid 2089	. . . 4 |- ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) )
31		eqid 2089	. . . 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 ) ) )
32	21	a1i 9	. . . 4 |- ( A e. ( 0 (,] 1 ) -> 4 e. NN )
33		absmul 10556	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
34	1, 7, 33	sylancr 406	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
35		absi 10546	. . . . . . . 8 |- ( abs ` _i ) = 1
36	35	oveq1i 5676	. . . . . . 7 |- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) )
37	5	simp2bi 960	. . . . . . . . . 10 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
38	6, 37	elrpd 9225	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) -> A e. RR+ )
39		rpre 9194	. . . . . . . . . 10 |- ( A e. RR+ -> A e. RR )
40		rpge0 9200	. . . . . . . . . 10 |- ( A e. RR+ -> 0 <_ A )
41	39, 40	absidd 10654	. . . . . . . . 9 |- ( A e. RR+ -> ( abs ` A ) = A )
42	38, 41	syl 14	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> ( abs ` A ) = A )
43	42	oveq2d 5682	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. ( abs ` A ) ) = ( 1 x. A ) )
44	36, 43	syl5eq 2133	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. A ) )
45	7	mulid2d 7560	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. A ) = A )
46	34, 44, 45	3eqtrd 2125	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = A )
47	5	simp3bi 961	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
48	46, 47	eqbrtrd 3871	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) <_ 1 )
49	11, 30, 31, 32, 9, 48	eftlub 11034	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( ( abs ` ( _i x. A ) ) ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
50	46	oveq1d 5681	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` ( _i x. A ) ) ^ 4 ) = ( A ^ 4 ) )
51	50	oveq1d 5681	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( ( abs ` ( _i x. A ) ) ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) = ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
52	49, 51	breqtrd 3875	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
53		3pos 8570	. . . . . . . . 9 |- 0 < 3
54		0re 7542	. . . . . . . . . 10 |- 0 e. RR
55		3re 8550	. . . . . . . . . 10 |- 3 e. RR
56		5re 8555	. . . . . . . . . 10 |- 5 e. RR
57	54, 55, 56	ltadd1i 8034	. . . . . . . . 9 |- ( 0 < 3 <-> ( 0 + 5 ) < ( 3 + 5 ) )
58	53, 57	mpbi 144	. . . . . . . 8 |- ( 0 + 5 ) < ( 3 + 5 )
59		5cn 8556	. . . . . . . . 9 |- 5 e. CC
60	59	addid2i 7679	. . . . . . . 8 |- ( 0 + 5 ) = 5
61		cu2 10107	. . . . . . . . 9 |- ( 2 ^ 3 ) = 8
62		5p3e8 8617	. . . . . . . . 9 |- ( 5 + 3 ) = 8
63		3cn 8551	. . . . . . . . . 10 |- 3 e. CC
64	59, 63	addcomi 7680	. . . . . . . . 9 |- ( 5 + 3 ) = ( 3 + 5 )
65	61, 62, 64	3eqtr2ri 2116	. . . . . . . 8 |- ( 3 + 5 ) = ( 2 ^ 3 )
66	58, 60, 65	3brtr3i 3878	. . . . . . 7 |- 5 < ( 2 ^ 3 )
67		2re 8546	. . . . . . . 8 |- 2 e. RR
68		1le2 8678	. . . . . . . 8 |- 1 <_ 2
69		4z 8834	. . . . . . . . 9 |- 4 e. ZZ
70		3lt4 8642	. . . . . . . . . 10 |- 3 < 4
71	55, 17, 70	ltleii 7641	. . . . . . . . 9 |- 3 <_ 4
72		3z 8833	. . . . . . . . . 10 |- 3 e. ZZ
73	72	eluz1i 9080	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
74	69, 71, 73	mpbir2an 889	. . . . . . . 8 |- 4 e. ( ZZ>= ` 3 )
75		leexp2a 10062	. . . . . . . 8 |- ( ( 2 e. RR /\ 1 <_ 2 /\ 4 e. ( ZZ>= ` 3 ) ) -> ( 2 ^ 3 ) <_ ( 2 ^ 4 ) )
76	67, 68, 74, 75	mp3an 1274	. . . . . . 7 |- ( 2 ^ 3 ) <_ ( 2 ^ 4 )
77		8re 8561	. . . . . . . . 9 |- 8 e. RR
78	61, 77	eqeltri 2161	. . . . . . . 8 |- ( 2 ^ 3 ) e. RR
79		2nn 8631	. . . . . . . . . 10 |- 2 e. NN
80		nnexpcl 10022	. . . . . . . . . 10 |- ( ( 2 e. NN /\ 4 e. NN0 ) -> ( 2 ^ 4 ) e. NN )
81	79, 10, 80	mp2an 418	. . . . . . . . 9 |- ( 2 ^ 4 ) e. NN
82	81	nnrei 8485	. . . . . . . 8 |- ( 2 ^ 4 ) e. RR
83	56, 78, 82	ltletri 7645	. . . . . . 7 |- ( ( 5 < ( 2 ^ 3 ) /\ ( 2 ^ 3 ) <_ ( 2 ^ 4 ) ) -> 5 < ( 2 ^ 4 ) )
84	66, 76, 83	mp2an 418	. . . . . 6 |- 5 < ( 2 ^ 4 )
85		6re 8557	. . . . . . . 8 |- 6 e. RR
86	85, 82	remulcli 7556	. . . . . . 7 |- ( 6 x. ( 2 ^ 4 ) ) e. RR
87		6pos 8577	. . . . . . . 8 |- 0 < 6
88	81	nngt0i 8506	. . . . . . . 8 |- 0 < ( 2 ^ 4 )
89	85, 82, 87, 88	mulgt0ii 7649	. . . . . . 7 |- 0 < ( 6 x. ( 2 ^ 4 ) )
90	56, 82, 86, 89	ltdiv1ii 8444	. . . . . 6 |- ( 5 < ( 2 ^ 4 ) <-> ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) )
91	84, 90	mpbi 144	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
92		df-5 8538	. . . . . 6 |- 5 = ( 4 + 1 )
93		df-4 8537	. . . . . . . . . . 11 |- 4 = ( 3 + 1 )
94	93	fveq2i 5321	. . . . . . . . . 10 |- ( ! ` 4 ) = ( ! ` ( 3 + 1 ) )
95		3nn0 8745	. . . . . . . . . . 11 |- 3 e. NN0
96		facp1 10192	. . . . . . . . . . 11 |- ( 3 e. NN0 -> ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) ) )
97	95, 96	ax-mp 7	. . . . . . . . . 10 |- ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) )
98		sq2 10104	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
99	98, 93	eqtr2i 2110	. . . . . . . . . . 11 |- ( 3 + 1 ) = ( 2 ^ 2 )
100	99	oveq2i 5677	. . . . . . . . . 10 |- ( ( ! ` 3 ) x. ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
101	94, 97, 100	3eqtri 2113	. . . . . . . . 9 |- ( ! ` 4 ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
102	101	oveq1i 5676	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) )
103	98	oveq2i 5677	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ! ` 4 ) x. 4 )
104		fac3 10194	. . . . . . . . . 10 |- ( ! ` 3 ) = 6
105		6cn 8558	. . . . . . . . . 10 |- 6 e. CC
106	104, 105	eqeltri 2161	. . . . . . . . 9 |- ( ! ` 3 ) e. CC
107	17	recni 7554	. . . . . . . . . 10 |- 4 e. CC
108	98, 107	eqeltri 2161	. . . . . . . . 9 |- ( 2 ^ 2 ) e. CC
109	106, 108, 108	mulassi 7551	. . . . . . . 8 |- ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
110	102, 103, 109	3eqtr3i 2117	. . . . . . 7 |- ( ( ! ` 4 ) x. 4 ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
111		2p2e4 8597	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
112	111	oveq2i 5677	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( 2 ^ 4 )
113		2cn 8547	. . . . . . . . . 10 |- 2 e. CC
114		2nn0 8744	. . . . . . . . . 10 |- 2 e. NN0
115		expadd 10051	. . . . . . . . . 10 |- ( ( 2 e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
116	113, 114, 114, 115	mp3an 1274	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
117	112, 116	eqtr3i 2111	. . . . . . . 8 |- ( 2 ^ 4 ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
118	117	oveq2i 5677	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
119	104	oveq1i 5676	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( 6 x. ( 2 ^ 4 ) )
120	110, 118, 119	3eqtr2ri 2116	. . . . . 6 |- ( 6 x. ( 2 ^ 4 ) ) = ( ( ! ` 4 ) x. 4 )
121	92, 120	oveq12i 5678	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) )
122	81	nncni 8486	. . . . . . . 8 |- ( 2 ^ 4 ) e. CC
123	122	mulid2i 7545	. . . . . . 7 |- ( 1 x. ( 2 ^ 4 ) ) = ( 2 ^ 4 )
124	123	oveq1i 5676	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
125	82, 88	gt0ap0ii 8158	. . . . . . . . 9 |- ( 2 ^ 4 ) # 0
126	122, 125	dividapi 8266	. . . . . . . 8 |- ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) = 1
127	126	oveq2i 5677	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 / 6 ) x. 1 )
128		ax-1cn 7492	. . . . . . . 8 |- 1 e. CC
129	85, 87	gt0ap0ii 8158	. . . . . . . 8 |- 6 # 0
130	128, 105, 122, 122, 129, 125	divmuldivapi 8293	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) )
131	85, 129	rerecclapi 8298	. . . . . . . . 9 |- ( 1 / 6 ) e. RR
132	131	recni 7554	. . . . . . . 8 |- ( 1 / 6 ) e. CC
133	132	mulid1i 7544	. . . . . . 7 |- ( ( 1 / 6 ) x. 1 ) = ( 1 / 6 )
134	127, 130, 133	3eqtr3i 2117	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
135	124, 134	eqtr3i 2111	. . . . 5 |- ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
136	91, 121, 135	3brtr3i 3878	. . . 4 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) < ( 1 / 6 )
137		rpexpcl 10028	. . . . . 6 |- ( ( A e. RR+ /\ 4 e. ZZ ) -> ( A ^ 4 ) e. RR+ )
138	38, 69, 137	sylancl 405	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR+ )
139		elrp 9190	. . . . . 6 |- ( ( A ^ 4 ) e. RR+ <-> ( ( A ^ 4 ) e. RR /\ 0 < ( A ^ 4 ) ) )
140		ltmul2 8371	. . . . . . 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 ) ) ) )
141	24, 131, 140	mp3an12 1264	. . . . . 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 ) ) ) )
142	139, 141	sylbi 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 ) ) ) )
143	138, 142	syl 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 ) ) ) )
144	136, 143	mpbii 147	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
145	16	recnd 7570	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. CC )
146		divrecap 8209	. . . . 5 |- ( ( ( A ^ 4 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
147	105, 129, 146	mp3an23 1266	. . . 4 |- ( ( A ^ 4 ) e. CC -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
148	145, 147	syl 14	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
149	144, 148	breqtrrd 3877	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) / 6 ) )
150	14, 26, 29, 52, 149	lelttrd 7662	1 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   = wceq 1290   e. wcel 1439   class class class wbr 3851   |-> cmpt 3905   ` cfv 5028  (class class class)co 5666   CCcc 7402   RRcr 7403   0cc0 7404   1c1 7405   _ici 7406   + caddc 7407   x. cmul 7409   RR*cxr 7575   < clt 7576   <_ cle 7577   # cap 8112   / cdiv 8193   NNcn 8476   2c2 8527   3c3 8528   4c4 8529   5c5 8530   6c6 8531   8c8 8533   NN0cn0 8727   ZZcz 8804   ZZ>=cuz 9073   RR+crp 9188   (,]cioc 9361   ^cexp 10008   !cfa 10187   abscabs 10484   sum_csu 10796

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517  ax-arch 7518  ax-caucvg 7519

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-isom 5037  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-frec 6170  df-1o 6195  df-oadd 6199  df-er 6306  df-en 6512  df-dom 6513  df-fin 6514  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-2 8535  df-3 8536  df-4 8537  df-5 8538  df-6 8539  df-7 8540  df-8 8541  df-n0 8728  df-z 8805  df-uz 9074  df-q 9159  df-rp 9189  df-ioc 9365  df-ico 9366  df-fz 9479  df-fzo 9608  df-iseq 9907  df-seq3 9908  df-exp 10009  df-fac 10188  df-ihash 10238  df-shft 10303  df-cj 10330  df-re 10331  df-im 10332  df-rsqrt 10485  df-abs 10486  df-clim 10721  df-isum 10797

This theorem is referenced by:  sin01bnd  11102  cos01bnd  11103