Theorem ef01bndlem 12442

Description: Lemma for sin01bnd 12443 and cos01bnd 12444. (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 8222	. . . . 5 |- _i e. CC
2		0xr 8320	. . . . . . . 8 |- 0 e. RR*
3		1re 8273	. . . . . . . 8 |- 1 e. RR
4		elioc2 10269	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
5	2, 3, 4	mp2an 426	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
6	5	simp1bi 1039	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
7	6	recnd 8302	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
8		mulcl 8254	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
9	1, 7, 8	sylancr 414	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( _i x. A ) e. CC )
10		4nn0 9515	. . . 4 |- 4 e. NN0
11		ef01bnd.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
12	11	eftlcl 12374	. . . 4 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
13	9, 10, 12	sylancl 413	. . 3 |- ( A e. ( 0 (,] 1 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
14	13	abscld 11866	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) e. RR )
15		reexpcl 10918	. . . 4 |- ( ( A e. RR /\ 4 e. NN0 ) -> ( A ^ 4 ) e. RR )
16	6, 10, 15	sylancl 413	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR )
17		4re 9314	. . . . 5 |- 4 e. RR
18	17, 3	readdcli 8287	. . . 4 |- ( 4 + 1 ) e. RR
19		faccl 11097	. . . . . 6 |- ( 4 e. NN0 -> ( ! ` 4 ) e. NN )
20	10, 19	ax-mp 5	. . . . 5 |- ( ! ` 4 ) e. NN
21		4nn 9401	. . . . 5 |- 4 e. NN
22	20, 21	nnmulcli 9259	. . . 4 |- ( ( ! ` 4 ) x. 4 ) e. NN
23		nndivre 9273	. . . 4 |- ( ( ( 4 + 1 ) e. RR /\ ( ( ! ` 4 ) x. 4 ) e. NN ) -> ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR )
24	18, 22, 23	mp2an 426	. . 3 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR
25		remulcl 8255	. . 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 413	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
27		6nn 9403	. . 3 |- 6 e. NN
28		nndivre 9273	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ 6 e. NN ) -> ( ( A ^ 4 ) / 6 ) e. RR )
29	16, 27, 28	sylancl 413	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) e. RR )
30		eqid 2232	. . . 4 |- ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) )
31		eqid 2232	. . . 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 11754	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
34	1, 7, 33	sylancr 414	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
35		absi 11744	. . . . . . . 8 |- ( abs ` _i ) = 1
36	35	oveq1i 6060	. . . . . . 7 |- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) )
37	5	simp2bi 1040	. . . . . . . . . 10 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
38	6, 37	elrpd 10026	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) -> A e. RR+ )
39		rpre 9993	. . . . . . . . . 10 |- ( A e. RR+ -> A e. RR )
40		rpge0 9999	. . . . . . . . . 10 |- ( A e. RR+ -> 0 <_ A )
41	39, 40	absidd 11852	. . . . . . . . 9 |- ( A e. RR+ -> ( abs ` A ) = A )
42	38, 41	syl 14	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> ( abs ` A ) = A )
43	42	oveq2d 6066	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. ( abs ` A ) ) = ( 1 x. A ) )
44	36, 43	eqtrid 2277	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. A ) )
45	7	mullidd 8292	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. A ) = A )
46	34, 44, 45	3eqtrd 2269	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = A )
47	5	simp3bi 1041	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
48	46, 47	eqbrtrd 4131	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) <_ 1 )
49	11, 30, 31, 32, 9, 48	eftlub 12376	. . 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 6065	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` ( _i x. A ) ) ^ 4 ) = ( A ^ 4 ) )
51	50	oveq1d 6065	. . 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 4135	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
53		3pos 9331	. . . . . . . . 9 |- 0 < 3
54		0re 8274	. . . . . . . . . 10 |- 0 e. RR
55		3re 9311	. . . . . . . . . 10 |- 3 e. RR
56		5re 9316	. . . . . . . . . 10 |- 5 e. RR
57	54, 55, 56	ltadd1i 8776	. . . . . . . . 9 |- ( 0 < 3 <-> ( 0 + 5 ) < ( 3 + 5 ) )
58	53, 57	mpbi 145	. . . . . . . 8 |- ( 0 + 5 ) < ( 3 + 5 )
59		5cn 9317	. . . . . . . . 9 |- 5 e. CC
60	59	addlidi 8416	. . . . . . . 8 |- ( 0 + 5 ) = 5
61		cu2 11000	. . . . . . . . 9 |- ( 2 ^ 3 ) = 8
62		5p3e8 9385	. . . . . . . . 9 |- ( 5 + 3 ) = 8
63		3cn 9312	. . . . . . . . . 10 |- 3 e. CC
64	59, 63	addcomi 8417	. . . . . . . . 9 |- ( 5 + 3 ) = ( 3 + 5 )
65	61, 62, 64	3eqtr2ri 2260	. . . . . . . 8 |- ( 3 + 5 ) = ( 2 ^ 3 )
66	58, 60, 65	3brtr3i 4138	. . . . . . 7 |- 5 < ( 2 ^ 3 )
67		2re 9307	. . . . . . . 8 |- 2 e. RR
68		1le2 9446	. . . . . . . 8 |- 1 <_ 2
69		4z 9607	. . . . . . . . 9 |- 4 e. ZZ
70		3lt4 9410	. . . . . . . . . 10 |- 3 < 4
71	55, 17, 70	ltleii 8376	. . . . . . . . 9 |- 3 <_ 4
72		3z 9606	. . . . . . . . . 10 |- 3 e. ZZ
73	72	eluz1i 9861	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
74	69, 71, 73	mpbir2an 951	. . . . . . . 8 |- 4 e. ( ZZ>= ` 3 )
75		leexp2a 10954	. . . . . . . 8 |- ( ( 2 e. RR /\ 1 <_ 2 /\ 4 e. ( ZZ>= ` 3 ) ) -> ( 2 ^ 3 ) <_ ( 2 ^ 4 ) )
76	67, 68, 74, 75	mp3an 1374	. . . . . . 7 |- ( 2 ^ 3 ) <_ ( 2 ^ 4 )
77		8re 9322	. . . . . . . . 9 |- 8 e. RR
78	61, 77	eqeltri 2305	. . . . . . . 8 |- ( 2 ^ 3 ) e. RR
79		2nn 9399	. . . . . . . . . 10 |- 2 e. NN
80		nnexpcl 10914	. . . . . . . . . 10 |- ( ( 2 e. NN /\ 4 e. NN0 ) -> ( 2 ^ 4 ) e. NN )
81	79, 10, 80	mp2an 426	. . . . . . . . 9 |- ( 2 ^ 4 ) e. NN
82	81	nnrei 9246	. . . . . . . 8 |- ( 2 ^ 4 ) e. RR
83	56, 78, 82	ltletri 8380	. . . . . . 7 |- ( ( 5 < ( 2 ^ 3 ) /\ ( 2 ^ 3 ) <_ ( 2 ^ 4 ) ) -> 5 < ( 2 ^ 4 ) )
84	66, 76, 83	mp2an 426	. . . . . 6 |- 5 < ( 2 ^ 4 )
85		6re 9318	. . . . . . . 8 |- 6 e. RR
86	85, 82	remulcli 8288	. . . . . . 7 |- ( 6 x. ( 2 ^ 4 ) ) e. RR
87		6pos 9338	. . . . . . . 8 |- 0 < 6
88	81	nngt0i 9267	. . . . . . . 8 |- 0 < ( 2 ^ 4 )
89	85, 82, 87, 88	mulgt0ii 8384	. . . . . . 7 |- 0 < ( 6 x. ( 2 ^ 4 ) )
90	56, 82, 86, 89	ltdiv1ii 9203	. . . . . 6 |- ( 5 < ( 2 ^ 4 ) <-> ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) )
91	84, 90	mpbi 145	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
92		df-5 9299	. . . . . 6 |- 5 = ( 4 + 1 )
93		df-4 9298	. . . . . . . . . . 11 |- 4 = ( 3 + 1 )
94	93	fveq2i 5673	. . . . . . . . . 10 |- ( ! ` 4 ) = ( ! ` ( 3 + 1 ) )
95		3nn0 9514	. . . . . . . . . . 11 |- 3 e. NN0
96		facp1 11092	. . . . . . . . . . 11 |- ( 3 e. NN0 -> ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) ) )
97	95, 96	ax-mp 5	. . . . . . . . . 10 |- ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) )
98		sq2 10997	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
99	98, 93	eqtr2i 2254	. . . . . . . . . . 11 |- ( 3 + 1 ) = ( 2 ^ 2 )
100	99	oveq2i 6061	. . . . . . . . . 10 |- ( ( ! ` 3 ) x. ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
101	94, 97, 100	3eqtri 2257	. . . . . . . . 9 |- ( ! ` 4 ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
102	101	oveq1i 6060	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) )
103	98	oveq2i 6061	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ! ` 4 ) x. 4 )
104		fac3 11094	. . . . . . . . . 10 |- ( ! ` 3 ) = 6
105		6cn 9319	. . . . . . . . . 10 |- 6 e. CC
106	104, 105	eqeltri 2305	. . . . . . . . 9 |- ( ! ` 3 ) e. CC
107	17	recni 8286	. . . . . . . . . 10 |- 4 e. CC
108	98, 107	eqeltri 2305	. . . . . . . . 9 |- ( 2 ^ 2 ) e. CC
109	106, 108, 108	mulassi 8283	. . . . . . . 8 |- ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
110	102, 103, 109	3eqtr3i 2261	. . . . . . 7 |- ( ( ! ` 4 ) x. 4 ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
111		2p2e4 9364	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
112	111	oveq2i 6061	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( 2 ^ 4 )
113		2cn 9308	. . . . . . . . . 10 |- 2 e. CC
114		2nn0 9513	. . . . . . . . . 10 |- 2 e. NN0
115		expadd 10943	. . . . . . . . . 10 |- ( ( 2 e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
116	113, 114, 114, 115	mp3an 1374	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
117	112, 116	eqtr3i 2255	. . . . . . . 8 |- ( 2 ^ 4 ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
118	117	oveq2i 6061	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
119	104	oveq1i 6060	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( 6 x. ( 2 ^ 4 ) )
120	110, 118, 119	3eqtr2ri 2260	. . . . . 6 |- ( 6 x. ( 2 ^ 4 ) ) = ( ( ! ` 4 ) x. 4 )
121	92, 120	oveq12i 6062	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) )
122	81	nncni 9247	. . . . . . . 8 |- ( 2 ^ 4 ) e. CC
123	122	mullidi 8277	. . . . . . 7 |- ( 1 x. ( 2 ^ 4 ) ) = ( 2 ^ 4 )
124	123	oveq1i 6060	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
125	82, 88	gt0ap0ii 8902	. . . . . . . . 9 |- ( 2 ^ 4 ) # 0
126	122, 125	dividapi 9019	. . . . . . . 8 |- ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) = 1
127	126	oveq2i 6061	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 / 6 ) x. 1 )
128		ax-1cn 8220	. . . . . . . 8 |- 1 e. CC
129	85, 87	gt0ap0ii 8902	. . . . . . . 8 |- 6 # 0
130	128, 105, 122, 122, 129, 125	divmuldivapi 9046	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) )
131	85, 129	rerecclapi 9051	. . . . . . . . 9 |- ( 1 / 6 ) e. RR
132	131	recni 8286	. . . . . . . 8 |- ( 1 / 6 ) e. CC
133	132	mulridi 8276	. . . . . . 7 |- ( ( 1 / 6 ) x. 1 ) = ( 1 / 6 )
134	127, 130, 133	3eqtr3i 2261	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
135	124, 134	eqtr3i 2255	. . . . 5 |- ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
136	91, 121, 135	3brtr3i 4138	. . . 4 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) < ( 1 / 6 )
137		rpexpcl 10920	. . . . . 6 |- ( ( A e. RR+ /\ 4 e. ZZ ) -> ( A ^ 4 ) e. RR+ )
138	38, 69, 137	sylancl 413	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR+ )
139		elrp 9988	. . . . . 6 |- ( ( A ^ 4 ) e. RR+ <-> ( ( A ^ 4 ) e. RR /\ 0 < ( A ^ 4 ) ) )
140		ltmul2 9130	. . . . . . 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 1364	. . . . . 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 121	. . . . 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 148	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
145	16	recnd 8302	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. CC )
146		divrecap 8962	. . . . 5 |- ( ( ( A ^ 4 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
147	105, 129, 146	mp3an23 1366	. . . 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 4137	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) / 6 ) )
150	14, 26, 29, 52, 149	lelttrd 8398	1 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   _ici 8129   + caddc 8130   x. cmul 8132   RR*cxr 8307   < clt 8308   <_ cle 8309   # cap 8855   / cdiv 8946   NNcn 9237   2c2 9288   3c3 9289   4c4 9290   5c5 9291   6c6 9292   8c8 9294   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   RR+crp 9986   (,]cioc 10222   ^cexp 10900   !cfa 11087   abscabs 11682   sum_csu 12038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ioc 10226  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-ihash 11139  df-shft 11500  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  sin01bnd  12443  cos01bnd  12444