Theorem ef01bndlem 12523

Description: Lemma for sin01bnd 12524 and cos01bnd 12525. (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 8274	. . . . 5 |- _i e. CC
2		0xr 8372	. . . . . . . 8 |- 0 e. RR*
3		1re 8325	. . . . . . . 8 |- 1 e. RR
4		elioc2 10338	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
5	2, 3, 4	mp2an 430	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
6	5	simp1bi 1043	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
7	6	recnd 8354	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
8		mulcl 8306	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
9	1, 7, 8	sylancr 418	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( _i x. A ) e. CC )
10		4nn0 9582	. . . 4 |- 4 e. NN0
11		ef01bnd.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
12	11	eftlcl 12455	. . . 4 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
13	9, 10, 12	sylancl 417	. . 3 |- ( A e. ( 0 (,] 1 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
14	13	abscld 11947	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) e. RR )
15		reexpcl 10993	. . . 4 |- ( ( A e. RR /\ 4 e. NN0 ) -> ( A ^ 4 ) e. RR )
16	6, 10, 15	sylancl 417	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR )
17		4re 9381	. . . . 5 |- 4 e. RR
18	17, 3	readdcli 8339	. . . 4 |- ( 4 + 1 ) e. RR
19		faccl 11173	. . . . . 6 |- ( 4 e. NN0 -> ( ! ` 4 ) e. NN )
20	10, 19	ax-mp 5	. . . . 5 |- ( ! ` 4 ) e. NN
21		4nn 9468	. . . . 5 |- 4 e. NN
22	20, 21	nnmulcli 9326	. . . 4 |- ( ( ! ` 4 ) x. 4 ) e. NN
23		nndivre 9340	. . . 4 |- ( ( ( 4 + 1 ) e. RR /\ ( ( ! ` 4 ) x. 4 ) e. NN ) -> ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR )
24	18, 22, 23	mp2an 430	. . 3 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR
25		remulcl 8307	. . 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 417	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
27		6nn 9470	. . 3 |- 6 e. NN
28		nndivre 9340	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ 6 e. NN ) -> ( ( A ^ 4 ) / 6 ) e. RR )
29	16, 27, 28	sylancl 417	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) e. RR )
30		eqid 2238	. . . 4 |- ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) )
31		eqid 2238	. . . 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 11835	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
34	1, 7, 33	sylancr 418	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
35		absi 11825	. . . . . . . 8 |- ( abs ` _i ) = 1
36	35	oveq1i 6095	. . . . . . 7 |- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) )
37	5	simp2bi 1044	. . . . . . . . . 10 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
38	6, 37	elrpd 10094	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) -> A e. RR+ )
39		rpre 10061	. . . . . . . . . 10 |- ( A e. RR+ -> A e. RR )
40		rpge0 10067	. . . . . . . . . 10 |- ( A e. RR+ -> 0 <_ A )
41	39, 40	absidd 11933	. . . . . . . . 9 |- ( A e. RR+ -> ( abs ` A ) = A )
42	38, 41	syl 14	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> ( abs ` A ) = A )
43	42	oveq2d 6101	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. ( abs ` A ) ) = ( 1 x. A ) )
44	36, 43	eqtrid 2283	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. A ) )
45	7	mullidd 8344	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. A ) = A )
46	34, 44, 45	3eqtrd 2275	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = A )
47	5	simp3bi 1045	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
48	46, 47	eqbrtrd 4152	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) <_ 1 )
49	11, 30, 31, 32, 9, 48	eftlub 12457	. . 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 6100	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` ( _i x. A ) ) ^ 4 ) = ( A ^ 4 ) )
51	50	oveq1d 6100	. . 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 4156	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
53		3pos 9398	. . . . . . . . 9 |- 0 < 3
54		0re 8326	. . . . . . . . . 10 |- 0 e. RR
55		3re 9378	. . . . . . . . . 10 |- 3 e. RR
56		5re 9383	. . . . . . . . . 10 |- 5 e. RR
57	54, 55, 56	ltadd1i 8830	. . . . . . . . 9 |- ( 0 < 3 <-> ( 0 + 5 ) < ( 3 + 5 ) )
58	53, 57	mpbi 145	. . . . . . . 8 |- ( 0 + 5 ) < ( 3 + 5 )
59		5cn 9384	. . . . . . . . 9 |- 5 e. CC
60	59	addlidi 8469	. . . . . . . 8 |- ( 0 + 5 ) = 5
61		cu2 11075	. . . . . . . . 9 |- ( 2 ^ 3 ) = 8
62		5p3e8 9452	. . . . . . . . 9 |- ( 5 + 3 ) = 8
63		3cn 9379	. . . . . . . . . 10 |- 3 e. CC
64	59, 63	addcomi 8470	. . . . . . . . 9 |- ( 5 + 3 ) = ( 3 + 5 )
65	61, 62, 64	3eqtr2ri 2266	. . . . . . . 8 |- ( 3 + 5 ) = ( 2 ^ 3 )
66	58, 60, 65	3brtr3i 4159	. . . . . . 7 |- 5 < ( 2 ^ 3 )
67		2re 9374	. . . . . . . 8 |- 2 e. RR
68		1le2 9513	. . . . . . . 8 |- 1 <_ 2
69		4z 9674	. . . . . . . . 9 |- 4 e. ZZ
70		3lt4 9477	. . . . . . . . . 10 |- 3 < 4
71	55, 17, 70	ltleii 8428	. . . . . . . . 9 |- 3 <_ 4
72		3z 9673	. . . . . . . . . 10 |- 3 e. ZZ
73	72	eluz1i 9929	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
74	69, 71, 73	mpbir2an 955	. . . . . . . 8 |- 4 e. ( ZZ>= ` 3 )
75		leexp2a 11029	. . . . . . . 8 |- ( ( 2 e. RR /\ 1 <_ 2 /\ 4 e. ( ZZ>= ` 3 ) ) -> ( 2 ^ 3 ) <_ ( 2 ^ 4 ) )
76	67, 68, 74, 75	mp3an 1378	. . . . . . 7 |- ( 2 ^ 3 ) <_ ( 2 ^ 4 )
77		8re 9389	. . . . . . . . 9 |- 8 e. RR
78	61, 77	eqeltri 2311	. . . . . . . 8 |- ( 2 ^ 3 ) e. RR
79		2nn 9466	. . . . . . . . . 10 |- 2 e. NN
80		nnexpcl 10989	. . . . . . . . . 10 |- ( ( 2 e. NN /\ 4 e. NN0 ) -> ( 2 ^ 4 ) e. NN )
81	79, 10, 80	mp2an 430	. . . . . . . . 9 |- ( 2 ^ 4 ) e. NN
82	81	nnrei 9313	. . . . . . . 8 |- ( 2 ^ 4 ) e. RR
83	56, 78, 82	ltletri 8432	. . . . . . 7 |- ( ( 5 < ( 2 ^ 3 ) /\ ( 2 ^ 3 ) <_ ( 2 ^ 4 ) ) -> 5 < ( 2 ^ 4 ) )
84	66, 76, 83	mp2an 430	. . . . . 6 |- 5 < ( 2 ^ 4 )
85		6re 9385	. . . . . . . 8 |- 6 e. RR
86	85, 82	remulcli 8340	. . . . . . 7 |- ( 6 x. ( 2 ^ 4 ) ) e. RR
87		6pos 9405	. . . . . . . 8 |- 0 < 6
88	81	nngt0i 9334	. . . . . . . 8 |- 0 < ( 2 ^ 4 )
89	85, 82, 87, 88	mulgt0ii 8436	. . . . . . 7 |- 0 < ( 6 x. ( 2 ^ 4 ) )
90	56, 82, 86, 89	ltdiv1ii 9259	. . . . . 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 9366	. . . . . 6 |- 5 = ( 4 + 1 )
93		df-4 9365	. . . . . . . . . . 11 |- 4 = ( 3 + 1 )
94	93	fveq2i 5698	. . . . . . . . . 10 |- ( ! ` 4 ) = ( ! ` ( 3 + 1 ) )
95		3nn0 9581	. . . . . . . . . . 11 |- 3 e. NN0
96		facp1 11168	. . . . . . . . . . 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 11072	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
99	98, 93	eqtr2i 2260	. . . . . . . . . . 11 |- ( 3 + 1 ) = ( 2 ^ 2 )
100	99	oveq2i 6096	. . . . . . . . . 10 |- ( ( ! ` 3 ) x. ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
101	94, 97, 100	3eqtri 2263	. . . . . . . . 9 |- ( ! ` 4 ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
102	101	oveq1i 6095	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) )
103	98	oveq2i 6096	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ! ` 4 ) x. 4 )
104		fac3 11170	. . . . . . . . . 10 |- ( ! ` 3 ) = 6
105		6cn 9386	. . . . . . . . . 10 |- 6 e. CC
106	104, 105	eqeltri 2311	. . . . . . . . 9 |- ( ! ` 3 ) e. CC
107	17	recni 8338	. . . . . . . . . 10 |- 4 e. CC
108	98, 107	eqeltri 2311	. . . . . . . . 9 |- ( 2 ^ 2 ) e. CC
109	106, 108, 108	mulassi 8335	. . . . . . . 8 |- ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
110	102, 103, 109	3eqtr3i 2267	. . . . . . 7 |- ( ( ! ` 4 ) x. 4 ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
111		2p2e4 9431	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
112	111	oveq2i 6096	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( 2 ^ 4 )
113		2cn 9375	. . . . . . . . . 10 |- 2 e. CC
114		2nn0 9580	. . . . . . . . . 10 |- 2 e. NN0
115		expadd 11018	. . . . . . . . . 10 |- ( ( 2 e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
116	113, 114, 114, 115	mp3an 1378	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
117	112, 116	eqtr3i 2261	. . . . . . . 8 |- ( 2 ^ 4 ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
118	117	oveq2i 6096	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
119	104	oveq1i 6095	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( 6 x. ( 2 ^ 4 ) )
120	110, 118, 119	3eqtr2ri 2266	. . . . . 6 |- ( 6 x. ( 2 ^ 4 ) ) = ( ( ! ` 4 ) x. 4 )
121	92, 120	oveq12i 6097	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) )
122	81	nncni 9314	. . . . . . . 8 |- ( 2 ^ 4 ) e. CC
123	122	mullidi 8329	. . . . . . 7 |- ( 1 x. ( 2 ^ 4 ) ) = ( 2 ^ 4 )
124	123	oveq1i 6095	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
125	82, 88	gt0ap0ii 8956	. . . . . . . . 9 |- ( 2 ^ 4 ) # 0
126	122, 125	dividapi 9075	. . . . . . . 8 |- ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) = 1
127	126	oveq2i 6096	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 / 6 ) x. 1 )
128		ax-1cn 8272	. . . . . . . 8 |- 1 e. CC
129	85, 87	gt0ap0ii 8956	. . . . . . . 8 |- 6 # 0
130	128, 105, 122, 122, 129, 125	divmuldivapi 9102	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) )
131	85, 129	rerecclapi 9107	. . . . . . . . 9 |- ( 1 / 6 ) e. RR
132	131	recni 8338	. . . . . . . 8 |- ( 1 / 6 ) e. CC
133	132	mulridi 8328	. . . . . . 7 |- ( ( 1 / 6 ) x. 1 ) = ( 1 / 6 )
134	127, 130, 133	3eqtr3i 2267	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
135	124, 134	eqtr3i 2261	. . . . 5 |- ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
136	91, 121, 135	3brtr3i 4159	. . . 4 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) < ( 1 / 6 )
137		rpexpcl 10995	. . . . . 6 |- ( ( A e. RR+ /\ 4 e. ZZ ) -> ( A ^ 4 ) e. RR+ )
138	38, 69, 137	sylancl 417	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR+ )
139		elrp 10056	. . . . . 6 |- ( ( A ^ 4 ) e. RR+ <-> ( ( A ^ 4 ) e. RR /\ 0 < ( A ^ 4 ) ) )
140		ltmul2 9186	. . . . . . 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 1368	. . . . . 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 8354	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. CC )
146		divrecap 9018	. . . . 5 |- ( ( ( A ^ 4 ) e. CC /\ 6 e. CC /\ 6 # 0 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
147	105, 129, 146	mp3an23 1370	. . . 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 4158	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) / 6 ) )
150	14, 26, 29, 52, 149	lelttrd 8451	1 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   _ici 8181   + caddc 8182   x. cmul 8184   RR*cxr 8359   < clt 8360   <_ cle 8361   # cap 8909   / cdiv 9002   NNcn 9304   2c2 9355   3c3 9356   4c4 9357   5c5 9358   6c6 9359   8c8 9361   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   RR+crp 10054   (,]cioc 10291   ^cexp 10975   !cfa 11163   abscabs 11763   sum_csu 12119

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-ioc 10295  df-ico 10296  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  sin01bnd  12524  cos01bnd  12525