Theorem ef01bndlem 14315

Description: Lemma for sin01bnd 14316 and cos01bnd 14317. (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 9616	. . . . 5 |- _i e. CC
2		0xr 9705	. . . . . . . 8 |- 0 e. RR*
3		1re 9660	. . . . . . . 8 |- 1 e. RR
4		elioc2 11722	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
5	2, 3, 4	mp2an 686	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
6	5	simp1bi 1045	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
7	6	recnd 9687	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
8		mulcl 9641	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
9	1, 7, 8	sylancr 676	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( _i x. A ) e. CC )
10		4nn0 10912	. . . 4 |- 4 e. NN0
11		ef01bnd.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
12	11	eftlcl 14238	. . . 4 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
13	9, 10, 12	sylancl 675	. . 3 |- ( A e. ( 0 (,] 1 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
14	13	abscld 13575	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) e. RR )
15		reexpcl 12327	. . . 4 |- ( ( A e. RR /\ 4 e. NN0 ) -> ( A ^ 4 ) e. RR )
16	6, 10, 15	sylancl 675	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR )
17		4re 10708	. . . . 5 |- 4 e. RR
18	17, 3	readdcli 9674	. . . 4 |- ( 4 + 1 ) e. RR
19		faccl 12507	. . . . . 6 |- ( 4 e. NN0 -> ( ! ` 4 ) e. NN )
20	10, 19	ax-mp 5	. . . . 5 |- ( ! ` 4 ) e. NN
21		4nn 10792	. . . . 5 |- 4 e. NN
22	20, 21	nnmulcli 10655	. . . 4 |- ( ( ! ` 4 ) x. 4 ) e. NN
23		nndivre 10667	. . . 4 |- ( ( ( 4 + 1 ) e. RR /\ ( ( ! ` 4 ) x. 4 ) e. NN ) -> ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR )
24	18, 22, 23	mp2an 686	. . 3 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR
25		remulcl 9642	. . 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 675	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
27		6nn 10794	. . 3 |- 6 e. NN
28		nndivre 10667	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ 6 e. NN ) -> ( ( A ^ 4 ) / 6 ) e. RR )
29	16, 27, 28	sylancl 675	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) e. RR )
30		eqid 2471	. . . 4 |- ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) )
31		eqid 2471	. . . 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 11	. . . 4 |- ( A e. ( 0 (,] 1 ) -> 4 e. NN )
33		absmul 13434	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
34	1, 7, 33	sylancr 676	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
35		absi 13426	. . . . . . . 8 |- ( abs ` _i ) = 1
36	35	oveq1i 6318	. . . . . . 7 |- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) )
37	5	simp2bi 1046	. . . . . . . . . 10 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
38	6, 37	elrpd 11361	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) -> A e. RR+ )
39		rpre 11331	. . . . . . . . . 10 |- ( A e. RR+ -> A e. RR )
40		rpge0 11337	. . . . . . . . . 10 |- ( A e. RR+ -> 0 <_ A )
41	39, 40	absidd 13561	. . . . . . . . 9 |- ( A e. RR+ -> ( abs ` A ) = A )
42	38, 41	syl 17	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> ( abs ` A ) = A )
43	42	oveq2d 6324	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. ( abs ` A ) ) = ( 1 x. A ) )
44	36, 43	syl5eq 2517	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. A ) )
45	7	mulid2d 9679	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. A ) = A )
46	34, 44, 45	3eqtrd 2509	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = A )
47	5	simp3bi 1047	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
48	46, 47	eqbrtrd 4416	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) <_ 1 )
49	11, 30, 31, 32, 9, 48	eftlub 14240	. . 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 6323	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` ( _i x. A ) ) ^ 4 ) = ( A ^ 4 ) )
51	50	oveq1d 6323	. . 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 4420	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
53		3pos 10725	. . . . . . . . 9 |- 0 < 3
54		0re 9661	. . . . . . . . . 10 |- 0 e. RR
55		3re 10705	. . . . . . . . . 10 |- 3 e. RR
56		5re 10710	. . . . . . . . . 10 |- 5 e. RR
57	54, 55, 56	ltadd1i 10189	. . . . . . . . 9 |- ( 0 < 3 <-> ( 0 + 5 ) < ( 3 + 5 ) )
58	53, 57	mpbi 213	. . . . . . . 8 |- ( 0 + 5 ) < ( 3 + 5 )
59		5cn 10711	. . . . . . . . 9 |- 5 e. CC
60	59	addid2i 9839	. . . . . . . 8 |- ( 0 + 5 ) = 5
61		cu2 12411	. . . . . . . . 9 |- ( 2 ^ 3 ) = 8
62		5p3e8 10771	. . . . . . . . 9 |- ( 5 + 3 ) = 8
63		3nn 10791	. . . . . . . . . . 11 |- 3 e. NN
64	63	nncni 10641	. . . . . . . . . 10 |- 3 e. CC
65	59, 64	addcomi 9842	. . . . . . . . 9 |- ( 5 + 3 ) = ( 3 + 5 )
66	61, 62, 65	3eqtr2ri 2500	. . . . . . . 8 |- ( 3 + 5 ) = ( 2 ^ 3 )
67	58, 60, 66	3brtr3i 4423	. . . . . . 7 |- 5 < ( 2 ^ 3 )
68		2re 10701	. . . . . . . 8 |- 2 e. RR
69		1le2 10846	. . . . . . . 8 |- 1 <_ 2
70		4z 10995	. . . . . . . . 9 |- 4 e. ZZ
71		3lt4 10802	. . . . . . . . . 10 |- 3 < 4
72	55, 17, 71	ltleii 9775	. . . . . . . . 9 |- 3 <_ 4
73	63	nnzi 10985	. . . . . . . . . 10 |- 3 e. ZZ
74	73	eluz1i 11190	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
75	70, 72, 74	mpbir2an 934	. . . . . . . 8 |- 4 e. ( ZZ>= ` 3 )
76		leexp2a 12366	. . . . . . . 8 |- ( ( 2 e. RR /\ 1 <_ 2 /\ 4 e. ( ZZ>= ` 3 ) ) -> ( 2 ^ 3 ) <_ ( 2 ^ 4 ) )
77	68, 69, 75, 76	mp3an 1390	. . . . . . 7 |- ( 2 ^ 3 ) <_ ( 2 ^ 4 )
78		8re 10716	. . . . . . . . 9 |- 8 e. RR
79	61, 78	eqeltri 2545	. . . . . . . 8 |- ( 2 ^ 3 ) e. RR
80		2nn 10790	. . . . . . . . . 10 |- 2 e. NN
81		nnexpcl 12323	. . . . . . . . . 10 |- ( ( 2 e. NN /\ 4 e. NN0 ) -> ( 2 ^ 4 ) e. NN )
82	80, 10, 81	mp2an 686	. . . . . . . . 9 |- ( 2 ^ 4 ) e. NN
83	82	nnrei 10640	. . . . . . . 8 |- ( 2 ^ 4 ) e. RR
84	56, 79, 83	ltletri 9780	. . . . . . 7 |- ( ( 5 < ( 2 ^ 3 ) /\ ( 2 ^ 3 ) <_ ( 2 ^ 4 ) ) -> 5 < ( 2 ^ 4 ) )
85	67, 77, 84	mp2an 686	. . . . . 6 |- 5 < ( 2 ^ 4 )
86		6re 10712	. . . . . . . 8 |- 6 e. RR
87	86, 83	remulcli 9675	. . . . . . 7 |- ( 6 x. ( 2 ^ 4 ) ) e. RR
88		6pos 10730	. . . . . . . 8 |- 0 < 6
89	82	nngt0i 10665	. . . . . . . 8 |- 0 < ( 2 ^ 4 )
90	86, 83, 88, 89	mulgt0ii 9785	. . . . . . 7 |- 0 < ( 6 x. ( 2 ^ 4 ) )
91	56, 83, 87, 90	ltdiv1ii 10558	. . . . . 6 |- ( 5 < ( 2 ^ 4 ) <-> ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) )
92	85, 91	mpbi 213	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
93		df-5 10693	. . . . . 6 |- 5 = ( 4 + 1 )
94		df-4 10692	. . . . . . . . . . 11 |- 4 = ( 3 + 1 )
95	94	fveq2i 5882	. . . . . . . . . 10 |- ( ! ` 4 ) = ( ! ` ( 3 + 1 ) )
96		3nn0 10911	. . . . . . . . . . 11 |- 3 e. NN0
97		facp1 12502	. . . . . . . . . . 11 |- ( 3 e. NN0 -> ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) ) )
98	96, 97	ax-mp 5	. . . . . . . . . 10 |- ( ! ` ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 3 + 1 ) )
99		sq2 12409	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
100	99, 94	eqtr2i 2494	. . . . . . . . . . 11 |- ( 3 + 1 ) = ( 2 ^ 2 )
101	100	oveq2i 6319	. . . . . . . . . 10 |- ( ( ! ` 3 ) x. ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
102	95, 98, 101	3eqtri 2497	. . . . . . . . 9 |- ( ! ` 4 ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
103	102	oveq1i 6318	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) )
104	99	oveq2i 6319	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ! ` 4 ) x. 4 )
105		fac3 12504	. . . . . . . . . 10 |- ( ! ` 3 ) = 6
106		6cn 10713	. . . . . . . . . 10 |- 6 e. CC
107	105, 106	eqeltri 2545	. . . . . . . . 9 |- ( ! ` 3 ) e. CC
108	17	recni 9673	. . . . . . . . . 10 |- 4 e. CC
109	99, 108	eqeltri 2545	. . . . . . . . 9 |- ( 2 ^ 2 ) e. CC
110	107, 109, 109	mulassi 9670	. . . . . . . 8 |- ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
111	103, 104, 110	3eqtr3i 2501	. . . . . . 7 |- ( ( ! ` 4 ) x. 4 ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
112		2p2e4 10750	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
113	112	oveq2i 6319	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( 2 ^ 4 )
114		2cn 10702	. . . . . . . . . 10 |- 2 e. CC
115		2nn0 10910	. . . . . . . . . 10 |- 2 e. NN0
116		expadd 12352	. . . . . . . . . 10 |- ( ( 2 e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
117	114, 115, 115, 116	mp3an 1390	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
118	113, 117	eqtr3i 2495	. . . . . . . 8 |- ( 2 ^ 4 ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
119	118	oveq2i 6319	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
120	105	oveq1i 6318	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( 6 x. ( 2 ^ 4 ) )
121	111, 119, 120	3eqtr2ri 2500	. . . . . 6 |- ( 6 x. ( 2 ^ 4 ) ) = ( ( ! ` 4 ) x. 4 )
122	93, 121	oveq12i 6320	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) )
123	82	nncni 10641	. . . . . . . 8 |- ( 2 ^ 4 ) e. CC
124	123	mulid2i 9664	. . . . . . 7 |- ( 1 x. ( 2 ^ 4 ) ) = ( 2 ^ 4 )
125	124	oveq1i 6318	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
126	82	nnne0i 10666	. . . . . . . . 9 |- ( 2 ^ 4 ) =/= 0
127	123, 126	dividi 10362	. . . . . . . 8 |- ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) = 1
128	127	oveq2i 6319	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 / 6 ) x. 1 )
129		ax-1cn 9615	. . . . . . . 8 |- 1 e. CC
130	86, 88	gt0ne0ii 10171	. . . . . . . 8 |- 6 =/= 0
131	129, 106, 123, 123, 130, 126	divmuldivi 10389	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) )
132	86, 130	rereccli 10394	. . . . . . . . 9 |- ( 1 / 6 ) e. RR
133	132	recni 9673	. . . . . . . 8 |- ( 1 / 6 ) e. CC
134	133	mulid1i 9663	. . . . . . 7 |- ( ( 1 / 6 ) x. 1 ) = ( 1 / 6 )
135	128, 131, 134	3eqtr3i 2501	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
136	125, 135	eqtr3i 2495	. . . . 5 |- ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
137	92, 122, 136	3brtr3i 4423	. . . 4 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) < ( 1 / 6 )
138		rpexpcl 12329	. . . . . 6 |- ( ( A e. RR+ /\ 4 e. ZZ ) -> ( A ^ 4 ) e. RR+ )
139	38, 70, 138	sylancl 675	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR+ )
140		elrp 11327	. . . . . 6 |- ( ( A ^ 4 ) e. RR+ <-> ( ( A ^ 4 ) e. RR /\ 0 < ( A ^ 4 ) ) )
141		ltmul2 10478	. . . . . . 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 ) ) ) )
142	24, 132, 141	mp3an12 1380	. . . . . 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 ) ) ) )
143	140, 142	sylbi 200	. . . . 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 ) ) ) )
144	139, 143	syl 17	. . . 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 ) ) ) )
145	137, 144	mpbii 216	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
146	16	recnd 9687	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. CC )
147		divrec 10308	. . . . 5 |- ( ( ( A ^ 4 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
148	106, 130, 147	mp3an23 1382	. . . 4 |- ( ( A ^ 4 ) e. CC -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
149	146, 148	syl 17	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
150	145, 149	breqtrrd 4422	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) / 6 ) )
151	14, 26, 29, 52, 150	lelttrd 9810	1 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   class class class wbr 4395   |-> cmpt 4454   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   _ici 9559   + caddc 9560   x. cmul 9562   RR*cxr 9692   < clt 9693   <_ cle 9694   / cdiv 10291   NNcn 10631   2c2 10681   3c3 10682   4c4 10683   5c5 10684   6c6 10685   8c8 10687   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   (,]cioc 11661   ^cexp 12310   !cfa 12497   abscabs 13374   sum_csu 13829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ioc 11665  df-ico 11666  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830

This theorem is referenced by:  sin01bnd  14316  cos01bnd  14317