Theorem ef01bndlem 13464

Description: Lemma for sin01bnd 13465 and cos01bnd 13466. (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 9337	. . . . 5 |- _i e. CC
2		0xr 9426	. . . . . . . 8 |- 0 e. RR*
3		1re 9381	. . . . . . . 8 |- 1 e. RR
4		elioc2 11354	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
5	2, 3, 4	mp2an 667	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
6	5	simp1bi 998	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
7	6	recnd 9408	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
8		mulcl 9362	. . . . 5 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
9	1, 7, 8	sylancr 658	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( _i x. A ) e. CC )
10		4nn0 10594	. . . 4 |- 4 e. NN0
11		ef01bnd.1	. . . . 5 |- F = ( n e. NN0 |-> ( ( ( _i x. A ) ^ n ) / ( ! ` n ) ) )
12	11	eftlcl 13387	. . . 4 |- ( ( ( _i x. A ) e. CC /\ 4 e. NN0 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
13	9, 10, 12	sylancl 657	. . 3 |- ( A e. ( 0 (,] 1 ) -> sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) e. CC )
14	13	abscld 12918	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) e. RR )
15		reexpcl 11878	. . . 4 |- ( ( A e. RR /\ 4 e. NN0 ) -> ( A ^ 4 ) e. RR )
16	6, 10, 15	sylancl 657	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR )
17		4re 10394	. . . . 5 |- 4 e. RR
18	17, 3	readdcli 9395	. . . 4 |- ( 4 + 1 ) e. RR
19		faccl 12057	. . . . . 6 |- ( 4 e. NN0 -> ( ! ` 4 ) e. NN )
20	10, 19	ax-mp 5	. . . . 5 |- ( ! ` 4 ) e. NN
21		4nn 10477	. . . . 5 |- 4 e. NN
22	20, 21	nnmulcli 10342	. . . 4 |- ( ( ! ` 4 ) x. 4 ) e. NN
23		nndivre 10353	. . . 4 |- ( ( ( 4 + 1 ) e. RR /\ ( ( ! ` 4 ) x. 4 ) e. NN ) -> ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR )
24	18, 22, 23	mp2an 667	. . 3 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) e. RR
25		remulcl 9363	. . 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 657	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) e. RR )
27		6nn 10479	. . 3 |- 6 e. NN
28		nndivre 10353	. . 3 |- ( ( ( A ^ 4 ) e. RR /\ 6 e. NN ) -> ( ( A ^ 4 ) / 6 ) e. RR )
29	16, 27, 28	sylancl 657	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) e. RR )
30		eqid 2441	. . . 4 |- ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) ) = ( n e. NN0 |-> ( ( ( abs ` ( _i x. A ) ) ^ n ) / ( ! ` n ) ) )
31		eqid 2441	. . . 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 12779	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
34	1, 7, 33	sylancr 658	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = ( ( abs ` _i ) x. ( abs ` A ) ) )
35		absi 12771	. . . . . . . 8 |- ( abs ` _i ) = 1
36	35	oveq1i 6100	. . . . . . 7 |- ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. ( abs ` A ) )
37	5	simp2bi 999	. . . . . . . . . 10 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
38	6, 37	elrpd 11021	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) -> A e. RR+ )
39		rpre 10993	. . . . . . . . . 10 |- ( A e. RR+ -> A e. RR )
40		rpge0 10999	. . . . . . . . . 10 |- ( A e. RR+ -> 0 <_ A )
41	39, 40	absidd 12905	. . . . . . . . 9 |- ( A e. RR+ -> ( abs ` A ) = A )
42	38, 41	syl 16	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> ( abs ` A ) = A )
43	42	oveq2d 6106	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. ( abs ` A ) ) = ( 1 x. A ) )
44	36, 43	syl5eq 2485	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` _i ) x. ( abs ` A ) ) = ( 1 x. A ) )
45	7	mulid2d 9400	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( 1 x. A ) = A )
46	34, 44, 45	3eqtrd 2477	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) = A )
47	5	simp3bi 1000	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
48	46, 47	eqbrtrd 4309	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( abs ` ( _i x. A ) ) <_ 1 )
49	11, 30, 31, 32, 9, 48	eftlub 13389	. . 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 6105	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( abs ` ( _i x. A ) ) ^ 4 ) = ( A ^ 4 ) )
51	50	oveq1d 6105	. . 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 4313	. 2 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) <_ ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) )
53		3pos 10411	. . . . . . . . 9 |- 0 < 3
54		0re 9382	. . . . . . . . . 10 |- 0 e. RR
55		3re 10391	. . . . . . . . . 10 |- 3 e. RR
56		5re 10396	. . . . . . . . . 10 |- 5 e. RR
57	54, 55, 56	ltadd1i 9890	. . . . . . . . 9 |- ( 0 < 3 <-> ( 0 + 5 ) < ( 3 + 5 ) )
58	53, 57	mpbi 208	. . . . . . . 8 |- ( 0 + 5 ) < ( 3 + 5 )
59		5cn 10397	. . . . . . . . 9 |- 5 e. CC
60	59	addid2i 9553	. . . . . . . 8 |- ( 0 + 5 ) = 5
61		cu2 11960	. . . . . . . . 9 |- ( 2 ^ 3 ) = 8
62		5p3e8 10456	. . . . . . . . 9 |- ( 5 + 3 ) = 8
63		3nn 10476	. . . . . . . . . . 11 |- 3 e. NN
64	63	nncni 10328	. . . . . . . . . 10 |- 3 e. CC
65	59, 64	addcomi 9556	. . . . . . . . 9 |- ( 5 + 3 ) = ( 3 + 5 )
66	61, 62, 65	3eqtr2ri 2468	. . . . . . . 8 |- ( 3 + 5 ) = ( 2 ^ 3 )
67	58, 60, 66	3brtr3i 4316	. . . . . . 7 |- 5 < ( 2 ^ 3 )
68		2re 10387	. . . . . . . 8 |- 2 e. RR
69		1le2 10531	. . . . . . . 8 |- 1 <_ 2
70	21	nnzi 10666	. . . . . . . . 9 |- 4 e. ZZ
71		3lt4 10487	. . . . . . . . . 10 |- 3 < 4
72	55, 17, 71	ltleii 9493	. . . . . . . . 9 |- 3 <_ 4
73	63	nnzi 10666	. . . . . . . . . 10 |- 3 e. ZZ
74	73	eluz1i 10864	. . . . . . . . 9 |- ( 4 e. ( ZZ>= ` 3 ) <-> ( 4 e. ZZ /\ 3 <_ 4 ) )
75	70, 72, 74	mpbir2an 906	. . . . . . . 8 |- 4 e. ( ZZ>= ` 3 )
76		leexp2a 11915	. . . . . . . 8 |- ( ( 2 e. RR /\ 1 <_ 2 /\ 4 e. ( ZZ>= ` 3 ) ) -> ( 2 ^ 3 ) <_ ( 2 ^ 4 ) )
77	68, 69, 75, 76	mp3an 1309	. . . . . . 7 |- ( 2 ^ 3 ) <_ ( 2 ^ 4 )
78		8re 10402	. . . . . . . . 9 |- 8 e. RR
79	61, 78	eqeltri 2511	. . . . . . . 8 |- ( 2 ^ 3 ) e. RR
80		2nn 10475	. . . . . . . . . 10 |- 2 e. NN
81		nnexpcl 11874	. . . . . . . . . 10 |- ( ( 2 e. NN /\ 4 e. NN0 ) -> ( 2 ^ 4 ) e. NN )
82	80, 10, 81	mp2an 667	. . . . . . . . 9 |- ( 2 ^ 4 ) e. NN
83	82	nnrei 10327	. . . . . . . 8 |- ( 2 ^ 4 ) e. RR
84	56, 79, 83	ltletri 9498	. . . . . . 7 |- ( ( 5 < ( 2 ^ 3 ) /\ ( 2 ^ 3 ) <_ ( 2 ^ 4 ) ) -> 5 < ( 2 ^ 4 ) )
85	67, 77, 84	mp2an 667	. . . . . 6 |- 5 < ( 2 ^ 4 )
86		6re 10398	. . . . . . . 8 |- 6 e. RR
87	86, 83	remulcli 9396	. . . . . . 7 |- ( 6 x. ( 2 ^ 4 ) ) e. RR
88		6pos 10416	. . . . . . . 8 |- 0 < 6
89	82	nngt0i 10351	. . . . . . . 8 |- 0 < ( 2 ^ 4 )
90	86, 83, 88, 89	mulgt0ii 9503	. . . . . . 7 |- 0 < ( 6 x. ( 2 ^ 4 ) )
91	56, 83, 87, 90	ltdiv1ii 10258	. . . . . 6 |- ( 5 < ( 2 ^ 4 ) <-> ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) )
92	85, 91	mpbi 208	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) < ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
93		df-5 10379	. . . . . 6 |- 5 = ( 4 + 1 )
94		df-4 10378	. . . . . . . . . . 11 |- 4 = ( 3 + 1 )
95	94	fveq2i 5691	. . . . . . . . . 10 |- ( ! ` 4 ) = ( ! ` ( 3 + 1 ) )
96		3nn0 10593	. . . . . . . . . . 11 |- 3 e. NN0
97		facp1 12052	. . . . . . . . . . 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 11958	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
100	99, 94	eqtr2i 2462	. . . . . . . . . . 11 |- ( 3 + 1 ) = ( 2 ^ 2 )
101	100	oveq2i 6101	. . . . . . . . . 10 |- ( ( ! ` 3 ) x. ( 3 + 1 ) ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
102	95, 98, 101	3eqtri 2465	. . . . . . . . 9 |- ( ! ` 4 ) = ( ( ! ` 3 ) x. ( 2 ^ 2 ) )
103	102	oveq1i 6100	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) )
104	99	oveq2i 6101	. . . . . . . 8 |- ( ( ! ` 4 ) x. ( 2 ^ 2 ) ) = ( ( ! ` 4 ) x. 4 )
105		fac3 12054	. . . . . . . . . 10 |- ( ! ` 3 ) = 6
106		6cn 10399	. . . . . . . . . 10 |- 6 e. CC
107	105, 106	eqeltri 2511	. . . . . . . . 9 |- ( ! ` 3 ) e. CC
108	17	recni 9394	. . . . . . . . . 10 |- 4 e. CC
109	99, 108	eqeltri 2511	. . . . . . . . 9 |- ( 2 ^ 2 ) e. CC
110	107, 109, 109	mulassi 9391	. . . . . . . 8 |- ( ( ( ! ` 3 ) x. ( 2 ^ 2 ) ) x. ( 2 ^ 2 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
111	103, 104, 110	3eqtr3i 2469	. . . . . . 7 |- ( ( ! ` 4 ) x. 4 ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
112		2p2e4 10435	. . . . . . . . . 10 |- ( 2 + 2 ) = 4
113	112	oveq2i 6101	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( 2 ^ 4 )
114		2cn 10388	. . . . . . . . . 10 |- 2 e. CC
115		2nn0 10592	. . . . . . . . . 10 |- 2 e. NN0
116		expadd 11902	. . . . . . . . . 10 |- ( ( 2 e. CC /\ 2 e. NN0 /\ 2 e. NN0 ) -> ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
117	114, 115, 115, 116	mp3an 1309	. . . . . . . . 9 |- ( 2 ^ ( 2 + 2 ) ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
118	113, 117	eqtr3i 2463	. . . . . . . 8 |- ( 2 ^ 4 ) = ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) )
119	118	oveq2i 6101	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( ( ! ` 3 ) x. ( ( 2 ^ 2 ) x. ( 2 ^ 2 ) ) )
120	105	oveq1i 6100	. . . . . . 7 |- ( ( ! ` 3 ) x. ( 2 ^ 4 ) ) = ( 6 x. ( 2 ^ 4 ) )
121	111, 119, 120	3eqtr2ri 2468	. . . . . 6 |- ( 6 x. ( 2 ^ 4 ) ) = ( ( ! ` 4 ) x. 4 )
122	93, 121	oveq12i 6102	. . . . 5 |- ( 5 / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) )
123	82	nncni 10328	. . . . . . . 8 |- ( 2 ^ 4 ) e. CC
124	123	mulid2i 9385	. . . . . . 7 |- ( 1 x. ( 2 ^ 4 ) ) = ( 2 ^ 4 )
125	124	oveq1i 6100	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) )
126	82	nnne0i 10352	. . . . . . . . 9 |- ( 2 ^ 4 ) =/= 0
127	123, 126	dividi 10060	. . . . . . . 8 |- ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) = 1
128	127	oveq2i 6101	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 / 6 ) x. 1 )
129		ax-1cn 9336	. . . . . . . 8 |- 1 e. CC
130	86, 88	gt0ne0ii 9872	. . . . . . . 8 |- 6 =/= 0
131	129, 106, 123, 123, 130, 126	divmuldivi 10087	. . . . . . 7 |- ( ( 1 / 6 ) x. ( ( 2 ^ 4 ) / ( 2 ^ 4 ) ) ) = ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) )
132	86, 130	rereccli 10092	. . . . . . . . 9 |- ( 1 / 6 ) e. RR
133	132	recni 9394	. . . . . . . 8 |- ( 1 / 6 ) e. CC
134	133	mulid1i 9384	. . . . . . 7 |- ( ( 1 / 6 ) x. 1 ) = ( 1 / 6 )
135	128, 131, 134	3eqtr3i 2469	. . . . . 6 |- ( ( 1 x. ( 2 ^ 4 ) ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
136	125, 135	eqtr3i 2463	. . . . 5 |- ( ( 2 ^ 4 ) / ( 6 x. ( 2 ^ 4 ) ) ) = ( 1 / 6 )
137	92, 122, 136	3brtr3i 4316	. . . 4 |- ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) < ( 1 / 6 )
138		rpexpcl 11880	. . . . . 6 |- ( ( A e. RR+ /\ 4 e. ZZ ) -> ( A ^ 4 ) e. RR+ )
139	38, 70, 138	sylancl 657	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. RR+ )
140		elrp 10989	. . . . . 6 |- ( ( A ^ 4 ) e. RR+ <-> ( ( A ^ 4 ) e. RR /\ 0 < ( A ^ 4 ) ) )
141		ltmul2 10176	. . . . . . 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 1299	. . . . . 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 195	. . . . 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 16	. . . 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 211	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
146	16	recnd 9408	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 4 ) e. CC )
147		divrec 10006	. . . . 5 |- ( ( ( A ^ 4 ) e. CC /\ 6 e. CC /\ 6 =/= 0 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
148	106, 130, 147	mp3an23 1301	. . . 4 |- ( ( A ^ 4 ) e. CC -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
149	146, 148	syl 16	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) / 6 ) = ( ( A ^ 4 ) x. ( 1 / 6 ) ) )
150	145, 149	breqtrrd 4315	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 4 ) x. ( ( 4 + 1 ) / ( ( ! ` 4 ) x. 4 ) ) ) < ( ( A ^ 4 ) / 6 ) )
151	14, 26, 29, 52, 150	lelttrd 9525	1 |- ( A e. ( 0 (,] 1 ) -> ( abs ` sum_ k e. ( ZZ>= ` 4 ) ( F ` k ) ) < ( ( A ^ 4 ) / 6 ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   class class class wbr 4289   e. cmpt 4347   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280   + caddc 9281   x. cmul 9283   RR*cxr 9413   < clt 9414   <_ cle 9415   / cdiv 9989   NNcn 10318   2c2 10367   3c3 10368   4c4 10369   5c5 10370   6c6 10371   8c8 10373   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   RR+crp 10987   (,]cioc 11297   ^cexp 11861   !cfa 12047   abscabs 12719   sum_csu 13159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-ioc 11301  df-ico 11302  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-fac 12048  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160

This theorem is referenced by:  sin01bnd  13465  cos01bnd  13466