Theorem sin01bndlem3 8735

Description: Lemma for sin01bnd 8738.

Hypothesis

Ref	Expression
sin01bndlem2.1	|- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}

Assertion

Ref	Expression
sin01bndlem3	|- (A e. (0(,]1) -> ((A - ((A^3) / 3)) < (sin` A) /\ (sin` A) < A))

Distinct variable group:   A,j,y

Proof of Theorem sin01bndlem3

Step	Hyp	Ref	Expression
1		0re 6603	. . . . . . . 8 |- 0 e. RR
2		1re 6598	. . . . . . . 8 |- 1 e. RR
3		elioc2 7558	. . . . . . . 8 |- ((0 e. RR /\ 1 e. RR) -> (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1)))
4	1, 2, 3	mp2an 761	. . . . . . 7 |- (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1))
5	4	simp1bi 891	. . . . . 6 |- (A e. (0(,]1) -> A e. RR)
6	5	recnd 6468	. . . . 5 |- (A e. (0(,]1) -> A e. CC)
7		reexpcl 7823	. . . . . . . 8 |- ((A e. RR /\ 3 e. NN0) -> (A^3) e. RR)
8		6re 7168	. . . . . . . . 9 |- 6 e. RR
9		6pos 7178	. . . . . . . . . 10 |- 0 < 6
10	8, 9	gt0ne0ii 6799	. . . . . . . . 9 |- 6 =/= 0
11		redivcl 6978	. . . . . . . . 9 |- (((A^3) e. RR /\ 6 e. RR /\ 6 =/= 0) -> ((A^3) / 6) e. RR)
12	8, 10, 11	mp3an23 1183	. . . . . . . 8 |- ((A^3) e. RR -> ((A^3) / 6) e. RR)
13	7, 12	syl 12	. . . . . . 7 |- ((A e. RR /\ 3 e. NN0) -> ((A^3) / 6) e. RR)
14		3nn 7184	. . . . . . . 8 |- 3 e. NN
15	14	nnnn0i 7316	. . . . . . 7 |- 3 e. NN0
16	13, 5, 15	sylancl 525	. . . . . 6 |- (A e. (0(,]1) -> ((A^3) / 6) e. RR)
17	16	recnd 6468	. . . . 5 |- (A e. (0(,]1) -> ((A^3) / 6) e. CC)
18		renegcl 6600	. . . . . . 7 |- (((A^3) / 6) e. RR -> -u((A^3) / 6) e. RR)
19	16, 18	syl 12	. . . . . 6 |- (A e. (0(,]1) -> -u((A^3) / 6) e. RR)
20	19	recnd 6468	. . . . 5 |- (A e. (0(,]1) -> -u((A^3) / 6) e. CC)
21		subsub 6627	. . . . 5 |- ((A e. CC /\ ((A^3) / 6) e. CC /\ -u((A^3) / 6) e. CC) -> (A - (((A^3) / 6) - -u((A^3) / 6))) = ((A - ((A^3) / 6)) + -u((A^3) / 6)))
22	6, 17, 20, 21	syl111anc 1100	. . . 4 |- (A e. (0(,]1) -> (A - (((A^3) / 6) - -u((A^3) / 6))) = ((A - ((A^3) / 6)) + -u((A^3) / 6)))
23		subneg 6554	. . . . . . 7 |- ((((A^3) / 6) e. CC /\ ((A^3) / 6) e. CC) -> (((A^3) / 6) - -u((A^3) / 6)) = (((A^3) / 6) + ((A^3) / 6)))
24	17, 17, 23	syl11anc 524	. . . . . 6 |- (A e. (0(,]1) -> (((A^3) / 6) - -u((A^3) / 6)) = (((A^3) / 6) + ((A^3) / 6)))
25		2times 7188	. . . . . . 7 |- (((A^3) / 6) e. CC -> (2 x. ((A^3) / 6)) = (((A^3) / 6) + ((A^3) / 6)))
26	17, 25	syl 12	. . . . . 6 |- (A e. (0(,]1) -> (2 x. ((A^3) / 6)) = (((A^3) / 6) + ((A^3) / 6)))
27		2cn 7164	. . . . . . . . . . 11 |- 2 e. CC
28	27	mulid1i 6485	. . . . . . . . . 10 |- (2 x. 1) = 2
29		3re 7165	. . . . . . . . . . . . 13 |- 3 e. RR
30	29	recni 6467	. . . . . . . . . . . 12 |- 3 e. CC
31	30, 27	mulcomi 6476	. . . . . . . . . . 11 |- (3 x. 2) = (2 x. 3)
32		3t2e6 7207	. . . . . . . . . . 11 |- (3 x. 2) = 6
33	31, 32	eqtr3i 1910	. . . . . . . . . 10 |- (2 x. 3) = 6
34	28, 33	opreq12i 4894	. . . . . . . . 9 |- ((2 x. 1) / (2 x. 3)) = (2 / 6)
35		2ne0 7174	. . . . . . . . . 10 |- 2 =/= 0
36		3pos 7175	. . . . . . . . . . . 12 |- 0 < 3
37	29, 36	gt0ne0ii 6799	. . . . . . . . . . 11 |- 3 =/= 0
38		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
39		divcan5 6957	. . . . . . . . . . . 12 |- ((1 e. CC /\ (3 e. CC /\ 3 =/= 0) /\ (2 e. CC /\ 2 =/= 0)) -> ((2 x. 1) / (2 x. 3)) = (1 / 3))
40	38, 39	mp3an1 1178	. . . . . . . . . . 11 |- (((3 e. CC /\ 3 =/= 0) /\ (2 e. CC /\ 2 =/= 0)) -> ((2 x. 1) / (2 x. 3)) = (1 / 3))
41	30, 37, 40	mpanl12 773	. . . . . . . . . 10 |- ((2 e. CC /\ 2 =/= 0) -> ((2 x. 1) / (2 x. 3)) = (1 / 3))
42	27, 35, 41	mp2an 761	. . . . . . . . 9 |- ((2 x. 1) / (2 x. 3)) = (1 / 3)
43	34, 42	eqtr3i 1910	. . . . . . . 8 |- (2 / 6) = (1 / 3)
44	43	opreq2i 4893	. . . . . . 7 |- ((A^3) x. (2 / 6)) = ((A^3) x. (1 / 3))
45	7, 5, 15	sylancl 525	. . . . . . . . 9 |- (A e. (0(,]1) -> (A^3) e. RR)
46	45	recnd 6468	. . . . . . . 8 |- (A e. (0(,]1) -> (A^3) e. CC)
47	8	recni 6467	. . . . . . . . . 10 |- 6 e. CC
48	47, 10	pm3.2i 307	. . . . . . . . 9 |- (6 e. CC /\ 6 =/= 0)
49		div12 6927	. . . . . . . . 9 |- ((2 e. CC /\ (A^3) e. CC /\ (6 e. CC /\ 6 =/= 0)) -> (2 x. ((A^3) / 6)) = ((A^3) x. (2 / 6)))
50	27, 48, 49	mp3an13 1182	. . . . . . . 8 |- ((A^3) e. CC -> (2 x. ((A^3) / 6)) = ((A^3) x. (2 / 6)))
51	46, 50	syl 12	. . . . . . 7 |- (A e. (0(,]1) -> (2 x. ((A^3) / 6)) = ((A^3) x. (2 / 6)))
52		divrec 6922	. . . . . . . . 9 |- (((A^3) e. CC /\ 3 e. CC /\ 3 =/= 0) -> ((A^3) / 3) = ((A^3) x. (1 / 3)))
53	30, 37, 52	mp3an23 1183	. . . . . . . 8 |- ((A^3) e. CC -> ((A^3) / 3) = ((A^3) x. (1 / 3)))
54	46, 53	syl 12	. . . . . . 7 |- (A e. (0(,]1) -> ((A^3) / 3) = ((A^3) x. (1 / 3)))
55	44, 51, 54	3eqtr4a 1954	. . . . . 6 |- (A e. (0(,]1) -> (2 x. ((A^3) / 6)) = ((A^3) / 3))
56	24, 26, 55	3eqtr2d 1932	. . . . 5 |- (A e. (0(,]1) -> (((A^3) / 6) - -u((A^3) / 6)) = ((A^3) / 3))
57	56	opreq2d 4898	. . . 4 |- (A e. (0(,]1) -> (A - (((A^3) / 6) - -u((A^3) / 6))) = (A - ((A^3) / 3)))
58	22, 57	eqtr3d 1927	. . 3 |- (A e. (0(,]1) -> ((A - ((A^3) / 6)) + -u((A^3) / 6)) = (A - ((A^3) / 3)))
59		sin01bndlem2.1	. . . . . . . 8 |- F = {<.j, y>. | (j e. NN0 /\ y = (((_i x. A)^j) / (!` j)))}
60	59	sin01bndlem2 8734	. . . . . . 7 |- (A e. (0(,]1) -> (abs` (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A^3) / 6))
61	59	eftlcl 8641	. . . . . . . . . 10 |- (((_i x. A) e. CC /\ 4 e. NN) -> sum_k e. (ZZ>=` 4)(F` k) e. CC)
62		mulcl 6456	. . . . . . . . . . 11 |- ((_i e. CC /\ A e. CC) -> (_i x. A) e. CC)
63		axicn 6423	. . . . . . . . . . 11 |- _i e. CC
64	62, 63, 6	sylancr 526	. . . . . . . . . 10 |- (A e. (0(,]1) -> (_i x. A) e. CC)
65		4nn 7186	. . . . . . . . . 10 |- 4 e. NN
66	61, 64, 65	sylancl 525	. . . . . . . . 9 |- (A e. (0(,]1) -> sum_k e. (ZZ>=` 4)(F` k) e. CC)
67		imcl 8008	. . . . . . . . 9 |- (sum_k e. (ZZ>=` 4)(F` k) e. CC -> (Im` sum_k e. (ZZ>=` 4)(F` k)) e. RR)
68	66, 67	syl 12	. . . . . . . 8 |- (A e. (0(,]1) -> (Im` sum_k e. (ZZ>=` 4)(F` k)) e. RR)
69		abslt 8132	. . . . . . . 8 |- (((Im` sum_k e. (ZZ>=` 4)(F` k)) e. RR /\ ((A^3) / 6) e. RR) -> ((abs` (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A^3) / 6) <-> (-u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)) /\ (Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6))))
70	68, 16, 69	syl11anc 524	. . . . . . 7 |- (A e. (0(,]1) -> ((abs` (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A^3) / 6) <-> (-u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)) /\ (Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6))))
71	60, 70	mpbid 212	. . . . . 6 |- (A e. (0(,]1) -> (-u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)) /\ (Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6)))
72	71	simplld 348	. . . . 5 |- (A e. (0(,]1) -> -u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)))
73		resubcl 6601	. . . . . . 7 |- ((A e. RR /\ ((A^3) / 6) e. RR) -> (A - ((A^3) / 6)) e. RR)
74	5, 16, 73	syl11anc 524	. . . . . 6 |- (A e. (0(,]1) -> (A - ((A^3) / 6)) e. RR)
75		ltadd2 6807	. . . . . 6 |- ((-u((A^3) / 6) e. RR /\ (Im` sum_k e. (ZZ>=` 4)(F` k)) e. RR /\ (A - ((A^3) / 6)) e. RR) -> (-u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)) <-> ((A - ((A^3) / 6)) + -u((A^3) / 6)) < ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k)))))
76	19, 68, 74, 75	syl111anc 1100	. . . . 5 |- (A e. (0(,]1) -> (-u((A^3) / 6) < (Im` sum_k e. (ZZ>=` 4)(F` k)) <-> ((A - ((A^3) / 6)) + -u((A^3) / 6)) < ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k)))))
77	72, 76	mpbid 212	. . . 4 |- (A e. (0(,]1) -> ((A - ((A^3) / 6)) + -u((A^3) / 6)) < ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))))
78	59	resin4p 8701	. . . . 5 |- (A e. RR -> (sin` A) = ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))))
79	5, 78	syl 12	. . . 4 |- (A e. (0(,]1) -> (sin` A) = ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))))
80	77, 79	breqtrrd 3363	. . 3 |- (A e. (0(,]1) -> ((A - ((A^3) / 6)) + -u((A^3) / 6)) < (sin` A))
81	58, 80	eqbrtrrd 3359	. 2 |- (A e. (0(,]1) -> (A - ((A^3) / 3)) < (sin` A))
82	71	simprd 352	. . . . 5 |- (A e. (0(,]1) -> (Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6))
83		ltadd2 6807	. . . . . 6 |- (((Im` sum_k e. (ZZ>=` 4)(F` k)) e. RR /\ ((A^3) / 6) e. RR /\ (A - ((A^3) / 6)) e. RR) -> ((Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6) <-> ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A - ((A^3) / 6)) + ((A^3) / 6))))
84	68, 16, 74, 83	syl111anc 1100	. . . . 5 |- (A e. (0(,]1) -> ((Im` sum_k e. (ZZ>=` 4)(F` k)) < ((A^3) / 6) <-> ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A - ((A^3) / 6)) + ((A^3) / 6))))
85	82, 84	mpbid 212	. . . 4 |- (A e. (0(,]1) -> ((A - ((A^3) / 6)) + (Im` sum_k e. (ZZ>=` 4)(F` k))) < ((A - ((A^3) / 6)) + ((A^3) / 6)))
86	79, 85	eqbrtrd 3357	. . 3 |- (A e. (0(,]1) -> (sin` A) < ((A - ((A^3) / 6)) + ((A^3) / 6)))
87		npcan 6559	. . . 4 |- ((A e. CC /\ ((A^3) / 6) e. CC) -> ((A - ((A^3) / 6)) + ((A^3) / 6)) = A)
88	6, 17, 87	syl11anc 524	. . 3 |- (A e. (0(,]1) -> ((A - ((A^3) / 6)) + ((A^3) / 6)) = A)
89	86, 88	breqtrd 3361	. 2 |- (A e. (0(,]1) -> (sin` A) < A)
90	81, 89	jca 310	1 |- (A e. (0(,]1) -> ((A - ((A^3) / 3)) < (sin` A) /\ (sin` A) < A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  {copab 3395  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   <_ cle 6448  NNcn 6449  NN0cn0 6450   < clt 6653  2c2 7145  3c3 7146  4c4 7147  6c6 7149  (,]cioc 7525  ZZ>=cuz 7586  ^cexp 7811  Imcim 7998  abscabs 8000  !cfa 8183  sum_csu 8239  sincsin 8557

This theorem is referenced by:  sin01bnd 8738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-n0 7309  df-z 7345  df-fl 7463  df-ioc 7529  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-clim 8235  df-sum 8240  df-ef 8560  df-sin 8562

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