Theorem pilem3 10022

Description: Lemma for pire 10026, pigt2lt4 10024 and sinpi 10025.

Hypotheses

Ref	Expression
pilem1.1	|- S = {x e. (-u4[,)0) | (sin` x) = 0}
pilem1.2	|- C = sup(S, RR, < )
pilem3.3	|- T = {x e. RR | (x < 0 /\ (sin` x) = 0)}
pilem3.4	|- P = {x e. RR | (0 < x /\ (sin` x) = 0)}

Assertion

Ref	Expression
pilem3	|- (pi e. RR+ /\ pi e. (2(,)4) /\ (sin` pi) = 0)

Distinct variable groups:   x,C   x,P

Proof of Theorem pilem3

Step	Hyp	Ref	Expression
1		pilem3.4	. . . . . . 7 |- P = {x e. RR | (0 < x /\ (sin` x) = 0)}
2		ssrab2 2692	. . . . . . 7 |- {x e. RR | (0 < x /\ (sin` x) = 0)} C_ RR
3	1, 2	eqsstri 2647	. . . . . 6 |- P C_ RR
4		pilem1.1	. . . . . . . . . . . . . 14 |- S = {x e. (-u4[,)0) | (sin` x) = 0}
5		pilem1.2	. . . . . . . . . . . . . 14 |- C = sup(S, RR, < )
6		eqid 1884	. . . . . . . . . . . . . 14 |- {x e. (-u4[,]-u2) | (sin` x) = 0} = {x e. (-u4[,]-u2) | (sin` x) = 0}
7	4, 5, 6	pilem1 10020	. . . . . . . . . . . . 13 |- (C e. (-u4(,)-u2) /\ (sin` C) = 0)
8	7	simpli 347	. . . . . . . . . . . 12 |- C e. (-u4(,)-u2)
9		4re 7166	. . . . . . . . . . . . . 14 |- 4 e. RR
10	9	renegcli 6576	. . . . . . . . . . . . 13 |- -u4 e. RR
11		2re 7163	. . . . . . . . . . . . . 14 |- 2 e. RR
12	11	renegcli 6576	. . . . . . . . . . . . 13 |- -u2 e. RR
13		elioo2 7546	. . . . . . . . . . . . . 14 |- ((-u4 e. RR* /\ -u2 e. RR*) -> (C e. (-u4(,)-u2) <-> (C e. RR /\ -u4 < C /\ C < -u2)))
14		rexr 6668	. . . . . . . . . . . . . 14 |- (-u4 e. RR -> -u4 e. RR*)
15		rexr 6668	. . . . . . . . . . . . . 14 |- (-u2 e. RR -> -u2 e. RR*)
16	13, 14, 15	syl2an 503	. . . . . . . . . . . . 13 |- ((-u4 e. RR /\ -u2 e. RR) -> (C e. (-u4(,)-u2) <-> (C e. RR /\ -u4 < C /\ C < -u2)))
17	10, 12, 16	mp2an 761	. . . . . . . . . . . 12 |- (C e. (-u4(,)-u2) <-> (C e. RR /\ -u4 < C /\ C < -u2))
18	8, 17	mpbi 206	. . . . . . . . . . 11 |- (C e. RR /\ -u4 < C /\ C < -u2)
19	18	simp3i 887	. . . . . . . . . 10 |- C < -u2
20		2pos 7173	. . . . . . . . . . 11 |- 0 < 2
21		lt0neg2 6858	. . . . . . . . . . . 12 |- (2 e. RR -> (0 < 2 <-> -u2 < 0))
22	11, 21	ax-mp 7	. . . . . . . . . . 11 |- (0 < 2 <-> -u2 < 0)
23	20, 22	mpbi 206	. . . . . . . . . 10 |- -u2 < 0
24	18	simp1i 885	. . . . . . . . . . 11 |- C e. RR
25		0re 6603	. . . . . . . . . . 11 |- 0 e. RR
26	24, 12, 25	lttri 6760	. . . . . . . . . 10 |- ((C < -u2 /\ -u2 < 0) -> C < 0)
27	19, 23, 26	mp2an 761	. . . . . . . . 9 |- C < 0
28		lt0neg1 6857	. . . . . . . . . 10 |- (C e. RR -> (C < 0 <-> 0 < -uC))
29	24, 28	ax-mp 7	. . . . . . . . 9 |- (C < 0 <-> 0 < -uC)
30	27, 29	mpbi 206	. . . . . . . 8 |- 0 < -uC
31	24	recni 6467	. . . . . . . . . 10 |- C e. CC
32		sinneg 8707	. . . . . . . . . 10 |- (C e. CC -> (sin` -uC) = -u(sin` C))
33	31, 32	ax-mp 7	. . . . . . . . 9 |- (sin` -uC) = -u(sin` C)
34	7	simpri 351	. . . . . . . . . 10 |- (sin` C) = 0
35	34	negeqi 6515	. . . . . . . . 9 |- -u(sin` C) = -u0
36		neg0 6575	. . . . . . . . 9 |- -u0 = 0
37	33, 35, 36	3eqtri 1912	. . . . . . . 8 |- (sin` -uC) = 0
38	24	renegcli 6576	. . . . . . . . 9 |- -uC e. RR
39		breq2 3342	. . . . . . . . . . . 12 |- (x = -uC -> (0 < x <-> 0 < -uC))
40		fveq2 4681	. . . . . . . . . . . . 13 |- (x = -uC -> (sin` x) = (sin` -uC))
41	40	eqeq1d 1892	. . . . . . . . . . . 12 |- (x = -uC -> ((sin` x) = 0 <-> (sin` -uC) = 0))
42	39, 41	anbi12d 690	. . . . . . . . . . 11 |- (x = -uC -> ((0 < x /\ (sin` x) = 0) <-> (0 < -uC /\ (sin` -uC) = 0)))
43	42, 1	elrab2 2416	. . . . . . . . . 10 |- (-uC e. P <-> (-uC e. RR /\ (0 < -uC /\ (sin` -uC) = 0)))
44	43	biimpri 169	. . . . . . . . 9 |- ((-uC e. RR /\ (0 < -uC /\ (sin` -uC) = 0)) -> -uC e. P)
45	38, 44	mpan 759	. . . . . . . 8 |- ((0 < -uC /\ (sin` -uC) = 0) -> -uC e. P)
46	30, 37, 45	mp2an 761	. . . . . . 7 |- -uC e. P
47		ne0i 2881	. . . . . . 7 |- (-uC e. P -> P =/= (/))
48	46, 47	ax-mp 7	. . . . . 6 |- P =/= (/)
49		breq2 3342	. . . . . . . . . . . 12 |- (x = w -> (0 < x <-> 0 < w))
50		fveq2 4681	. . . . . . . . . . . . 13 |- (x = w -> (sin` x) = (sin` w))
51	50	eqeq1d 1892	. . . . . . . . . . . 12 |- (x = w -> ((sin` x) = 0 <-> (sin` w) = 0))
52	49, 51	anbi12d 690	. . . . . . . . . . 11 |- (x = w -> ((0 < x /\ (sin` x) = 0) <-> (0 < w /\ (sin` w) = 0)))
53	52, 1	elrab2 2416	. . . . . . . . . 10 |- (w e. P <-> (w e. RR /\ (0 < w /\ (sin` w) = 0)))
54	53	simplbi 349	. . . . . . . . 9 |- (w e. P -> w e. RR)
55		3anass 862	. . . . . . . . . . 11 |- ((w e. RR /\ 0 < w /\ (sin` w) = 0) <-> (w e. RR /\ (0 < w /\ (sin` w) = 0)))
56	53, 55	bitr4i 193	. . . . . . . . . 10 |- (w e. P <-> (w e. RR /\ 0 < w /\ (sin` w) = 0))
57	56	simp2bi 892	. . . . . . . . 9 |- (w e. P -> 0 < w)
58		ltle 6690	. . . . . . . . . 10 |- ((0 e. RR /\ w e. RR) -> (0 < w -> 0 <_ w))
59	25, 58	mpan 759	. . . . . . . . 9 |- (w e. RR -> (0 < w -> 0 <_ w))
60	54, 57, 59	sylc 83	. . . . . . . 8 |- (w e. P -> 0 <_ w)
61	60	rgen 2159	. . . . . . 7 |- A.w e. P 0 <_ w
62		breq1 3341	. . . . . . . . 9 |- (z = 0 -> (z <_ w <-> 0 <_ w))
63	62	ralbidv 2123	. . . . . . . 8 |- (z = 0 -> (A.w e. P z <_ w <-> A.w e. P 0 <_ w))
64	63	rcla4ev 2381	. . . . . . 7 |- ((0 e. RR /\ A.w e. P 0 <_ w) -> E.z e. RR A.w e. P z <_ w)
65	25, 61, 64	mp2an 761	. . . . . 6 |- E.z e. RR A.w e. P z <_ w
66		infmsup 7277	. . . . . . 7 |- ((P C_ RR /\ P =/= (/) /\ E.z e. RR A.w e. P z <_ w) -> sup(P, RR, `' < ) = -usup({v e. RR | -uv e. P}, RR, < ))
67		df-pi 8564	. . . . . . . 8 |- pi = sup({x e. RR | (0 < x /\ (sin` x) = 0)}, RR, `' < )
68		supeq1 5665	. . . . . . . . 9 |- (P = {x e. RR | (0 < x /\ (sin` x) = 0)} -> sup(P, RR, `' < ) = sup({x e. RR | (0 < x /\ (sin` x) = 0)}, RR, `' < ))
69	1, 68	ax-mp 7	. . . . . . . 8 |- sup(P, RR, `' < ) = sup({x e. RR | (0 < x /\ (sin` x) = 0)}, RR, `' < )
70	67, 69	eqtr4i 1911	. . . . . . 7 |- pi = sup(P, RR, `' < )
71	66, 70	syl5eq 1940	. . . . . 6 |- ((P C_ RR /\ P =/= (/) /\ E.z e. RR A.w e. P z <_ w) -> pi = -usup({v e. RR | -uv e. P}, RR, < ))
72	3, 48, 65, 71	mp3an 1191	. . . . 5 |- pi = -usup({v e. RR | -uv e. P}, RR, < )
73		pilem3.3	. . . . . . . 8 |- T = {x e. RR | (x < 0 /\ (sin` x) = 0)}
74	4, 5, 73	pilem2 10021	. . . . . . 7 |- C = sup(T, RR, < )
75		breq1 3341	. . . . . . . . . . 11 |- (x = v -> (x < 0 <-> v < 0))
76		fveq2 4681	. . . . . . . . . . . 12 |- (x = v -> (sin` x) = (sin` v))
77	76	eqeq1d 1892	. . . . . . . . . . 11 |- (x = v -> ((sin` x) = 0 <-> (sin` v) = 0))
78	75, 77	anbi12d 690	. . . . . . . . . 10 |- (x = v -> ((x < 0 /\ (sin` x) = 0) <-> (v < 0 /\ (sin` v) = 0)))
79	78	cbvrabv 2422	. . . . . . . . 9 |- {x e. RR | (x < 0 /\ (sin` x) = 0)} = {v e. RR | (v < 0 /\ (sin` v) = 0)}
80		lt0neg1 6857	. . . . . . . . . . . . 13 |- (v e. RR -> (v < 0 <-> 0 < -uv))
81		resincl 8703	. . . . . . . . . . . . . . . 16 |- (v e. RR -> (sin` v) e. RR)
82	81	recnd 6468	. . . . . . . . . . . . . . 15 |- (v e. RR -> (sin` v) e. CC)
83		negeq0 6984	. . . . . . . . . . . . . . 15 |- ((sin` v) e. CC -> ((sin` v) = 0 <-> -u(sin` v) = 0))
84	82, 83	syl 12	. . . . . . . . . . . . . 14 |- (v e. RR -> ((sin` v) = 0 <-> -u(sin` v) = 0))
85		recn 6466	. . . . . . . . . . . . . . . 16 |- (v e. RR -> v e. CC)
86		sinneg 8707	. . . . . . . . . . . . . . . 16 |- (v e. CC -> (sin` -uv) = -u(sin` v))
87	85, 86	syl 12	. . . . . . . . . . . . . . 15 |- (v e. RR -> (sin` -uv) = -u(sin` v))
88	87	eqeq1d 1892	. . . . . . . . . . . . . 14 |- (v e. RR -> ((sin` -uv) = 0 <-> -u(sin` v) = 0))
89	84, 88	bitr4d 590	. . . . . . . . . . . . 13 |- (v e. RR -> ((sin` v) = 0 <-> (sin` -uv) = 0))
90	80, 89	anbi12d 690	. . . . . . . . . . . 12 |- (v e. RR -> ((v < 0 /\ (sin` v) = 0) <-> (0 < -uv /\ (sin` -uv) = 0)))
91		renegcl 6600	. . . . . . . . . . . . 13 |- (v e. RR -> -uv e. RR)
92	91	biantrurd 796	. . . . . . . . . . . 12 |- (v e. RR -> ((0 < -uv /\ (sin` -uv) = 0) <-> (-uv e. RR /\ (0 < -uv /\ (sin` -uv) = 0))))
93	90, 92	bitrd 587	. . . . . . . . . . 11 |- (v e. RR -> ((v < 0 /\ (sin` v) = 0) <-> (-uv e. RR /\ (0 < -uv /\ (sin` -uv) = 0))))
94		breq2 3342	. . . . . . . . . . . . 13 |- (x = -uv -> (0 < x <-> 0 < -uv))
95		fveq2 4681	. . . . . . . . . . . . . 14 |- (x = -uv -> (sin` x) = (sin` -uv))
96	95	eqeq1d 1892	. . . . . . . . . . . . 13 |- (x = -uv -> ((sin` x) = 0 <-> (sin` -uv) = 0))
97	94, 96	anbi12d 690	. . . . . . . . . . . 12 |- (x = -uv -> ((0 < x /\ (sin` x) = 0) <-> (0 < -uv /\ (sin` -uv) = 0)))
98	97, 1	elrab2 2416	. . . . . . . . . . 11 |- (-uv e. P <-> (-uv e. RR /\ (0 < -uv /\ (sin` -uv) = 0)))
99	93, 98	syl6bbr 597	. . . . . . . . . 10 |- (v e. RR -> ((v < 0 /\ (sin` v) = 0) <-> -uv e. P))
100	99	rabbiia 2285	. . . . . . . . 9 |- {v e. RR | (v < 0 /\ (sin` v) = 0)} = {v e. RR | -uv e. P}
101	73, 79, 100	3eqtri 1912	. . . . . . . 8 |- T = {v e. RR | -uv e. P}
102		supeq1 5665	. . . . . . . 8 |- (T = {v e. RR | -uv e. P} -> sup(T, RR, < ) = sup({v e. RR | -uv e. P}, RR, < ))
103	101, 102	ax-mp 7	. . . . . . 7 |- sup(T, RR, < ) = sup({v e. RR | -uv e. P}, RR, < )
104	74, 103	eqtri 1908	. . . . . 6 |- C = sup({v e. RR | -uv e. P}, RR, < )
105	104	negeqi 6515	. . . . 5 |- -uC = -usup({v e. RR | -uv e. P}, RR, < )
106	72, 105	eqtr4i 1911	. . . 4 |- pi = -uC
107	106, 38	eqeltri 1967	. . 3 |- pi e. RR
108	30, 106	breqtrri 3362	. . 3 |- 0 < pi
109	107, 108	elrpii 7234	. 2 |- pi e. RR+
110		iooneg 7575	. . . . . 6 |- ((2 e. RR /\ 4 e. RR /\ -uC e. RR) -> (-uC e. (2(,)4) <-> -u-uC e. (-u4(,)-u2)))
111	11, 9, 38, 110	mp3an 1191	. . . . 5 |- (-uC e. (2(,)4) <-> -u-uC e. (-u4(,)-u2))
112	31	negnegi 6549	. . . . . 6 |- -u-uC = C
113	112	eleq1i 1960	. . . . 5 |- (-u-uC e. (-u4(,)-u2) <-> C e. (-u4(,)-u2))
114	111, 113	bitr2i 191	. . . 4 |- (C e. (-u4(,)-u2) <-> -uC e. (2(,)4))
115	8, 114	mpbi 206	. . 3 |- -uC e. (2(,)4)
116	106, 115	eqeltri 1967	. 2 |- pi e. (2(,)4)
117	106	fveq2i 4684	. . 3 |- (sin` pi) = (sin` -uC)
118	117, 37	eqtri 1908	. 2 |- (sin` pi) = 0
119	109, 116, 118	3pm3.2i 1048	1 |- (pi e. RR+ /\ pi e. (2(,)4) /\ (sin` pi) = 0)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   C_ wss 2593  (/)c0 2875   class class class wbr 3338  `'ccnv 3985  ` cfv 3998  (class class class)co 4884  supcsup 5663  CCcc 6384  RRcr 6385  0cc0 6386  -ucneg 6446   <_ cle 6448  RR+crp 6453  RR*cxr 6652   < clt 6653  2c2 7145  4c4 7147  (,)cioo 7524  [,)cico 7526  [,]cicc 7527  sincsin 8557  picpi 8559

This theorem is referenced by:  pilem4 10023

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-reg 5695  ax-inf2 5731  ax-ac 5906

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-iin 3258  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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  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-9 7161  df-rp 7232  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-ioc 7529  df-ico 7530  df-icc 7531  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-bc 8209  df-clim 8235  df-sum 8240  df-cncf 8525  df-ef 8560  df-sin 8562  df-cos 8563  df-pi 8564  df-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200

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