Theorem sincnlem 10015

Description: Lemma for sincn 10018 and coscn 10019.

Hypotheses

Ref	Expression
sinco.1	|- F = {<.x, y>. | (x e. CC /\ y = (_i x. x))}
sinco.2	|- G = {<.x, y>. | (x e. CC /\ y = (-u_i x. x))}
sincolem.3	|- J = {<.x, y>. | (x e. CC /\ y = (x / A))}
sincolem.4	|- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w)O((exp o. G)` w)))}
sincnlem.5	|- A e. CC
sincnlem.6	|- A =/= 0
sincnlem.7	|- C = (abs o. - )
sincnlem.8	|- D = {<.<.p, q>., r>. | ((p e. (CC X. CC) /\ q e. (CC X. CC)) /\ r = sup({((1st` p)C(1st` q)), ((2nd` p)C(2nd` q))}, RR, < ))}
sincnlem.9	|- O e. ((Open` D) Cn (Open` C))

Assertion

Ref	Expression
sincnlem	|- (J o. H) e. (CC-cn->CC)

Distinct variable groups:   F,p,q,r,v,w   G,p,q,r,v,w   v,O,w,x,y   x,A,y   C,p,q,r,w

Proof of Theorem sincnlem

Step	Hyp	Ref	Expression
1		axicn 6423	. . . . . . 7 |- _i e. CC
2		sinco.1	. . . . . . . 8 |- F = {<.x, y>. | (x e. CC /\ y = (_i x. x))}
3	2	mulc1cncf 8541	. . . . . . 7 |- (_i e. CC -> F e. (CC-cn->CC))
4	1, 3	ax-mp 7	. . . . . 6 |- F e. (CC-cn->CC)
5		sincnlem.7	. . . . . . 7 |- C = (abs o. - )
6		eqid 1884	. . . . . . 7 |- (Open` C) = (Open` C)
7	5, 6	cncfmet1 9184	. . . . . 6 |- (CC-cn->CC) = ((Open` C) Cn (Open` C))
8	4, 7	eleqtri 1969	. . . . 5 |- F e. ((Open` C) Cn (Open` C))
9		efcn 8688	. . . . . 6 |- exp e. (CC-cn->CC)
10	9, 7	eleqtri 1969	. . . . 5 |- exp e. ((Open` C) Cn (Open` C))
11	5	cnmet 9182	. . . . . . 7 |- C e. Met
12	11, 11, 11	3pm3.2i 1048	. . . . . 6 |- (C e. Met /\ C e. Met /\ C e. Met)
13	6, 6, 6	metcnco 9175	. . . . . 6 |- (((C e. Met /\ C e. Met /\ C e. Met) /\ (F e. ((Open` C) Cn (Open` C)) /\ exp e. ((Open` C) Cn (Open` C)))) -> (exp o. F) e. ((Open` C) Cn (Open` C)))
14	12, 13	mpan 759	. . . . 5 |- ((F e. ((Open` C) Cn (Open` C)) /\ exp e. ((Open` C) Cn (Open` C))) -> (exp o. F) e. ((Open` C) Cn (Open` C)))
15	8, 10, 14	mp2an 761	. . . 4 |- (exp o. F) e. ((Open` C) Cn (Open` C))
16	1	negcli 6526	. . . . . . 7 |- -u_i e. CC
17		sinco.2	. . . . . . . 8 |- G = {<.x, y>. | (x e. CC /\ y = (-u_i x. x))}
18	17	mulc1cncf 8541	. . . . . . 7 |- (-u_i e. CC -> G e. (CC-cn->CC))
19	16, 18	ax-mp 7	. . . . . 6 |- G e. (CC-cn->CC)
20	19, 7	eleqtri 1969	. . . . 5 |- G e. ((Open` C) Cn (Open` C))
21	6, 6, 6	metcnco 9175	. . . . . 6 |- (((C e. Met /\ C e. Met /\ C e. Met) /\ (G e. ((Open` C) Cn (Open` C)) /\ exp e. ((Open` C) Cn (Open` C)))) -> (exp o. G) e. ((Open` C) Cn (Open` C)))
22	12, 21	mpan 759	. . . . 5 |- ((G e. ((Open` C) Cn (Open` C)) /\ exp e. ((Open` C) Cn (Open` C))) -> (exp o. G) e. ((Open` C) Cn (Open` C)))
23	20, 10, 22	mp2an 761	. . . 4 |- (exp o. G) e. ((Open` C) Cn (Open` C))
24	5	cnmetba 9181	. . . . 5 |- CC = dom dom C
25		eqid 1884	. . . . 5 |- (Open` D) = (Open` D)
26		sincnlem.8	. . . . 5 |- D = {<.<.p, q>., r>. | ((p e. (CC X. CC) /\ q e. (CC X. CC)) /\ r = sup({((1st` p)C(1st` q)), ((2nd` p)C(2nd` q))}, RR, < ))}
27		sincnlem.9	. . . . 5 |- O e. ((Open` D) Cn (Open` C))
28		sincolem.4	. . . . 5 |- H = {<.w, v>. | (w e. CC /\ v = (((exp o. F)` w)O((exp o. G)` w)))}
29	24, 24, 24, 11, 11, 11, 11, 6, 6, 6, 25, 6, 26, 27, 28	oprcn 9255	. . . 4 |- (((exp o. F) e. ((Open` C) Cn (Open` C)) /\ (exp o. G) e. ((Open` C) Cn (Open` C))) -> H e. ((Open` C) Cn (Open` C)))
30	15, 23, 29	mp2an 761	. . 3 |- H e. ((Open` C) Cn (Open` C))
31		sincnlem.5	. . . . 5 |- A e. CC
32		sincnlem.6	. . . . 5 |- A =/= 0
33		sincolem.3	. . . . . 6 |- J = {<.x, y>. | (x e. CC /\ y = (x / A))}
34	33	divccncf 8542	. . . . 5 |- ((A e. CC /\ A =/= 0) -> J e. (CC-cn->CC))
35	31, 32, 34	mp2an 761	. . . 4 |- J e. (CC-cn->CC)
36	35, 7	eleqtri 1969	. . 3 |- J e. ((Open` C) Cn (Open` C))
37	6, 6, 6	metcnco 9175	. . . 4 |- (((C e. Met /\ C e. Met /\ C e. Met) /\ (H e. ((Open` C) Cn (Open` C)) /\ J e. ((Open` C) Cn (Open` C)))) -> (J o. H) e. ((Open` C) Cn (Open` C)))
38	12, 37	mpan 759	. . 3 |- ((H e. ((Open` C) Cn (Open` C)) /\ J e. ((Open` C) Cn (Open` C))) -> (J o. H) e. ((Open` C) Cn (Open` C)))
39	30, 36, 38	mp2an 761	. 2 |- (J o. H) e. ((Open` C) Cn (Open` C))
40	39, 7	eleqtrri 1970	1 |- (J o. H) e. (CC-cn->CC)

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  {cpr 3045  {copab 3395   X. cxp 3984   o. ccom 3990  ` cfv 3998  (class class class)co 4884  {copab2 4885  1stc1st 5018  2ndc2nd 5019  supcsup 5663  CCcc 6384  RRcr 6385  0cc0 6386  _ici 6388   x. cmul 6391   - cmin 6445  -ucneg 6446   / cdiv 6447   < clt 6653  abscabs 8000  -cn->ccncf 8524  expce 8555   Cn ccn 9028  Metcme 9066  Opencopn 9069

This theorem is referenced by:  sincn 10018  coscn 10019

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-map 5383  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-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  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-top 8861  df-cn 9030  df-cnp 9031  df-met 9070  df-bl 9072  df-opn 9073

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