Theorem sincnlem 8666

Description: Lemma for sincn 8669 and coscn 8670.

Hypotheses

Ref	Expression
sinco.1	|- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}
sinco.2	|- G = {<.x, y>. | (x e. CC /\ y = (-ui 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 5270	. . . . . . 7 |- i e. CC
2		sinco.1	. . . . . . . 8 |- F = {<.x, y>. | (x e. CC /\ y = (i x. x))}
3	2	mulc1cncf 7279	. . . . . . 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 1475	. . . . . . 7 |- (Open` C) = (Open` C)
7	5, 6	cncfmet1 7906	. . . . . 6 |- (CC-cn->CC) = ((Open` C) Cn (Open` C))
8	4, 7	eleqtr 1546	. . . . 5 |- F e. ((Open` C) Cn (Open` C))
9		efcn 7423	. . . . . 6 |- exp e. (CC-cn->CC)
10	9, 7	eleqtr 1546	. . . . 5 |- exp e. ((Open` C) Cn (Open` C))
11	5	cnmet 7904	. . . . . . 7 |- C e. Met
12	11, 11, 11	3pm3.2i 818	. . . . . 6 |- (C e. Met /\ C e. Met /\ C e. Met)
13	6, 6, 6	metcnco 7897	. . . . . 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 695	. . . . 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 697	. . . 4 |- (exp o. F) e. ((Open` C) Cn (Open` C))
16	1	negcl 5369	. . . . . . 7 |- -ui e. CC
17		sinco.2	. . . . . . . 8 |- G = {<.x, y>. | (x e. CC /\ y = (-ui x. x))}
18	17	mulc1cncf 7279	. . . . . . 7 |- (-ui e. CC -> G e. (CC-cn->CC))
19	16, 18	ax-mp 7	. . . . . 6 |- G e. (CC-cn->CC)
20	19, 7	eleqtr 1546	. . . . 5 |- G e. ((Open` C) Cn (Open` C))
21	6, 6, 6	metcnco 7897	. . . . . 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 695	. . . . 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 697	. . . 4 |- (exp o. G) e. ((Open` C) Cn (Open` C))
24	5	cnmetba 7903	. . . . 5 |- CC = dom dom C
25		eqid 1475	. . . . 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 7977	. . . 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 697	. . 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 7280	. . . . 5 |- ((A e. CC /\ A =/= 0) -> J e. (CC-cn->CC))
35	31, 32, 34	mp2an 697	. . . 4 |- J e. (CC-cn->CC)
36	35, 7	eleqtr 1546	. . 3 |- J e. ((Open` C) Cn (Open` C))
37	6, 6, 6	metcnco 7897	. . . 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 695	. . 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 697	. 2 |- (J o. H) e. ((Open` C) Cn (Open` C))
40	39, 7	eleqtrr 1547	1 |- (J o. H) e. (CC-cn->CC)

Syntax hints:   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  {cpr 2410  {copab 2666   X. cxp 3168   o. ccom 3174  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  supcsup 4573  CCcc 5232  RRcr 5233  0cc0 5234  ici 5236   x. cmul 5239   - cmin 5292  -ucneg 5293   / cdiv 5294   < clt 5486  abscabs 6750  -cn->ccncf 7262  expce 7293   Cn ccn 7752  Metcme 7789  Opencopn 7792

This theorem is referenced by:  sincn 8669  coscn 8670

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-map 4324  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-4 5972  df-n0 6100  df-z 6136  df-fl 6224  df-rp 6281  df-seq1 6308  df-shft 6341  df-uz 6418  df-fz 6468  df-seqz 6533  df-seq0 6534  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-fac 6932  df-bc 6957  df-clim 6975  df-sum 6980  df-cncf 7263  df-ef 7298  df-top 7592  df-cn 7754  df-cnp 7755  df-met 7793  df-bl 7795  df-opn 7796

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