Theorem sincnlem 8661

Description: Lemma for sincn 8664 and coscn 8665.

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 5282	. . . . . . 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 1478	. . . . . . 7 |- (Open` C) = (Open` C)
7	5, 6	cncfmet1 7903	. . . . . 6 |- (CC-cn->CC) = ((Open` C) Cn (Open` C))
8	4, 7	eleqtr 1549	. . . . 5 |- F e. ((Open` C) Cn (Open` C))
9		efcn 7423	. . . . . 6 |- exp e. (CC-cn->CC)
10	9, 7	eleqtr 1549	. . . . 5 |- exp e. ((Open` C) Cn (Open` C))
11	5	cnmet 7901	. . . . . . 7 |- C e. Met
12	11, 11, 11	3pm3.2i 820	. . . . . 6 |- (C e. Met /\ C e. Met /\ C e. Met)
13	6, 6, 6	metcnco 7894	. . . . . 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 697	. . . . 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 699	. . . 4 |- (exp o. F) e. ((Open` C) Cn (Open` C))
16	1	negcl 5381	. . . . . . 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 1549	. . . . 5 |- G e. ((Open` C) Cn (Open` C))
21	6, 6, 6	metcnco 7894	. . . . . 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 697	. . . . 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 699	. . . 4 |- (exp o. G) e. ((Open` C) Cn (Open` C))
24	5	cnmetba 7900	. . . . 5 |- CC = dom dom C
25		eqid 1478	. . . . 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 7974	. . . 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 699	. . 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 699	. . . 4 |- J e. (CC-cn->CC)
36	35, 7	eleqtr 1549	. . 3 |- J e. ((Open` C) Cn (Open` C))
37	6, 6, 6	metcnco 7894	. . . 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 697	. . 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 699	. 2 |- (J o. H) e. ((Open` C) Cn (Open` C))
40	39, 7	eleqtrr 1550	1 |- (J o. H) e. (CC-cn->CC)

Syntax hints:   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  {cpr 2414  {copab 2671   X. cxp 3174   o. ccom 3180  ` cfv 3188  (class class class)co 3969  {copab2 3970  1stc1st 4083  2ndc2nd 4084  supcsup 4582  CCcc 5244  RRcr 5245  0cc0 5246  ici 5248   x. cmul 5251   - cmin 5304  -ucneg 5305   / cdiv 5306   < clt 5498  abscabs 6751  -cn->ccncf 7262  expce 7293   Cn ccn 7749  Metcme 7786  Opencopn 7789

This theorem is referenced by:  sincn 8664  coscn 8665

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-map 4330  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-3 5973  df-4 5974  df-n0 6102  df-z 6138  df-fl 6226  df-rp 6282  df-seq1 6309  df-shft 6342  df-uz 6419  df-fz 6469  df-seqz 6534  df-seq0 6535  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-fac 6932  df-bc 6957  df-clim 6975  df-sum 6980  df-cncf 7263  df-ef 7298  df-top 7594  df-cn 7751  df-cnp 7752  df-met 7790  df-bl 7792  df-opn 7793

Copyright terms: Public domain