Theorem efcn 8688

Description: The exponential function is continuous. (Contributed by Paul Chapman, 15-Sep-2007.)

Assertion

Ref	Expression
efcn	|- exp e. (CC-cn->CC)

Proof of Theorem efcn

Step	Hyp	Ref	Expression
1		ssid 2634	. . 3 |- CC C_ CC
2		elcncf 8527	. . 3 |- ((CC C_ CC /\ CC C_ CC) -> (exp e. (CC-cn->CC) <-> (exp:CC-->CC /\ A.x e. CC A.y e. RR+ E.z e. RR+ A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y))))
3	1, 1, 2	mp2an 761	. 2 |- (exp e. (CC-cn->CC) <-> (exp:CC-->CC /\ A.x e. CC A.y e. RR+ E.z e. RR+ A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y)))
4		eff 8575	. 2 |- exp:CC-->CC
5		elrp 7233	. . . . 5 |- ((y / ((abs` (exp` x)) + y)) e. RR+ <-> ((y / ((abs` (exp` x)) + y)) e. RR /\ 0 < (y / ((abs` (exp` x)) + y))))
6		simpr 350	. . . . . . . 8 |- ((x e. CC /\ y e. RR+) -> y e. RR+)
7		elrp 7233	. . . . . . . 8 |- (y e. RR+ <-> (y e. RR /\ 0 < y))
8	6, 7	sylib 215	. . . . . . 7 |- ((x e. CC /\ y e. RR+) -> (y e. RR /\ 0 < y))
9	8	simplld 348	. . . . . 6 |- ((x e. CC /\ y e. RR+) -> y e. RR)
10		efcl 8574	. . . . . . . . 9 |- (x e. CC -> (exp` x) e. CC)
11		abscl 8084	. . . . . . . . 9 |- ((exp` x) e. CC -> (abs` (exp` x)) e. RR)
12	10, 11	syl 12	. . . . . . . 8 |- (x e. CC -> (abs` (exp` x)) e. RR)
13	12	adantr 425	. . . . . . 7 |- ((x e. CC /\ y e. RR+) -> (abs` (exp` x)) e. RR)
14		readdcl 6455	. . . . . . 7 |- (((abs` (exp` x)) e. RR /\ y e. RR) -> ((abs` (exp` x)) + y) e. RR)
15	13, 9, 14	syl11anc 524	. . . . . 6 |- ((x e. CC /\ y e. RR+) -> ((abs` (exp` x)) + y) e. RR)
16		efne0 8631	. . . . . . . . . 10 |- (x e. CC -> (exp` x) =/= 0)
17		absgt0 8145	. . . . . . . . . . 11 |- ((exp` x) e. CC -> ((exp` x) =/= 0 <-> 0 < (abs` (exp` x))))
18	10, 17	syl 12	. . . . . . . . . 10 |- (x e. CC -> ((exp` x) =/= 0 <-> 0 < (abs` (exp` x))))
19	16, 18	mpbid 212	. . . . . . . . 9 |- (x e. CC -> 0 < (abs` (exp` x)))
20	19	adantr 425	. . . . . . . 8 |- ((x e. CC /\ y e. RR+) -> 0 < (abs` (exp` x)))
21	8	simprd 352	. . . . . . . 8 |- ((x e. CC /\ y e. RR+) -> 0 < y)
22		addgt0 6831	. . . . . . . 8 |- ((((abs` (exp` x)) e. RR /\ y e. RR) /\ (0 < (abs` (exp` x)) /\ 0 < y)) -> 0 < ((abs` (exp` x)) + y))
23	13, 9, 20, 21, 22	syl22anc 1101	. . . . . . 7 |- ((x e. CC /\ y e. RR+) -> 0 < ((abs` (exp` x)) + y))
24		gt0ne0 6800	. . . . . . 7 |- ((((abs` (exp` x)) + y) e. RR /\ 0 < ((abs` (exp` x)) + y)) -> ((abs` (exp` x)) + y) =/= 0)
25	15, 23, 24	syl11anc 524	. . . . . 6 |- ((x e. CC /\ y e. RR+) -> ((abs` (exp` x)) + y) =/= 0)
26		redivcl 6978	. . . . . 6 |- ((y e. RR /\ ((abs` (exp` x)) + y) e. RR /\ ((abs` (exp` x)) + y) =/= 0) -> (y / ((abs` (exp` x)) + y)) e. RR)
27	9, 15, 25, 26	syl111anc 1100	. . . . 5 |- ((x e. CC /\ y e. RR+) -> (y / ((abs` (exp` x)) + y)) e. RR)
28		divgt0 7037	. . . . . 6 |- (((y e. RR /\ 0 < y) /\ (((abs` (exp` x)) + y) e. RR /\ 0 < ((abs` (exp` x)) + y))) -> 0 < (y / ((abs` (exp` x)) + y)))
29	9, 21, 15, 23, 28	syl22anc 1101	. . . . 5 |- ((x e. CC /\ y e. RR+) -> 0 < (y / ((abs` (exp` x)) + y)))
30	5, 27, 29	sylanbrc 527	. . . 4 |- ((x e. CC /\ y e. RR+) -> (y / ((abs` (exp` x)) + y)) e. RR+)
31		3ancoma 865	. . . . . . . 8 |- ((x e. CC /\ w e. CC /\ y e. RR+) <-> (w e. CC /\ x e. CC /\ y e. RR+))
32	7	3anbi3i 1060	. . . . . . . 8 |- ((x e. CC /\ w e. CC /\ y e. RR+) <-> (x e. CC /\ w e. CC /\ (y e. RR /\ 0 < y)))
33		3anass 862	. . . . . . . 8 |- ((w e. CC /\ x e. CC /\ y e. RR+) <-> (w e. CC /\ (x e. CC /\ y e. RR+)))
34	31, 32, 33	3bitr3i 198	. . . . . . 7 |- ((x e. CC /\ w e. CC /\ (y e. RR /\ 0 < y)) <-> (w e. CC /\ (x e. CC /\ y e. RR+)))
35		efcnlem4 8687	. . . . . . 7 |- ((x e. CC /\ w e. CC /\ (y e. RR /\ 0 < y)) -> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y))
36	34, 35	sylbir 218	. . . . . 6 |- ((w e. CC /\ (x e. CC /\ y e. RR+)) -> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y))
37	36	expcom 403	. . . . 5 |- ((x e. CC /\ y e. RR+) -> (w e. CC -> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y)))
38	37	r19.21aiv 2175	. . . 4 |- ((x e. CC /\ y e. RR+) -> A.w e. CC ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y))
39		breq2 3342	. . . . . . 7 |- (z = (y / ((abs` (exp` x)) + y)) -> ((abs` (x - w)) < z <-> (abs` (x - w)) < (y / ((abs` (exp` x)) + y))))
40		imbi1 685	. . . . . . 7 |- (((abs` (x - w)) < z <-> (abs` (x - w)) < (y / ((abs` (exp` x)) + y))) -> (((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y) <-> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y)))
41		imbi2 686	. . . . . . 7 |- ((((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y) <-> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y)) -> ((w e. CC -> ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y)) <-> (w e. CC -> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y))))
42	39, 40, 41	3syl 24	. . . . . 6 |- (z = (y / ((abs` (exp` x)) + y)) -> ((w e. CC -> ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y)) <-> (w e. CC -> ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y))))
43	42	ralbidv2 2125	. . . . 5 |- (z = (y / ((abs` (exp` x)) + y)) -> (A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y) <-> A.w e. CC ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y)))
44	43	rcla4ev 2381	. . . 4 |- (((y / ((abs` (exp` x)) + y)) e. RR+ /\ A.w e. CC ((abs` (x - w)) < (y / ((abs` (exp` x)) + y)) -> (abs` ((exp` x) - (exp` w))) < y)) -> E.z e. RR+ A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y))
45	30, 38, 44	syl11anc 524	. . 3 |- ((x e. CC /\ y e. RR+) -> E.z e. RR+ A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y))
46	45	rgen2 2186	. 2 |- A.x e. CC A.y e. RR+ E.z e. RR+ A.w e. CC ((abs` (x - w)) < z -> (abs` ((exp` x) - (exp` w))) < y)
47	3, 4, 46	mpbir2an 800	1 |- exp e. (CC-cn->CC)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   C_ wss 2593   class class class wbr 3338  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   - cmin 6445   / cdiv 6447  RR+crp 6453   < clt 6653  abscabs 8000  -cn->ccncf 8524  expce 8555

This theorem is referenced by:  reeff1olem1 8689  sincnlem 10015

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-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

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