Theorem reeff1o 8691

Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.)

Assertion

Ref	Expression
reeff1o	|- (exp |` RR):RR-1-1-onto->(0(,) +oo)

Proof of Theorem reeff1o

Step	Hyp	Ref	Expression
1		df-f1o 4013	. 2 |- ((exp |` RR):RR-1-1-onto->(0(,) +oo) <-> ((exp |` RR):RR-1-1->(0(,) +oo) /\ (exp |` RR):RR-onto->(0(,) +oo)))
2		reeff1 8675	. 2 |- (exp |` RR):RR-1-1->(0(,) +oo)
3		df-fo 4012	. . 3 |- ((exp |` RR):RR-onto->(0(,) +oo) <-> ((exp |` RR) Fn RR /\ ran (exp |` RR) = (0(,) +oo)))
4		axresscn 6420	. . . 4 |- RR C_ CC
5		sumex 8241	. . . . . 6 |- sum_k e. NN0 ((p^k) / (!` k)) e. _V
6		df-ef 8560	. . . . . 6 |- exp = {<.p, q>. | (p e. CC /\ q = sum_k e. NN0 ((p^k) / (!` k)))}
7	5, 6	fnopab2 4549	. . . . 5 |- exp Fn CC
8		fnssresb 4525	. . . . 5 |- (exp Fn CC -> ((exp |` RR) Fn RR <-> RR C_ CC))
9	7, 8	ax-mp 7	. . . 4 |- ((exp |` RR) Fn RR <-> RR C_ CC)
10	4, 9	mpbir 207	. . 3 |- (exp |` RR) Fn RR
11		df-f1 4011	. . . . . . . 8 |- ((exp |` RR):RR-1-1->(0(,) +oo) <-> ((exp |` RR):RR-->(0(,) +oo) /\ Fun `'(exp |` RR)))
12	2, 11	mpbi 206	. . . . . . 7 |- ((exp |` RR):RR-->(0(,) +oo) /\ Fun `'(exp |` RR))
13	12	simpli 347	. . . . . 6 |- (exp |` RR):RR-->(0(,) +oo)
14		df-f 4010	. . . . . 6 |- ((exp |` RR):RR-->(0(,) +oo) <-> ((exp |` RR) Fn RR /\ ran (exp |` RR) C_ (0(,) +oo)))
15	13, 14	mpbi 206	. . . . 5 |- ((exp |` RR) Fn RR /\ ran (exp |` RR) C_ (0(,) +oo))
16	15	simpri 351	. . . 4 |- ran (exp |` RR) C_ (0(,) +oo)
17		1re 6598	. . . . . . . . . 10 |- 1 e. RR
18		lelttric 6805	. . . . . . . . . . 11 |- ((z e. RR /\ 1 e. RR) -> (z <_ 1 \/ 1 < z))
19		leloe 6688	. . . . . . . . . . . 12 |- ((z e. RR /\ 1 e. RR) -> (z <_ 1 <-> (z < 1 \/ z = 1)))
20	19	orbi1d 677	. . . . . . . . . . 11 |- ((z e. RR /\ 1 e. RR) -> ((z <_ 1 \/ 1 < z) <-> ((z < 1 \/ z = 1) \/ 1 < z)))
21	18, 20	mpbid 212	. . . . . . . . . 10 |- ((z e. RR /\ 1 e. RR) -> ((z < 1 \/ z = 1) \/ 1 < z))
22	17, 21	mpan2 760	. . . . . . . . 9 |- (z e. RR -> ((z < 1 \/ z = 1) \/ 1 < z))
23	22	adantr 425	. . . . . . . 8 |- ((z e. RR /\ 0 < z) -> ((z < 1 \/ z = 1) \/ 1 < z))
24		reclt1 7081	. . . . . . . . . . . 12 |- ((z e. RR /\ 0 < z) -> (z < 1 <-> 1 < (1 / z)))
25		reeff1olem2 8690	. . . . . . . . . . . . . . . 16 |- (((1 / z) e. RR /\ 1 < (1 / z)) -> E.y e. RR (exp` y) = (1 / z))
26		gt0ne0 6800	. . . . . . . . . . . . . . . . 17 |- ((z e. RR /\ 0 < z) -> z =/= 0)
27		rereccl 6981	. . . . . . . . . . . . . . . . 17 |- ((z e. RR /\ z =/= 0) -> (1 / z) e. RR)
28	26, 27	syldan 516	. . . . . . . . . . . . . . . 16 |- ((z e. RR /\ 0 < z) -> (1 / z) e. RR)
29	25, 28	sylan 497	. . . . . . . . . . . . . . 15 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.y e. RR (exp` y) = (1 / z))
30		rec11r 6955	. . . . . . . . . . . . . . . . . . . . 21 |- (((z e. CC /\ z =/= 0) /\ ((exp` y) e. CC /\ (exp` y) =/= 0)) -> ((1 / z) = (exp` y) <-> (1 / (exp` y)) = z))
31		recn 6466	. . . . . . . . . . . . . . . . . . . . . . 23 |- (z e. RR -> z e. CC)
32	31	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- ((z e. RR /\ 0 < z) -> z e. CC)
33	32, 26	jca 310	. . . . . . . . . . . . . . . . . . . . 21 |- ((z e. RR /\ 0 < z) -> (z e. CC /\ z =/= 0))
34		recn 6466	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. RR -> y e. CC)
35		efcl 8574	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (exp` y) e. CC)
36	34, 35	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (exp` y) e. CC)
37		efne0 8631	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (exp` y) =/= 0)
38	34, 37	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (exp` y) =/= 0)
39	36, 38	jca 310	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> ((exp` y) e. CC /\ (exp` y) =/= 0))
40	30, 33, 39	syl2an 503	. . . . . . . . . . . . . . . . . . . 20 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / z) = (exp` y) <-> (1 / (exp` y)) = z))
41		efcan 8630	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. CC -> ((exp` y) x. (exp` -uy)) = 1)
42	41	eqcomd 1889	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. CC -> 1 = ((exp` y) x. (exp` -uy)))
43		negcl 6525	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y e. CC -> -uy e. CC)
44		efcl 8574	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (-uy e. CC -> (exp` -uy) e. CC)
45	43, 44	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. CC -> (exp` -uy) e. CC)
46		ax1cn 6422	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. CC
47		divmul2 6897	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((1 e. CC /\ (exp` -uy) e. CC /\ ((exp` y) e. CC /\ (exp` y) =/= 0)) -> ((1 / (exp` y)) = (exp` -uy) <-> 1 = ((exp` y) x. (exp` -uy))))
48	46, 47	mp3an1 1178	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((exp` -uy) e. CC /\ ((exp` y) e. CC /\ (exp` y) =/= 0)) -> ((1 / (exp` y)) = (exp` -uy) <-> 1 = ((exp` y) x. (exp` -uy))))
49	45, 35, 37, 48	syl12anc 1098	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. CC -> ((1 / (exp` y)) = (exp` -uy) <-> 1 = ((exp` y) x. (exp` -uy))))
50	42, 49	mpbird 213	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (1 / (exp` y)) = (exp` -uy))
51	34, 50	syl 12	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (1 / (exp` y)) = (exp` -uy))
52	51	eqeq1d 1892	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> ((1 / (exp` y)) = z <-> (exp` -uy) = z))
53	52	adantl 424	. . . . . . . . . . . . . . . . . . . 20 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / (exp` y)) = z <-> (exp` -uy) = z))
54	40, 53	bitrd 587	. . . . . . . . . . . . . . . . . . 19 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / z) = (exp` y) <-> (exp` -uy) = z))
55		eqcom 1886	. . . . . . . . . . . . . . . . . . 19 |- ((1 / z) = (exp` y) <-> (exp` y) = (1 / z))
56	54, 55	syl5bbr 593	. . . . . . . . . . . . . . . . . 18 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((exp` y) = (1 / z) <-> (exp` -uy) = z))
57	56	biimpd 170	. . . . . . . . . . . . . . . . 17 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((exp` y) = (1 / z) -> (exp` -uy) = z))
58	57	reximdva 2203	. . . . . . . . . . . . . . . 16 |- ((z e. RR /\ 0 < z) -> (E.y e. RR (exp` y) = (1 / z) -> E.y e. RR (exp` -uy) = z))
59	58	adantr 425	. . . . . . . . . . . . . . 15 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> (E.y e. RR (exp` y) = (1 / z) -> E.y e. RR (exp` -uy) = z))
60	29, 59	mpd 29	. . . . . . . . . . . . . 14 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.y e. RR (exp` -uy) = z)
61		renegcl 6600	. . . . . . . . . . . . . . 15 |- (y e. RR -> -uy e. RR)
62		infm3lem 7262	. . . . . . . . . . . . . . 15 |- (x e. RR -> E.y e. RR x = -uy)
63		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (x = -uy -> (exp` x) = (exp` -uy))
64	63	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (x = -uy -> ((exp` x) = z <-> (exp` -uy) = z))
65	61, 62, 64	rexxfr 3841	. . . . . . . . . . . . . 14 |- (E.x e. RR (exp` x) = z <-> E.y e. RR (exp` -uy) = z)
66	60, 65	sylibr 217	. . . . . . . . . . . . 13 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.x e. RR (exp` x) = z)
67	66	ex 402	. . . . . . . . . . . 12 |- ((z e. RR /\ 0 < z) -> (1 < (1 / z) -> E.x e. RR (exp` x) = z))
68	24, 67	sylbid 220	. . . . . . . . . . 11 |- ((z e. RR /\ 0 < z) -> (z < 1 -> E.x e. RR (exp` x) = z))
69	68	imp 377	. . . . . . . . . 10 |- (((z e. RR /\ 0 < z) /\ z < 1) -> E.x e. RR (exp` x) = z)
70		ef0 8597	. . . . . . . . . . . . 13 |- (exp` 0) = 1
71	70	eqeq2i 1894	. . . . . . . . . . . 12 |- (z = (exp` 0) <-> z = 1)
72		0re 6603	. . . . . . . . . . . . . 14 |- 0 e. RR
73		fveq2 4681	. . . . . . . . . . . . . . . 16 |- (x = 0 -> (exp` x) = (exp` 0))
74	73	eqeq1d 1892	. . . . . . . . . . . . . . 15 |- (x = 0 -> ((exp` x) = z <-> (exp` 0) = z))
75	74	rcla4ev 2381	. . . . . . . . . . . . . 14 |- ((0 e. RR /\ (exp` 0) = z) -> E.x e. RR (exp` x) = z)
76	72, 75	mpan 759	. . . . . . . . . . . . 13 |- ((exp` 0) = z -> E.x e. RR (exp` x) = z)
77	76	eqcoms 1887	. . . . . . . . . . . 12 |- (z = (exp` 0) -> E.x e. RR (exp` x) = z)
78	71, 77	sylbir 218	. . . . . . . . . . 11 |- (z = 1 -> E.x e. RR (exp` x) = z)
79	78	adantl 424	. . . . . . . . . 10 |- (((z e. RR /\ 0 < z) /\ z = 1) -> E.x e. RR (exp` x) = z)
80	69, 79	jaodan 471	. . . . . . . . 9 |- (((z e. RR /\ 0 < z) /\ (z < 1 \/ z = 1)) -> E.x e. RR (exp` x) = z)
81		reeff1olem2 8690	. . . . . . . . . 10 |- ((z e. RR /\ 1 < z) -> E.x e. RR (exp` x) = z)
82	81	adantlr 429	. . . . . . . . 9 |- (((z e. RR /\ 0 < z) /\ 1 < z) -> E.x e. RR (exp` x) = z)
83	80, 82	jaodan 471	. . . . . . . 8 |- (((z e. RR /\ 0 < z) /\ ((z < 1 \/ z = 1) \/ 1 < z)) -> E.x e. RR (exp` x) = z)
84	23, 83	mpdan 768	. . . . . . 7 |- ((z e. RR /\ 0 < z) -> E.x e. RR (exp` x) = z)
85		repos 7568	. . . . . . 7 |- (z e. (0(,) +oo) <-> (z e. RR /\ 0 < z))
86		fvres 4691	. . . . . . . . 9 |- (x e. RR -> ((exp |` RR)` x) = (exp` x))
87	86	eqeq1d 1892	. . . . . . . 8 |- (x e. RR -> (((exp |` RR)` x) = z <-> (exp` x) = z))
88	87	rexbiia 2134	. . . . . . 7 |- (E.x e. RR ((exp |` RR)` x) = z <-> E.x e. RR (exp` x) = z)
89	84, 85, 88	3imtr4i 236	. . . . . 6 |- (z e. (0(,) +oo) -> E.x e. RR ((exp |` RR)` x) = z)
90		fvelrnb 4719	. . . . . . 7 |- ((exp |` RR) Fn RR -> (z e. ran (exp |` RR) <-> E.x e. RR ((exp |` RR)` x) = z))
91	10, 90	ax-mp 7	. . . . . 6 |- (z e. ran (exp |` RR) <-> E.x e. RR ((exp |` RR)` x) = z)
92	89, 91	sylibr 217	. . . . 5 |- (z e. (0(,) +oo) -> z e. ran (exp |` RR))
93	92	ssriv 2621	. . . 4 |- (0(,) +oo) C_ ran (exp |` RR)
94	16, 93	eqssi 2632	. . 3 |- ran (exp |` RR) = (0(,) +oo)
95	3, 10, 94	mpbir2an 800	. 2 |- (exp |` RR):RR-onto->(0(,) +oo)
96	1, 2, 95	mpbir2an 800	1 |- (exp |` RR):RR-1-1-onto->(0(,) +oo)

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   C_ wss 2593   class class class wbr 3338  `'ccnv 3985  ran crn 3987   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391  -ucneg 6446   / cdiv 6447   <_ cle 6448  NN0cn0 6450   +oocpnf 6650   < clt 6653  (,)cioo 7524  ^cexp 7811  !cfa 8183  sum_csu 8239  expce 8555

This theorem is referenced by:  reeff1o2 8692

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-q 7436  df-fl 7463  df-ioo 7528  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

Copyright terms: Public domain