Theorem reeff1o 7634

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 3278	. 2 |- ((exp |` RR):RR-1-1-onto->(0(,) +oo) <-> ((exp |` RR):RR-1-1->(0(,) +oo) /\ (exp |` RR):RR-onto->(0(,) +oo)))
2		reeff1 7618	. 2 |- (exp |` RR):RR-1-1->(0(,) +oo)
3		df-fo 3277	. . 3 |- ((exp |` RR):RR-onto->(0(,) +oo) <-> ((exp |` RR) Fn RR /\ ran (exp |` RR) = (0(,) +oo)))
4		axresscn 5422	. . . 4 |- RR (_ CC
5		sumex 7184	. . . . . 6 |- sum_k e. NN0 ((p^k) / (!` k)) e. V
6		df-ef 7503	. . . . . 6 |- exp = {<.p, q>. | (p e. CC /\ q = sum_k e. NN0 ((p^k) / (!` k)))}
7	5, 6	fnopab2 3725	. . . . 5 |- exp Fn CC
8		fnssresb 3705	. . . . 5 |- (exp Fn CC -> ((exp |` RR) Fn RR <-> RR (_ CC))
9	7, 8	ax-mp 7	. . . 4 |- ((exp |` RR) Fn RR <-> RR (_ CC)
10	4, 9	mpbir 188	. . 3 |- (exp |` RR) Fn RR
11		df-f1 3276	. . . . . . . 8 |- ((exp |` RR):RR-1-1->(0(,) +oo) <-> ((exp |` RR):RR-->(0(,) +oo) /\ Fun `'(exp |` RR)))
12	2, 11	mpbi 187	. . . . . . 7 |- ((exp |` RR):RR-->(0(,) +oo) /\ Fun `'(exp |` RR))
13	12	pm3.26i 318	. . . . . 6 |- (exp |` RR):RR-->(0(,) +oo)
14		df-f 3275	. . . . . 6 |- ((exp |` RR):RR-->(0(,) +oo) <-> ((exp |` RR) Fn RR /\ ran (exp |` RR) (_ (0(,) +oo)))
15	13, 14	mpbi 187	. . . . 5 |- ((exp |` RR) Fn RR /\ ran (exp |` RR) (_ (0(,) +oo))
16	15	pm3.27i 322	. . . 4 |- ran (exp |` RR) (_ (0(,) +oo)
17		1re 5589	. . . . . . . . . 10 |- 1 e. RR
18		lelttric 5776	. . . . . . . . . . 11 |- ((z e. RR /\ 1 e. RR) -> (z <_ 1 \/ 1 < z))
19		leloe 5672	. . . . . . . . . . . 12 |- ((z e. RR /\ 1 e. RR) -> (z <_ 1 <-> (z < 1 \/ z = 1)))
20	19	orbi1d 618	. . . . . . . . . . 11 |- ((z e. RR /\ 1 e. RR) -> ((z <_ 1 \/ 1 < z) <-> ((z < 1 \/ z = 1) \/ 1 < z)))
21	18, 20	mpbid 193	. . . . . . . . . 10 |- ((z e. RR /\ 1 e. RR) -> ((z < 1 \/ z = 1) \/ 1 < z))
22	17, 21	mpan2 700	. . . . . . . . 9 |- (z e. RR -> ((z < 1 \/ z = 1) \/ 1 < z))
23	22	adantr 389	. . . . . . . 8 |- ((z e. RR /\ 0 < z) -> ((z < 1 \/ z = 1) \/ 1 < z))
24		reclt1 6043	. . . . . . . . . . . 12 |- ((z e. RR /\ 0 < z) -> (z < 1 <-> 1 < (1 / z)))
25		reeff1olem2 7633	. . . . . . . . . . . . . . . 16 |- (((1 / z) e. RR /\ 1 < (1 / z)) -> E.y e. RR (exp` y) = (1 / z))
26		gt0ne0 5772	. . . . . . . . . . . . . . . . 17 |- ((z e. RR /\ 0 < z) -> z =/= 0)
27		rereccl 5943	. . . . . . . . . . . . . . . . 17 |- ((z e. RR /\ z =/= 0) -> (1 / z) e. RR)
28	26, 27	syldan 469	. . . . . . . . . . . . . . . 16 |- ((z e. RR /\ 0 < z) -> (1 / z) e. RR)
29	25, 28	sylan 450	. . . . . . . . . . . . . . 15 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.y e. RR (exp` y) = (1 / z))
30		rec11r 5918	. . . . . . . . . . . . . . . . . . . . 21 |- (((z e. CC /\ z =/= 0) /\ ((exp` y) e. CC /\ (exp` y) =/= 0)) -> ((1 / z) = (exp` y) <-> (1 / (exp` y)) = z))
31		recn 5467	. . . . . . . . . . . . . . . . . . . . . . 23 |- (z e. RR -> z e. CC)
32	31	adantr 389	. . . . . . . . . . . . . . . . . . . . . 22 |- ((z e. RR /\ 0 < z) -> z e. CC)
33	32, 26	jca 286	. . . . . . . . . . . . . . . . . . . . 21 |- ((z e. RR /\ 0 < z) -> (z e. CC /\ z =/= 0))
34		recn 5467	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. RR -> y e. CC)
35		efcl 7517	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (exp` y) e. CC)
36	34, 35	syl 10	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (exp` y) e. CC)
37		efne0 7574	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (exp` y) =/= 0)
38	34, 37	syl 10	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (exp` y) =/= 0)
39	36, 38	jca 286	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> ((exp` y) e. CC /\ (exp` y) =/= 0))
40	30, 33, 39	syl2an 456	. . . . . . . . . . . . . . . . . . . 20 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / z) = (exp` y) <-> (1 / (exp` y)) = z))
41		efcan 7573	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. CC -> ((exp` y) x. (exp` -uy)) = 1)
42	41	eqcomd 1523	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. CC -> 1 = ((exp` y) x. (exp` -uy)))
43		ax1cn 5423	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- 1 e. CC
44		divmul2 5860	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
45	43, 44	mp3an1 909	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((exp` -uy) e. CC /\ ((exp` y) e. CC /\ (exp` y) =/= 0)) -> ((1 / (exp` y)) = (exp` -uy) <-> 1 = ((exp` y) x. (exp` -uy))))
46		negcl 5522	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (y e. CC -> -uy e. CC)
47		efcl 7517	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (-uy e. CC -> (exp` -uy) e. CC)
48	46, 47	syl 10	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. CC -> (exp` -uy) e. CC)
49	35, 37	jca 286	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y e. CC -> ((exp` y) e. CC /\ (exp` y) =/= 0))
50	45, 48, 49	sylanc 473	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y e. CC -> ((1 / (exp` y)) = (exp` -uy) <-> 1 = ((exp` y) x. (exp` -uy))))
51	42, 50	mpbird 194	. . . . . . . . . . . . . . . . . . . . . . 23 |- (y e. CC -> (1 / (exp` y)) = (exp` -uy))
52	34, 51	syl 10	. . . . . . . . . . . . . . . . . . . . . 22 |- (y e. RR -> (1 / (exp` y)) = (exp` -uy))
53	52	eqeq1d 1526	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. RR -> ((1 / (exp` y)) = z <-> (exp` -uy) = z))
54	53	adantl 388	. . . . . . . . . . . . . . . . . . . 20 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / (exp` y)) = z <-> (exp` -uy) = z))
55	40, 54	bitrd 531	. . . . . . . . . . . . . . . . . . 19 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((1 / z) = (exp` y) <-> (exp` -uy) = z))
56		eqcom 1520	. . . . . . . . . . . . . . . . . . 19 |- ((1 / z) = (exp` y) <-> (exp` y) = (1 / z))
57	55, 56	syl5bbr 537	. . . . . . . . . . . . . . . . . 18 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((exp` y) = (1 / z) <-> (exp` -uy) = z))
58	57	biimpd 151	. . . . . . . . . . . . . . . . 17 |- (((z e. RR /\ 0 < z) /\ y e. RR) -> ((exp` y) = (1 / z) -> (exp` -uy) = z))
59	58	r19.22dva 1785	. . . . . . . . . . . . . . . 16 |- ((z e. RR /\ 0 < z) -> (E.y e. RR (exp` y) = (1 / z) -> E.y e. RR (exp` -uy) = z))
60	59	adantr 389	. . . . . . . . . . . . . . 15 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> (E.y e. RR (exp` y) = (1 / z) -> E.y e. RR (exp` -uy) = z))
61	29, 60	mpd 26	. . . . . . . . . . . . . 14 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.y e. RR (exp` -uy) = z)
62		renegcl 5591	. . . . . . . . . . . . . . 15 |- (y e. RR -> -uy e. RR)
63		infm3lem 6221	. . . . . . . . . . . . . . 15 |- (x e. RR -> E.y e. RR x = -uy)
64		fveq2 3835	. . . . . . . . . . . . . . . 16 |- (x = -uy -> (exp` x) = (exp` -uy))
65	64	eqeq1d 1526	. . . . . . . . . . . . . . 15 |- (x = -uy -> ((exp` x) = z <-> (exp` -uy) = z))
66	62, 63, 65	rexxfr 3124	. . . . . . . . . . . . . 14 |- (E.x e. RR (exp` x) = z <-> E.y e. RR (exp` -uy) = z)
67	61, 66	sylibr 198	. . . . . . . . . . . . 13 |- (((z e. RR /\ 0 < z) /\ 1 < (1 / z)) -> E.x e. RR (exp` x) = z)
68	67	ex 371	. . . . . . . . . . . 12 |- ((z e. RR /\ 0 < z) -> (1 < (1 / z) -> E.x e. RR (exp` x) = z))
69	24, 68	sylbid 201	. . . . . . . . . . 11 |- ((z e. RR /\ 0 < z) -> (z < 1 -> E.x e. RR (exp` x) = z))
70	69	imp 348	. . . . . . . . . 10 |- (((z e. RR /\ 0 < z) /\ z < 1) -> E.x e. RR (exp` x) = z)
71		ef0 7540	. . . . . . . . . . . . 13 |- (exp` 0) = 1
72	71	eqeq2i 1528	. . . . . . . . . . . 12 |- (z = (exp` 0) <-> z = 1)
73		0re 5594	. . . . . . . . . . . . . 14 |- 0 e. RR
74		fveq2 3835	. . . . . . . . . . . . . . . 16 |- (x = 0 -> (exp` x) = (exp` 0))
75	74	eqeq1d 1526	. . . . . . . . . . . . . . 15 |- (x = 0 -> ((exp` x) = z <-> (exp` 0) = z))
76	75	rcla4ev 1923	. . . . . . . . . . . . . 14 |- ((0 e. RR /\ (exp` 0) = z) -> E.x e. RR (exp` x) = z)
77	73, 76	mpan 699	. . . . . . . . . . . . 13 |- ((exp` 0) = z -> E.x e. RR (exp` x) = z)
78	77	eqcoms 1521	. . . . . . . . . . . 12 |- (z = (exp` 0) -> E.x e. RR (exp` x) = z)
79	72, 78	sylbir 199	. . . . . . . . . . 11 |- (z = 1 -> E.x e. RR (exp` x) = z)
80	79	adantl 388	. . . . . . . . . 10 |- (((z e. RR /\ 0 < z) /\ z = 1) -> E.x e. RR (exp` x) = z)
81	70, 80	jaodan 426	. . . . . . . . 9 |- (((z e. RR /\ 0 < z) /\ (z < 1 \/ z = 1)) -> E.x e. RR (exp` x) = z)
82		reeff1olem2 7633	. . . . . . . . . 10 |- ((z e. RR /\ 1 < z) -> E.x e. RR (exp` x) = z)
83	82	adantlr 393	. . . . . . . . 9 |- (((z e. RR /\ 0 < z) /\ 1 < z) -> E.x e. RR (exp` x) = z)
84	81, 83	jaodan 426	. . . . . . . 8 |- (((z e. RR /\ 0 < z) /\ ((z < 1 \/ z = 1) \/ 1 < z)) -> E.x e. RR (exp` x) = z)
85	23, 84	mpdan 708	. . . . . . 7 |- ((z e. RR /\ 0 < z) -> E.x e. RR (exp` x) = z)
86		repos 6526	. . . . . . 7 |- (z e. (0(,) +oo) <-> (z e. RR /\ 0 < z))
87		fvres 3845	. . . . . . . . 9 |- (x e. RR -> ((exp |` RR)` x) = (exp` x))
88	87	eqeq1d 1526	. . . . . . . 8 |- (x e. RR -> (((exp |` RR)` x) = z <-> (exp` x) = z))
89	88	rexbiia 1720	. . . . . . 7 |- (E.x e. RR ((exp |` RR)` x) = z <-> E.x e. RR (exp` x) = z)
90	85, 86, 89	3imtr4i 217	. . . . . 6 |- (z e. (0(,) +oo) -> E.x e. RR ((exp |` RR)` x) = z)
91		fvelrnb 3871	. . . . . . 7 |- ((exp |` RR) Fn RR -> (z e. ran (exp |` RR) <-> E.x e. RR ((exp |` RR)` x) = z))
92	10, 91	ax-mp 7	. . . . . 6 |- (z e. ran (exp |` RR) <-> E.x e. RR ((exp |` RR)` x) = z)
93	90, 92	sylibr 198	. . . . 5 |- (z e. (0(,) +oo) -> z e. ran (exp |` RR))
94	93	ssriv 2121	. . . 4 |- (0(,) +oo) (_ ran (exp |` RR)
95	16, 94	eqssi 2130	. . 3 |- ran (exp |` RR) = (0(,) +oo)
96	3, 10, 95	mpbir2an 735	. 2 |- (exp |` RR):RR-onto->(0(,) +oo)
97	1, 2, 96	mpbir2an 735	1 |- (exp |` RR):RR-1-1-onto->(0(,) +oo)

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628  E.wrex 1692   (_ wss 2099   class class class wbr 2692  `'ccnv 3250  ran crn 3252   |` cres 3253  Fun wfun 3257   Fn wfn 3258  -->wf 3259  -1-1->wf1 3260  -onto->wfo 3261  -1-1-onto->wf1o 3262  ` cfv 3263  (class class class)co 4021  CCcc 5386  RRcr 5387  0cc0 5388  1c1 5389   x. cmul 5393  -ucneg 5447   / cdiv 5448   <_ cle 5449  NN0cn0 5451   +oocpnf 5637   < clt 5640  (,)cioo 6483  ^cexp 6763  !cfa 7134  sum_csu 7182  expce 7498

This theorem is referenced by:  reeff1o2 7635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-sup 4717  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-div 5855  df-n 6070  df-2 6116  df-3 6117  df-4 6118  df-rp 6191  df-n0 6268  df-z 6304  df-q 6395  df-fl 6422  df-ioo 6487  df-icc 6490  df-uz 6545  df-fz 6596  df-seq1 6673  df-shft 6706  df-seqz 6728  df-seq0 6729  df-exp 6764  df-sqr 6871  df-re 6952  df-im 6953  df-cj 6954  df-abs 6955  df-fac 7135  df-bc 7160  df-clim 7178  df-sum 7183  df-cncf 7468  df-ef 7503

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