Theorem reeff1o 24201

Description: The real exponential function is one-to-one onto. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 10-Nov-2013.)

Assertion

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

Proof of Theorem reeff1o

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reeff1 14850	. 2 |- ( exp |` RR ) : RR -1-1-> RR+
2		f1f 6101	. . . 4 |- ( ( exp |` RR ) : RR -1-1-> RR+ -> ( exp |` RR ) : RR --> RR+ )
3		ffn 6045	. . . 4 |- ( ( exp |` RR ) : RR --> RR+ -> ( exp |` RR ) Fn RR )
4	1, 2, 3	mp2b 10	. . 3 |- ( exp |` RR ) Fn RR
5		frn 6053	. . . . 5 |- ( ( exp |` RR ) : RR --> RR+ -> ran ( exp |` RR ) C_ RR+ )
6	1, 2, 5	mp2b 10	. . . 4 |- ran ( exp |` RR ) C_ RR+
7		elrp 11834	. . . . . . . . . . 11 |- ( z e. RR+ <-> ( z e. RR /\ 0 < z ) )
8		reclt1 10918	. . . . . . . . . . 11 |- ( ( z e. RR /\ 0 < z ) -> ( z < 1 <-> 1 < ( 1 / z ) ) )
9	7, 8	sylbi 207	. . . . . . . . . 10 |- ( z e. RR+ -> ( z < 1 <-> 1 < ( 1 / z ) ) )
10		rpre 11839	. . . . . . . . . . . . . . 15 |- ( z e. RR+ -> z e. RR )
11		rpne0 11848	. . . . . . . . . . . . . . 15 |- ( z e. RR+ -> z =/= 0 )
12	10, 11	rereccld 10852	. . . . . . . . . . . . . 14 |- ( z e. RR+ -> ( 1 / z ) e. RR )
13		reeff1olem 24200	. . . . . . . . . . . . . 14 |- ( ( ( 1 / z ) e. RR /\ 1 < ( 1 / z ) ) -> E. y e. RR ( exp ` y ) = ( 1 / z ) )
14	12, 13	sylan 488	. . . . . . . . . . . . 13 |- ( ( z e. RR+ /\ 1 < ( 1 / z ) ) -> E. y e. RR ( exp ` y ) = ( 1 / z ) )
15		eqcom 2629	. . . . . . . . . . . . . . . . 17 |- ( ( 1 / z ) = ( exp ` y ) <-> ( exp ` y ) = ( 1 / z ) )
16		rpcnne0 11850	. . . . . . . . . . . . . . . . . . 19 |- ( z e. RR+ -> ( z e. CC /\ z =/= 0 ) )
17		recn 10026	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. RR -> y e. CC )
18		efcl 14813	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. CC -> ( exp ` y ) e. CC )
19	17, 18	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. RR -> ( exp ` y ) e. CC )
20		efne0 14827	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. CC -> ( exp ` y ) =/= 0 )
21	17, 20	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. RR -> ( exp ` y ) =/= 0 )
22	19, 21	jca 554	. . . . . . . . . . . . . . . . . . 19 |- ( y e. RR -> ( ( exp ` y ) e. CC /\ ( exp ` y ) =/= 0 ) )
23		rec11r 10724	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( z e. CC /\ z =/= 0 ) /\ ( ( exp ` y ) e. CC /\ ( exp ` y ) =/= 0 ) ) -> ( ( 1 / z ) = ( exp ` y ) <-> ( 1 / ( exp ` y ) ) = z ) )
24	16, 22, 23	syl2an 494	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. RR+ /\ y e. RR ) -> ( ( 1 / z ) = ( exp ` y ) <-> ( 1 / ( exp ` y ) ) = z ) )
25		efcan 14826	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( y e. CC -> ( ( exp ` y ) x. ( exp ` -u y ) ) = 1 )
26	25	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y e. CC -> 1 = ( ( exp ` y ) x. ( exp ` -u y ) ) )
27		negcl 10281	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( y e. CC -> -u y e. CC )
28		efcl 14813	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( -u y e. CC -> ( exp ` -u y ) e. CC )
29	27, 28	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( y e. CC -> ( exp ` -u y ) e. CC )
30		ax-1cn 9994	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 1 e. CC
31		divmul2 10689	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( 1 e. CC /\ ( exp ` -u y ) e. CC /\ ( ( exp ` y ) e. CC /\ ( exp ` y ) =/= 0 ) ) -> ( ( 1 / ( exp ` y ) ) = ( exp ` -u y ) <-> 1 = ( ( exp ` y ) x. ( exp ` -u y ) ) ) )
32	30, 31	mp3an1 1411	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( exp ` -u y ) e. CC /\ ( ( exp ` y ) e. CC /\ ( exp ` y ) =/= 0 ) ) -> ( ( 1 / ( exp ` y ) ) = ( exp ` -u y ) <-> 1 = ( ( exp ` y ) x. ( exp ` -u y ) ) ) )
33	29, 18, 20, 32	syl12anc 1324	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y e. CC -> ( ( 1 / ( exp ` y ) ) = ( exp ` -u y ) <-> 1 = ( ( exp ` y ) x. ( exp ` -u y ) ) ) )
34	26, 33	mpbird 247	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. CC -> ( 1 / ( exp ` y ) ) = ( exp ` -u y ) )
35	17, 34	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. RR -> ( 1 / ( exp ` y ) ) = ( exp ` -u y ) )
36	35	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( y e. RR -> ( ( 1 / ( exp ` y ) ) = z <-> ( exp ` -u y ) = z ) )
37	36	adantl 482	. . . . . . . . . . . . . . . . . 18 |- ( ( z e. RR+ /\ y e. RR ) -> ( ( 1 / ( exp ` y ) ) = z <-> ( exp ` -u y ) = z ) )
38	24, 37	bitrd 268	. . . . . . . . . . . . . . . . 17 |- ( ( z e. RR+ /\ y e. RR ) -> ( ( 1 / z ) = ( exp ` y ) <-> ( exp ` -u y ) = z ) )
39	15, 38	syl5bbr 274	. . . . . . . . . . . . . . . 16 |- ( ( z e. RR+ /\ y e. RR ) -> ( ( exp ` y ) = ( 1 / z ) <-> ( exp ` -u y ) = z ) )
40	39	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( z e. RR+ /\ y e. RR ) -> ( ( exp ` y ) = ( 1 / z ) -> ( exp ` -u y ) = z ) )
41	40	reximdva 3017	. . . . . . . . . . . . . 14 |- ( z e. RR+ -> ( E. y e. RR ( exp ` y ) = ( 1 / z ) -> E. y e. RR ( exp ` -u y ) = z ) )
42	41	adantr 481	. . . . . . . . . . . . 13 |- ( ( z e. RR+ /\ 1 < ( 1 / z ) ) -> ( E. y e. RR ( exp ` y ) = ( 1 / z ) -> E. y e. RR ( exp ` -u y ) = z ) )
43	14, 42	mpd 15	. . . . . . . . . . . 12 |- ( ( z e. RR+ /\ 1 < ( 1 / z ) ) -> E. y e. RR ( exp ` -u y ) = z )
44		renegcl 10344	. . . . . . . . . . . . 13 |- ( y e. RR -> -u y e. RR )
45		infm3lem 10981	. . . . . . . . . . . . 13 |- ( x e. RR -> E. y e. RR x = -u y )
46		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = -u y -> ( exp ` x ) = ( exp ` -u y ) )
47	46	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( x = -u y -> ( ( exp ` x ) = z <-> ( exp ` -u y ) = z ) )
48	44, 45, 47	rexxfr 4888	. . . . . . . . . . . 12 |- ( E. x e. RR ( exp ` x ) = z <-> E. y e. RR ( exp ` -u y ) = z )
49	43, 48	sylibr 224	. . . . . . . . . . 11 |- ( ( z e. RR+ /\ 1 < ( 1 / z ) ) -> E. x e. RR ( exp ` x ) = z )
50	49	ex 450	. . . . . . . . . 10 |- ( z e. RR+ -> ( 1 < ( 1 / z ) -> E. x e. RR ( exp ` x ) = z ) )
51	9, 50	sylbid 230	. . . . . . . . 9 |- ( z e. RR+ -> ( z < 1 -> E. x e. RR ( exp ` x ) = z ) )
52	51	imp 445	. . . . . . . 8 |- ( ( z e. RR+ /\ z < 1 ) -> E. x e. RR ( exp ` x ) = z )
53		ef0 14821	. . . . . . . . . . 11 |- ( exp ` 0 ) = 1
54	53	eqeq2i 2634	. . . . . . . . . 10 |- ( z = ( exp ` 0 ) <-> z = 1 )
55		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
56		fveq2 6191	. . . . . . . . . . . . . 14 |- ( x = 0 -> ( exp ` x ) = ( exp ` 0 ) )
57	56	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( x = 0 -> ( ( exp ` x ) = z <-> ( exp ` 0 ) = z ) )
58	57	rspcev 3309	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ ( exp ` 0 ) = z ) -> E. x e. RR ( exp ` x ) = z )
59	55, 58	mpan 706	. . . . . . . . . . 11 |- ( ( exp ` 0 ) = z -> E. x e. RR ( exp ` x ) = z )
60	59	eqcoms 2630	. . . . . . . . . 10 |- ( z = ( exp ` 0 ) -> E. x e. RR ( exp ` x ) = z )
61	54, 60	sylbir 225	. . . . . . . . 9 |- ( z = 1 -> E. x e. RR ( exp ` x ) = z )
62	61	adantl 482	. . . . . . . 8 |- ( ( z e. RR+ /\ z = 1 ) -> E. x e. RR ( exp ` x ) = z )
63		reeff1olem 24200	. . . . . . . . 9 |- ( ( z e. RR /\ 1 < z ) -> E. x e. RR ( exp ` x ) = z )
64	10, 63	sylan 488	. . . . . . . 8 |- ( ( z e. RR+ /\ 1 < z ) -> E. x e. RR ( exp ` x ) = z )
65		1re 10039	. . . . . . . . 9 |- 1 e. RR
66		lttri4 10122	. . . . . . . . 9 |- ( ( z e. RR /\ 1 e. RR ) -> ( z < 1 \/ z = 1 \/ 1 < z ) )
67	10, 65, 66	sylancl 694	. . . . . . . 8 |- ( z e. RR+ -> ( z < 1 \/ z = 1 \/ 1 < z ) )
68	52, 62, 64, 67	mpjao3dan 1395	. . . . . . 7 |- ( z e. RR+ -> E. x e. RR ( exp ` x ) = z )
69		fvres 6207	. . . . . . . . 9 |- ( x e. RR -> ( ( exp |` RR ) ` x ) = ( exp ` x ) )
70	69	eqeq1d 2624	. . . . . . . 8 |- ( x e. RR -> ( ( ( exp |` RR ) ` x ) = z <-> ( exp ` x ) = z ) )
71	70	rexbiia 3040	. . . . . . 7 |- ( E. x e. RR ( ( exp |` RR ) ` x ) = z <-> E. x e. RR ( exp ` x ) = z )
72	68, 71	sylibr 224	. . . . . 6 |- ( z e. RR+ -> E. x e. RR ( ( exp |` RR ) ` x ) = z )
73		fvelrnb 6243	. . . . . . 7 |- ( ( exp |` RR ) Fn RR -> ( z e. ran ( exp |` RR ) <-> E. x e. RR ( ( exp |` RR ) ` x ) = z ) )
74	4, 73	ax-mp 5	. . . . . 6 |- ( z e. ran ( exp |` RR ) <-> E. x e. RR ( ( exp |` RR ) ` x ) = z )
75	72, 74	sylibr 224	. . . . 5 |- ( z e. RR+ -> z e. ran ( exp |` RR ) )
76	75	ssriv 3607	. . . 4 |- RR+ C_ ran ( exp |` RR )
77	6, 76	eqssi 3619	. . 3 |- ran ( exp |` RR ) = RR+
78		df-fo 5894	. . 3 |- ( ( exp |` RR ) : RR -onto-> RR+ <-> ( ( exp |` RR ) Fn RR /\ ran ( exp |` RR ) = RR+ ) )
79	4, 77, 78	mpbir2an 955	. 2 |- ( exp |` RR ) : RR -onto-> RR+
80		df-f1o 5895	. 2 |- ( ( exp |` RR ) : RR -1-1-onto-> RR+ <-> ( ( exp |` RR ) : RR -1-1-> RR+ /\ ( exp |` RR ) : RR -onto-> RR+ ) )
81	1, 79, 80	mpbir2an 955	1 |- ( exp |` RR ) : RR -1-1-onto-> RR+

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   class class class wbr 4653   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   -ucneg 10267   / cdiv 10684   RR+crp 11832   expce 14792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  reefiso  24202  efcvx  24203  reefgim  24204  eff1olem  24294  dfrelog  24312  relogf1o  24313  dvrelog  24383