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Theorem reeff1o 22571
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.
StepHypRef Expression
1 reeff1 13707 . 2  |-  ( exp  |`  RR ) : RR -1-1-> RR+
2 f1f 5774 . . . 4  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
3 ffn 5724 . . . 4  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR )  Fn  RR )
41, 2, 3mp2b 10 . . 3  |-  ( exp  |`  RR )  Fn  RR
5 frn 5730 . . . . 5  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ran  ( exp  |`  RR )  C_  RR+ )
61, 2, 5mp2b 10 . . . 4  |-  ran  ( exp  |`  RR )  C_  RR+
7 rpre 11217 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e.  RR )
8 1re 9586 . . . . . . . . 9  |-  1  e.  RR
9 lttri4 9660 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
107, 8, 9sylancl 662 . . . . . . . 8  |-  ( z  e.  RR+  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
11 elrp 11213 . . . . . . . . . . . 12  |-  ( z  e.  RR+  <->  ( z  e.  RR  /\  0  < 
z ) )
12 reclt1 10431 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  0  <  z )  -> 
( z  <  1  <->  1  <  ( 1  / 
z ) ) )
1311, 12sylbi 195 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( z  <  1  <->  1  <  ( 1  /  z ) ) )
14 rpne0 11226 . . . . . . . . . . . . . . . 16  |-  ( z  e.  RR+  ->  z  =/=  0 )
157, 14rereccld 10362 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( 1  /  z )  e.  RR )
16 reeff1olem 22570 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  z
)  e.  RR  /\  1  <  ( 1  / 
z ) )  ->  E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z ) )
1715, 16sylan 471 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  y
)  =  ( 1  /  z ) )
18 eqcom 2471 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  /  z )  =  ( exp `  y
)  <->  ( exp `  y
)  =  ( 1  /  z ) )
19 rpcnne0 11228 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  RR+  ->  ( z  e.  CC  /\  z  =/=  0 ) )
20 recn 9573 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  RR  ->  y  e.  CC )
21 efcl 13671 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2220, 21syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  e.  CC )
23 efne0 13684 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  =/=  0 )
2420, 23syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  =/=  0 )
2522, 24jca 532 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )
26 rec11r 10234 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( z  e.  CC  /\  z  =/=  0 )  /\  ( ( exp `  y )  e.  CC  /\  ( exp `  y
)  =/=  0 ) )  ->  ( (
1  /  z )  =  ( exp `  y
)  <->  ( 1  / 
( exp `  y
) )  =  z ) )
2719, 25, 26syl2an 477 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( 1  /  ( exp `  y
) )  =  z ) )
28 efcan 13683 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  (
( exp `  y
)  x.  ( exp `  -u y ) )  =  1 )
2928eqcomd 2470 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  1  =  ( ( exp `  y )  x.  ( exp `  -u y ) ) )
30 negcl 9811 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  e.  CC  ->  -u y  e.  CC )
31 efcl 13671 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( -u y  e.  CC  ->  ( exp `  -u y
)  e.  CC )
3230, 31syl 16 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  ( exp `  -u y )  e.  CC )
33 ax-1cn 9541 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  CC
34 divmul2 10202 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( 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 ) ) ) )
3533, 34mp3an1 1306 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 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 ) ) ) )
3632, 21, 23, 35syl12anc 1221 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  (
( 1  /  ( exp `  y ) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3729, 36mpbird 232 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3820, 37syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3938eqeq1d 2464 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4039adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4127, 40bitrd 253 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( exp `  -u y )  =  z ) )
4218, 41syl5bbr 259 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  <->  ( exp `  -u y )  =  z ) )
4342biimpd 207 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  -> 
( exp `  -u y
)  =  z ) )
4443reximdva 2933 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  /  z
)  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4544adantr 465 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4617, 45mpd 15 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z )
47 renegcl 9873 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  -u y  e.  RR )
48 infm3lem 10492 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
49 fveq2 5859 . . . . . . . . . . . . . . 15  |-  ( x  =  -u y  ->  ( exp `  x )  =  ( exp `  -u y
) )
5049eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( x  =  -u y  ->  (
( exp `  x
)  =  z  <->  ( exp `  -u y )  =  z ) )
5147, 48, 50rexxfr 4662 . . . . . . . . . . . . 13  |-  ( E. x  e.  RR  ( exp `  x )  =  z  <->  E. y  e.  RR  ( exp `  -u y
)  =  z )
5246, 51sylibr 212 . . . . . . . . . . . 12  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
5352ex 434 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( 1  <  ( 1  / 
z )  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5413, 53sylbid 215 . . . . . . . . . 10  |-  ( z  e.  RR+  ->  ( z  <  1  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5554imp 429 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  <  1 )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
56 ef0 13679 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
5756eqeq2i 2480 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  <->  z  =  1 )
58 0re 9587 . . . . . . . . . . . . 13  |-  0  e.  RR
59 fveq2 5859 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
6059eqeq1d 2464 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  (
( exp `  x
)  =  z  <->  ( exp `  0 )  =  z ) )
6160rspcev 3209 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( exp `  0 )  =  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6258, 61mpan 670 . . . . . . . . . . . 12  |-  ( ( exp `  0 )  =  z  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6362eqcoms 2474 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6457, 63sylbir 213 . . . . . . . . . 10  |-  ( z  =  1  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6564adantl 466 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  =  1 )  ->  E. x  e.  RR  ( exp `  x )  =  z )
66 reeff1olem 22570 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
677, 66sylan 471 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6855, 65, 673jaodan 1289 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  (
z  <  1  \/  z  =  1  \/  1  <  z ) )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6910, 68mpdan 668 . . . . . . 7  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( exp `  x
)  =  z )
70 fvres 5873 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
7170eqeq1d 2464 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( ( exp  |`  RR ) `
 x )  =  z  <->  ( exp `  x
)  =  z ) )
7271rexbiia 2959 . . . . . . 7  |-  ( E. x  e.  RR  (
( exp  |`  RR ) `
 x )  =  z  <->  E. x  e.  RR  ( exp `  x )  =  z )
7369, 72sylibr 212 . . . . . 6  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
74 fvelrnb 5908 . . . . . . 7  |-  ( ( exp  |`  RR )  Fn  RR  ->  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `
 x )  =  z ) )
754, 74ax-mp 5 . . . . . 6  |-  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
7673, 75sylibr 212 . . . . 5  |-  ( z  e.  RR+  ->  z  e. 
ran  ( exp  |`  RR ) )
7776ssriv 3503 . . . 4  |-  RR+  C_  ran  ( exp  |`  RR )
786, 77eqssi 3515 . . 3  |-  ran  ( exp  |`  RR )  = 
RR+
79 df-fo 5587 . . 3  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
ran  ( exp  |`  RR )  =  RR+ ) )
804, 78, 79mpbir2an 913 . 2  |-  ( exp  |`  RR ) : RR -onto-> RR+
81 df-f1o 5588 . 2  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  <->  ( ( exp  |`  RR ) : RR -1-1-> RR+ 
/\  ( exp  |`  RR ) : RR -onto-> RR+ )
)
821, 80, 81mpbir2an 913 1  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810    C_ wss 3471   class class class wbr 4442   ran crn 4995    |` cres 4996    Fn wfn 5576   -->wf 5577   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    < clt 9619   -ucneg 9797    / cdiv 10197   RR+crp 11211   expce 13650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001
This theorem is referenced by:  reefiso  22572  efcvx  22573  reefgim  22574  eff1olem  22663  dfrelog  22676  relogf1o  22677  dvrelog  22741
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