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Theorem reeff1o 20363
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 12721 . 2  |-  ( exp  |`  RR ) : RR -1-1-> RR+
2 f1f 5639 . . . 4  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
3 ffn 5591 . . . 4  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR )  Fn  RR )
41, 2, 3mp2b 10 . . 3  |-  ( exp  |`  RR )  Fn  RR
5 frn 5597 . . . . 5  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ran  ( exp  |`  RR )  C_  RR+ )
61, 2, 5mp2b 10 . . . 4  |-  ran  ( exp  |`  RR )  C_  RR+
7 rpre 10618 . . . . . . . . 9  |-  ( z  e.  RR+  ->  z  e.  RR )
8 1re 9090 . . . . . . . . 9  |-  1  e.  RR
9 lttri4 9159 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
107, 8, 9sylancl 644 . . . . . . . 8  |-  ( z  e.  RR+  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
11 elrp 10614 . . . . . . . . . . . 12  |-  ( z  e.  RR+  <->  ( z  e.  RR  /\  0  < 
z ) )
12 reclt1 9905 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  0  <  z )  -> 
( z  <  1  <->  1  <  ( 1  / 
z ) ) )
1311, 12sylbi 188 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( z  <  1  <->  1  <  ( 1  /  z ) ) )
14 rpne0 10627 . . . . . . . . . . . . . . . 16  |-  ( z  e.  RR+  ->  z  =/=  0 )
157, 14rereccld 9841 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( 1  /  z )  e.  RR )
16 reeff1olem 20362 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  /  z
)  e.  RR  /\  1  <  ( 1  / 
z ) )  ->  E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z ) )
1715, 16sylan 458 . . . . . . . . . . . . . 14  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  y
)  =  ( 1  /  z ) )
18 eqcom 2438 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  /  z )  =  ( exp `  y
)  <->  ( exp `  y
)  =  ( 1  /  z ) )
19 rpcnne0 10629 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  RR+  ->  ( z  e.  CC  /\  z  =/=  0 ) )
20 recn 9080 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  RR  ->  y  e.  CC )
21 efcl 12685 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
2220, 21syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  e.  CC )
23 efne0 12698 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  ( exp `  y )  =/=  0 )
2420, 23syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  ( exp `  y )  =/=  0 )
2522, 24jca 519 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )
26 rec11r 9713 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( z  e.  CC  /\  z  =/=  0 )  /\  ( ( exp `  y )  e.  CC  /\  ( exp `  y
)  =/=  0 ) )  ->  ( (
1  /  z )  =  ( exp `  y
)  <->  ( 1  / 
( exp `  y
) )  =  z ) )
2719, 25, 26syl2an 464 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( 1  /  ( exp `  y
) )  =  z ) )
28 efcan 12697 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  (
( exp `  y
)  x.  ( exp `  -u y ) )  =  1 )
2928eqcomd 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  1  =  ( ( exp `  y )  x.  ( exp `  -u y ) ) )
30 negcl 9306 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( y  e.  CC  ->  -u y  e.  CC )
31 efcl 12685 . . . . . . . . . . . . . . . . . . . . . . . . 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 9048 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  CC
34 divmul2 9682 . . . . . . . . . . . . . . . . . . . . . . . . 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 1266 . . . . . . . . . . . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  (
( 1  /  ( exp `  y ) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3729, 36mpbird 224 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3820, 37syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3938eqeq1d 2444 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4039adantl 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
4127, 40bitrd 245 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( exp `  -u y )  =  z ) )
4218, 41syl5bbr 251 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  <->  ( exp `  -u y )  =  z ) )
4342biimpd 199 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  -> 
( exp `  -u y
)  =  z ) )
4443reximdva 2818 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  /  z
)  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4544adantr 452 . . . . . . . . . . . . . 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 9364 . . . . . . . . . . . . . 14  |-  ( y  e.  RR  ->  -u y  e.  RR )
48 infm3lem 9966 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
49 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( x  =  -u y  ->  ( exp `  x )  =  ( exp `  -u y
) )
5049eqeq1d 2444 . . . . . . . . . . . . . 14  |-  ( x  =  -u y  ->  (
( exp `  x
)  =  z  <->  ( exp `  -u y )  =  z ) )
5147, 48, 50rexxfr 4743 . . . . . . . . . . . . 13  |-  ( E. x  e.  RR  ( exp `  x )  =  z  <->  E. y  e.  RR  ( exp `  -u y
)  =  z )
5246, 51sylibr 204 . . . . . . . . . . . 12  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
5352ex 424 . . . . . . . . . . 11  |-  ( z  e.  RR+  ->  ( 1  <  ( 1  / 
z )  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5413, 53sylbid 207 . . . . . . . . . 10  |-  ( z  e.  RR+  ->  ( z  <  1  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5554imp 419 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  <  1 )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
56 ef0 12693 . . . . . . . . . . . 12  |-  ( exp `  0 )  =  1
5756eqeq2i 2446 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  <->  z  =  1 )
58 0re 9091 . . . . . . . . . . . . 13  |-  0  e.  RR
59 fveq2 5728 . . . . . . . . . . . . . . 15  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
6059eqeq1d 2444 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  (
( exp `  x
)  =  z  <->  ( exp `  0 )  =  z ) )
6160rspcev 3052 . . . . . . . . . . . . 13  |-  ( ( 0  e.  RR  /\  ( exp `  0 )  =  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6258, 61mpan 652 . . . . . . . . . . . 12  |-  ( ( exp `  0 )  =  z  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6362eqcoms 2439 . . . . . . . . . . 11  |-  ( z  =  ( exp `  0
)  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6457, 63sylbir 205 . . . . . . . . . 10  |-  ( z  =  1  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6564adantl 453 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  z  =  1 )  ->  E. x  e.  RR  ( exp `  x )  =  z )
66 reeff1olem 20362 . . . . . . . . . 10  |-  ( ( z  e.  RR  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
677, 66sylan 458 . . . . . . . . 9  |-  ( ( z  e.  RR+  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6855, 65, 673jaodan 1250 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  (
z  <  1  \/  z  =  1  \/  1  <  z ) )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6910, 68mpdan 650 . . . . . . 7  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( exp `  x
)  =  z )
70 fvres 5745 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
7170eqeq1d 2444 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( ( exp  |`  RR ) `
 x )  =  z  <->  ( exp `  x
)  =  z ) )
7271rexbiia 2738 . . . . . . 7  |-  ( E. x  e.  RR  (
( exp  |`  RR ) `
 x )  =  z  <->  E. x  e.  RR  ( exp `  x )  =  z )
7369, 72sylibr 204 . . . . . 6  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
74 fvelrnb 5774 . . . . . . 7  |-  ( ( exp  |`  RR )  Fn  RR  ->  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `
 x )  =  z ) )
754, 74ax-mp 8 . . . . . 6  |-  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
7673, 75sylibr 204 . . . . 5  |-  ( z  e.  RR+  ->  z  e. 
ran  ( exp  |`  RR ) )
7776ssriv 3352 . . . 4  |-  RR+  C_  ran  ( exp  |`  RR )
786, 77eqssi 3364 . . 3  |-  ran  ( exp  |`  RR )  = 
RR+
79 df-fo 5460 . . 3  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
ran  ( exp  |`  RR )  =  RR+ ) )
804, 78, 79mpbir2an 887 . 2  |-  ( exp  |`  RR ) : RR -onto-> RR+
81 df-f1o 5461 . 2  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  <->  ( ( exp  |`  RR ) : RR -1-1-> RR+ 
/\  ( exp  |`  RR ) : RR -onto-> RR+ )
)
821, 80, 81mpbir2an 887 1  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320   class class class wbr 4212   ran crn 4879    |` cres 4880    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120   -ucneg 9292    / cdiv 9677   RR+crp 10612   expce 12664
This theorem is referenced by:  reefiso  20364  efcvx  20365  reefgim  20366  eff1olem  20450  dfrelog  20463  relogf1o  20464  dvrelog  20528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-fi 7416  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-ioo 10920  df-ico 10922  df-icc 10923  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-rest 13650  df-topn 13651  df-topgen 13667  df-pt 13668  df-prds 13671  df-xrs 13726  df-0g 13727  df-gsum 13728  df-qtop 13733  df-imas 13734  df-xps 13736  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-mulg 14815  df-cntz 15116  df-cmn 15414  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-fbas 16699  df-fg 16700  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-topsp 16967  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-lp 17200  df-perf 17201  df-cn 17291  df-cnp 17292  df-haus 17379  df-tx 17594  df-hmeo 17787  df-fil 17878  df-fm 17970  df-flim 17971  df-flf 17972  df-xms 18350  df-ms 18351  df-tms 18352  df-cncf 18908  df-limc 19753  df-dv 19754
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