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Theorem reeff1o 23136
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 14066 . 2  |-  ( exp  |`  RR ) : RR -1-1-> RR+
2 f1f 5766 . . . 4  |-  ( ( exp  |`  RR ) : RR -1-1-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
3 ffn 5716 . . . 4  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ( exp  |`  RR )  Fn  RR )
41, 2, 3mp2b 10 . . 3  |-  ( exp  |`  RR )  Fn  RR
5 frn 5722 . . . . 5  |-  ( ( exp  |`  RR ) : RR --> RR+  ->  ran  ( exp  |`  RR )  C_  RR+ )
61, 2, 5mp2b 10 . . . 4  |-  ran  ( exp  |`  RR )  C_  RR+
7 elrp 11269 . . . . . . . . . . 11  |-  ( z  e.  RR+  <->  ( z  e.  RR  /\  0  < 
z ) )
8 reclt1 10482 . . . . . . . . . . 11  |-  ( ( z  e.  RR  /\  0  <  z )  -> 
( z  <  1  <->  1  <  ( 1  / 
z ) ) )
97, 8sylbi 197 . . . . . . . . . 10  |-  ( z  e.  RR+  ->  ( z  <  1  <->  1  <  ( 1  /  z ) ) )
10 rpre 11273 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  z  e.  RR )
11 rpne0 11282 . . . . . . . . . . . . . . 15  |-  ( z  e.  RR+  ->  z  =/=  0 )
1210, 11rereccld 10414 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  ( 1  /  z )  e.  RR )
13 reeff1olem 23135 . . . . . . . . . . . . . 14  |-  ( ( ( 1  /  z
)  e.  RR  /\  1  <  ( 1  / 
z ) )  ->  E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z ) )
1412, 13sylan 471 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  y
)  =  ( 1  /  z ) )
15 eqcom 2413 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  /  z )  =  ( exp `  y
)  <->  ( exp `  y
)  =  ( 1  /  z ) )
16 rpcnne0 11284 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  RR+  ->  ( z  e.  CC  /\  z  =/=  0 ) )
17 recn 9614 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  RR  ->  y  e.  CC )
18 efcl 14029 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  CC  ->  ( exp `  y )  e.  CC )
1917, 18syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  ( exp `  y )  e.  CC )
20 efne0 14043 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  CC  ->  ( exp `  y )  =/=  0 )
2117, 20syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  ( exp `  y )  =/=  0 )
2219, 21jca 532 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  RR  ->  (
( exp `  y
)  e.  CC  /\  ( exp `  y )  =/=  0 ) )
23 rec11r 10286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( z  e.  CC  /\  z  =/=  0 )  /\  ( ( exp `  y )  e.  CC  /\  ( exp `  y
)  =/=  0 ) )  ->  ( (
1  /  z )  =  ( exp `  y
)  <->  ( 1  / 
( exp `  y
) )  =  z ) )
2416, 22, 23syl2an 477 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( 1  /  ( exp `  y
) )  =  z ) )
25 efcan 14042 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  (
( exp `  y
)  x.  ( exp `  -u y ) )  =  1 )
2625eqcomd 2412 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  1  =  ( ( exp `  y )  x.  ( exp `  -u y ) ) )
27 negcl 9858 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  CC  ->  -u y  e.  CC )
28 efcl 14029 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -u y  e.  CC  ->  ( exp `  -u y
)  e.  CC )
2927, 28syl 17 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  CC  ->  ( exp `  -u y )  e.  CC )
30 ax-1cn 9582 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  CC
31 divmul2 10254 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
3230, 31mp3an1 1315 . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
3329, 18, 20, 32syl12anc 1230 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  CC  ->  (
( 1  /  ( exp `  y ) )  =  ( exp `  -u y
)  <->  1  =  ( ( exp `  y
)  x.  ( exp `  -u y ) ) ) )
3426, 33mpbird 234 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  CC  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3517, 34syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  RR  ->  (
1  /  ( exp `  y ) )  =  ( exp `  -u y
) )
3635eqeq1d 2406 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  RR  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
3736adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  ( exp `  y ) )  =  z  <->  ( exp `  -u y )  =  z ) )
3824, 37bitrd 255 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( 1  /  z
)  =  ( exp `  y )  <->  ( exp `  -u y )  =  z ) )
3915, 38syl5bbr 261 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  <->  ( exp `  -u y )  =  z ) )
4039biimpd 209 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  RR+  /\  y  e.  RR )  ->  (
( exp `  y
)  =  ( 1  /  z )  -> 
( exp `  -u y
)  =  z ) )
4140reximdva 2881 . . . . . . . . . . . . . 14  |-  ( z  e.  RR+  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  /  z
)  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4241adantr 465 . . . . . . . . . . . . 13  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  ( E. y  e.  RR  ( exp `  y )  =  ( 1  / 
z )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z ) )
4314, 42mpd 15 . . . . . . . . . . . 12  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. y  e.  RR  ( exp `  -u y
)  =  z )
44 renegcl 9920 . . . . . . . . . . . . 13  |-  ( y  e.  RR  ->  -u y  e.  RR )
45 infm3lem 10543 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  E. y  e.  RR  x  =  -u y )
46 fveq2 5851 . . . . . . . . . . . . . 14  |-  ( x  =  -u y  ->  ( exp `  x )  =  ( exp `  -u y
) )
4746eqeq1d 2406 . . . . . . . . . . . . 13  |-  ( x  =  -u y  ->  (
( exp `  x
)  =  z  <->  ( exp `  -u y )  =  z ) )
4844, 45, 47rexxfr 4613 . . . . . . . . . . . 12  |-  ( E. x  e.  RR  ( exp `  x )  =  z  <->  E. y  e.  RR  ( exp `  -u y
)  =  z )
4943, 48sylibr 214 . . . . . . . . . . 11  |-  ( ( z  e.  RR+  /\  1  <  ( 1  /  z
) )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
5049ex 434 . . . . . . . . . 10  |-  ( z  e.  RR+  ->  ( 1  <  ( 1  / 
z )  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
519, 50sylbid 217 . . . . . . . . 9  |-  ( z  e.  RR+  ->  ( z  <  1  ->  E. x  e.  RR  ( exp `  x
)  =  z ) )
5251imp 429 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  z  <  1 )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
53 ef0 14037 . . . . . . . . . . 11  |-  ( exp `  0 )  =  1
5453eqeq2i 2422 . . . . . . . . . 10  |-  ( z  =  ( exp `  0
)  <->  z  =  1 )
55 0re 9628 . . . . . . . . . . . 12  |-  0  e.  RR
56 fveq2 5851 . . . . . . . . . . . . . 14  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
5756eqeq1d 2406 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
( exp `  x
)  =  z  <->  ( exp `  0 )  =  z ) )
5857rspcev 3162 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  ( exp `  0 )  =  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
5955, 58mpan 670 . . . . . . . . . . 11  |-  ( ( exp `  0 )  =  z  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6059eqcoms 2416 . . . . . . . . . 10  |-  ( z  =  ( exp `  0
)  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6154, 60sylbir 215 . . . . . . . . 9  |-  ( z  =  1  ->  E. x  e.  RR  ( exp `  x
)  =  z )
6261adantl 466 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  z  =  1 )  ->  E. x  e.  RR  ( exp `  x )  =  z )
63 reeff1olem 23135 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x )  =  z )
6410, 63sylan 471 . . . . . . . 8  |-  ( ( z  e.  RR+  /\  1  <  z )  ->  E. x  e.  RR  ( exp `  x
)  =  z )
65 1re 9627 . . . . . . . . 9  |-  1  e.  RR
66 lttri4 9702 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  1  e.  RR )  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
6710, 65, 66sylancl 662 . . . . . . . 8  |-  ( z  e.  RR+  ->  ( z  <  1  \/  z  =  1  \/  1  <  z ) )
6852, 62, 64, 67mpjao3dan 1299 . . . . . . 7  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( exp `  x
)  =  z )
69 fvres 5865 . . . . . . . . 9  |-  ( x  e.  RR  ->  (
( exp  |`  RR ) `
 x )  =  ( exp `  x
) )
7069eqeq1d 2406 . . . . . . . 8  |-  ( x  e.  RR  ->  (
( ( exp  |`  RR ) `
 x )  =  z  <->  ( exp `  x
)  =  z ) )
7170rexbiia 2907 . . . . . . 7  |-  ( E. x  e.  RR  (
( exp  |`  RR ) `
 x )  =  z  <->  E. x  e.  RR  ( exp `  x )  =  z )
7268, 71sylibr 214 . . . . . 6  |-  ( z  e.  RR+  ->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
73 fvelrnb 5898 . . . . . . 7  |-  ( ( exp  |`  RR )  Fn  RR  ->  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `
 x )  =  z ) )
744, 73ax-mp 5 . . . . . 6  |-  ( z  e.  ran  ( exp  |`  RR )  <->  E. x  e.  RR  ( ( exp  |`  RR ) `  x
)  =  z )
7572, 74sylibr 214 . . . . 5  |-  ( z  e.  RR+  ->  z  e. 
ran  ( exp  |`  RR ) )
7675ssriv 3448 . . . 4  |-  RR+  C_  ran  ( exp  |`  RR )
776, 76eqssi 3460 . . 3  |-  ran  ( exp  |`  RR )  = 
RR+
78 df-fo 5577 . . 3  |-  ( ( exp  |`  RR ) : RR -onto-> RR+  <->  ( ( exp  |`  RR )  Fn  RR  /\ 
ran  ( exp  |`  RR )  =  RR+ ) )
794, 77, 78mpbir2an 923 . 2  |-  ( exp  |`  RR ) : RR -onto-> RR+
80 df-f1o 5578 . 2  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  <->  ( ( exp  |`  RR ) : RR -1-1-> RR+ 
/\  ( exp  |`  RR ) : RR -onto-> RR+ )
)
811, 79, 80mpbir2an 923 1  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    \/ w3o 975    = wceq 1407    e. wcel 1844    =/= wne 2600   E.wrex 2757    C_ wss 3416   class class class wbr 4397   ran crn 4826    |` cres 4827    Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   -onto->wfo 5569   -1-1-onto->wf1o 5570   ` cfv 5571  (class class class)co 6280   CCcc 9522   RRcr 9523   0cc0 9524   1c1 9525    x. cmul 9529    < clt 9660   -ucneg 9844    / cdiv 10249   RR+crp 11267   expce 14008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-pre-sup 9602  ax-addf 9603  ax-mulf 9604
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-fal 1413  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-om 6686  df-1st 6786  df-2nd 6787  df-supp 6905  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-ixp 7510  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fsupp 7866  df-fi 7907  df-sup 7937  df-oi 7971  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-q 11230  df-rp 11268  df-xneg 11373  df-xadd 11374  df-xmul 11375  df-ioo 11588  df-ico 11590  df-icc 11591  df-fz 11729  df-fzo 11857  df-fl 11968  df-seq 12154  df-exp 12213  df-fac 12400  df-bc 12427  df-hash 12455  df-shft 13051  df-cj 13083  df-re 13084  df-im 13085  df-sqrt 13219  df-abs 13220  df-limsup 13445  df-clim 13462  df-rlim 13463  df-sum 13660  df-ef 14014  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-sca 14927  df-vsca 14928  df-ip 14929  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-hom 14935  df-cco 14936  df-rest 15039  df-topn 15040  df-0g 15058  df-gsum 15059  df-topgen 15060  df-pt 15061  df-prds 15064  df-xrs 15118  df-qtop 15123  df-imas 15124  df-xps 15126  df-mre 15202  df-mrc 15203  df-acs 15205  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-submnd 16293  df-mulg 16386  df-cntz 16681  df-cmn 17126  df-psmet 18733  df-xmet 18734  df-met 18735  df-bl 18736  df-mopn 18737  df-fbas 18738  df-fg 18739  df-cnfld 18743  df-top 19693  df-bases 19695  df-topon 19696  df-topsp 19697  df-cld 19814  df-ntr 19815  df-cls 19816  df-nei 19894  df-lp 19932  df-perf 19933  df-cn 20023  df-cnp 20024  df-haus 20111  df-tx 20357  df-hmeo 20550  df-fil 20641  df-fm 20733  df-flim 20734  df-flf 20735  df-xms 21117  df-ms 21118  df-tms 21119  df-cncf 21676  df-limc 22564  df-dv 22565
This theorem is referenced by:  reefiso  23137  efcvx  23138  reefgim  23139  eff1olem  23229  dfrelog  23247  relogf1o  23248  dvrelog  23314
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