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Theorem iooref1o 14438
Description: A one-to-one mapping from the real numbers onto the open unit interval. (Contributed by Jim Kingdon, 27-Jun-2024.)
Hypothesis
Ref Expression
iooref1o.f  |-  F  =  ( x  e.  RR  |->  ( 1  /  (
1  +  ( exp `  x ) ) ) )
Assertion
Ref Expression
iooref1o  |-  F : RR
-1-1-onto-> ( 0 (,) 1
)

Proof of Theorem iooref1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iooref1o.f . . 3  |-  F  =  ( x  e.  RR  |->  ( 1  /  (
1  +  ( exp `  x ) ) ) )
2 1rp 9644 . . . . . . . . 9  |-  1  e.  RR+
32a1i 9 . . . . . . . 8  |-  ( x  e.  RR  ->  1  e.  RR+ )
4 rpefcl 11677 . . . . . . . 8  |-  ( x  e.  RR  ->  ( exp `  x )  e.  RR+ )
53, 4rpaddcld 9699 . . . . . . 7  |-  ( x  e.  RR  ->  (
1  +  ( exp `  x ) )  e.  RR+ )
65rpreccld 9694 . . . . . 6  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR+ )
76rpred 9683 . . . . 5  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR )
86rpgt0d 9686 . . . . 5  |-  ( x  e.  RR  ->  0  <  ( 1  /  (
1  +  ( exp `  x ) ) ) )
9 1red 7963 . . . . . . 7  |-  ( x  e.  RR  ->  1  e.  RR )
109, 4ltaddrpd 9717 . . . . . 6  |-  ( x  e.  RR  ->  1  <  ( 1  +  ( exp `  x ) ) )
115recgt1d 9698 . . . . . 6  |-  ( x  e.  RR  ->  (
1  <  ( 1  +  ( exp `  x
) )  <->  ( 1  /  ( 1  +  ( exp `  x
) ) )  <  1 ) )
1210, 11mpbid 147 . . . . 5  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  <  1 )
13 0xr 7994 . . . . . 6  |-  0  e.  RR*
14 1re 7947 . . . . . . 7  |-  1  e.  RR
1514rexri 8005 . . . . . 6  |-  1  e.  RR*
16 elioo2 9908 . . . . . 6  |-  ( ( 0  e.  RR*  /\  1  e.  RR* )  ->  (
( 1  /  (
1  +  ( exp `  x ) ) )  e.  ( 0 (,) 1 )  <->  ( (
1  /  ( 1  +  ( exp `  x
) ) )  e.  RR  /\  0  < 
( 1  /  (
1  +  ( exp `  x ) ) )  /\  ( 1  / 
( 1  +  ( exp `  x ) ) )  <  1
) ) )
1713, 15, 16mp2an 426 . . . . 5  |-  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
)  <->  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  e.  RR  /\  0  < 
( 1  /  (
1  +  ( exp `  x ) ) )  /\  ( 1  / 
( 1  +  ( exp `  x ) ) )  <  1
) )
187, 8, 12, 17syl3anbrc 1181 . . . 4  |-  ( x  e.  RR  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
) )
1918adantl 277 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  (
1  /  ( 1  +  ( exp `  x
) ) )  e.  ( 0 (,) 1
) )
20 elioore 9899 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR )
21 eliooord 9915 . . . . . . . . . . 11  |-  ( y  e.  ( 0 (,) 1 )  ->  (
0  <  y  /\  y  <  1 ) )
2221simpld 112 . . . . . . . . . 10  |-  ( y  e.  ( 0 (,) 1 )  ->  0  <  y )
2320, 22elrpd 9680 . . . . . . . . 9  |-  ( y  e.  ( 0 (,) 1 )  ->  y  e.  RR+ )
2423rpreccld 9694 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR+ )
2524rpred 9683 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  /  y )  e.  RR )
26 1red 7963 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  e.  RR )
2725, 26resubcld 8328 . . . . . 6  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR )
2821simprd 114 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  y  <  1 )
2923reclt1d 9697 . . . . . . . 8  |-  ( y  e.  ( 0 (,) 1 )  ->  (
y  <  1  <->  1  <  ( 1  /  y ) ) )
3028, 29mpbid 147 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  1  <  ( 1  /  y
) )
3126, 25posdifd 8479 . . . . . . 7  |-  ( y  e.  ( 0 (,) 1 )  ->  (
1  <  ( 1  /  y )  <->  0  <  ( ( 1  /  y
)  -  1 ) ) )
3230, 31mpbid 147 . . . . . 6  |-  ( y  e.  ( 0 (,) 1 )  ->  0  <  ( ( 1  / 
y )  -  1 ) )
3327, 32elrpd 9680 . . . . 5  |-  ( y  e.  ( 0 (,) 1 )  ->  (
( 1  /  y
)  -  1 )  e.  RR+ )
3433relogcld 13970 . . . 4  |-  ( y  e.  ( 0 (,) 1 )  ->  ( log `  ( ( 1  /  y )  - 
1 ) )  e.  RR )
3534adantl 277 . . 3  |-  ( ( T.  /\  y  e.  ( 0 (,) 1
) )  ->  ( log `  ( ( 1  /  y )  - 
1 ) )  e.  RR )
36 1cnd 7964 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  1  e.  CC )
374adantr 276 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  x
)  e.  RR+ )
3837rpcnd 9685 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  x
)  e.  CC )
3936, 38addcld 7967 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) )  e.  CC )
4023adantl 277 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y  e.  RR+ )
4140rpcnd 9685 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y  e.  CC )
4240rpap0d 9689 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  y #  0 )
4336, 39, 41, 42divmulap2d 8770 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  y )  =  ( 1  +  ( exp `  x ) )  <->  1  =  ( y  x.  ( 1  +  ( exp `  x
) ) ) ) )
4424adantl 277 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  / 
y )  e.  RR+ )
4544rpcnd 9685 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  / 
y )  e.  CC )
4636, 38, 45addrsub 8318 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  +  ( exp `  x
) )  =  ( 1  /  y )  <-> 
( exp `  x
)  =  ( ( 1  /  y )  -  1 ) ) )
4733adantl 277 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  y )  - 
1 )  e.  RR+ )
4847reeflogd 13971 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( exp `  ( log `  ( ( 1  /  y )  - 
1 ) ) )  =  ( ( 1  /  y )  - 
1 ) )
4948eqeq2d 2189 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( exp `  x )  =  ( exp `  ( log `  ( ( 1  / 
y )  -  1 ) ) )  <->  ( exp `  x )  =  ( ( 1  /  y
)  -  1 ) ) )
50 reef11 11691 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  ( log `  ( ( 1  /  y )  -  1 ) )  e.  RR )  -> 
( ( exp `  x
)  =  ( exp `  ( log `  (
( 1  /  y
)  -  1 ) ) )  <->  x  =  ( log `  ( ( 1  /  y )  -  1 ) ) ) )
5134, 50sylan2 286 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( exp `  x )  =  ( exp `  ( log `  ( ( 1  / 
y )  -  1 ) ) )  <->  x  =  ( log `  ( ( 1  /  y )  -  1 ) ) ) )
5246, 49, 513bitr2rd 217 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  +  ( exp `  x
) )  =  ( 1  /  y ) ) )
53 eqcom 2179 . . . . . . 7  |-  ( ( 1  +  ( exp `  x ) )  =  ( 1  /  y
)  <->  ( 1  / 
y )  =  ( 1  +  ( exp `  x ) ) )
5452, 53bitrdi 196 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  / 
y )  =  ( 1  +  ( exp `  x ) ) ) )
555adantr 276 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) )  e.  RR+ )
5655rpap0d 9689 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( 1  +  ( exp `  x
) ) #  0 )
5736, 41, 39, 56divmulap3d 8771 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  =  y  <->  1  =  ( y  x.  ( 1  +  ( exp `  x
) ) ) ) )
5843, 54, 573bitr4d 220 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  ( 1  / 
( 1  +  ( exp `  x ) ) )  =  y ) )
59 eqcom 2179 . . . . 5  |-  ( ( 1  /  ( 1  +  ( exp `  x
) ) )  =  y  <->  y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) )
6058, 59bitrdi 196 . . . 4  |-  ( ( x  e.  RR  /\  y  e.  ( 0 (,) 1 ) )  ->  ( x  =  ( log `  (
( 1  /  y
)  -  1 ) )  <->  y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) ) )
6160adantl 277 . . 3  |-  ( ( T.  /\  ( x  e.  RR  /\  y  e.  ( 0 (,) 1
) ) )  -> 
( x  =  ( log `  ( ( 1  /  y )  -  1 ) )  <-> 
y  =  ( 1  /  ( 1  +  ( exp `  x
) ) ) ) )
621, 19, 35, 61f1o2d 6070 . 2  |-  ( T. 
->  F : RR -1-1-onto-> ( 0 (,) 1
) )
6362mptru 1362 1  |-  F : RR
-1-1-onto-> ( 0 (,) 1
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   T. wtru 1354    e. wcel 2148   class class class wbr 4000    |-> cmpt 4061   -1-1-onto->wf1o 5211   ` cfv 5212  (class class class)co 5869   RRcr 7801   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807   RR*cxr 7981    < clt 7982    - cmin 8118    / cdiv 8618   RR+crp 9640   (,)cioo 9875   expce 11634   logclog 13944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922  ax-pre-suploc 7923  ax-addf 7924  ax-mulf 7925
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-disj 3978  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-of 6077  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-map 6644  df-pm 6645  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-ioo 9879  df-ico 9881  df-icc 9882  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-bc 10712  df-ihash 10740  df-shft 10808  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-sumdc 11346  df-ef 11640  df-e 11641  df-rest 12638  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-met 13156  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cn 13355  df-cnp 13356  df-tx 13420  df-cncf 13725  df-limced 13792  df-dvap 13793  df-relog 13946
This theorem is referenced by:  iooreen  14439
  Copyright terms: Public domain W3C validator