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Theorem reeff1olem 12875
Description: Lemma for reeff1o 12877. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
Assertion
Ref Expression
reeff1olem  |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
Distinct variable group:    x, U

Proof of Theorem reeff1olem
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioossicc 9754 . . 3  |-  ( 0 (,) U )  C_  ( 0 [,] U
)
2 0re 7778 . . . . 5  |-  0  e.  RR
3 iccssre 9750 . . . . 5  |-  ( ( 0  e.  RR  /\  U  e.  RR )  ->  ( 0 [,] U
)  C_  RR )
42, 3mpan 420 . . . 4  |-  ( U  e.  RR  ->  (
0 [,] U ) 
C_  RR )
54adantr 274 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( 0 [,] U
)  C_  RR )
61, 5sstrid 3108 . 2  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( 0 (,) U
)  C_  RR )
72a1i 9 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
0  e.  RR )
8 simpl 108 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  ->  U  e.  RR )
9 0lt1 7901 . . . . 5  |-  0  <  1
10 1re 7777 . . . . . 6  |-  1  e.  RR
11 lttr 7850 . . . . . 6  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  U  e.  RR )  ->  (
( 0  <  1  /\  1  <  U )  ->  0  <  U
) )
122, 10, 11mp3an12 1305 . . . . 5  |-  ( U  e.  RR  ->  (
( 0  <  1  /\  1  <  U )  ->  0  <  U
) )
139, 12mpani 426 . . . 4  |-  ( U  e.  RR  ->  (
1  <  U  ->  0  <  U ) )
1413imp 123 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
0  <  U )
15 ax-resscn 7724 . . . 4  |-  RR  C_  CC
165, 15sstrdi 3109 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( 0 [,] U
)  C_  CC )
17 efcn 12872 . . . 4  |-  exp  e.  ( CC -cn-> CC )
1817a1i 9 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  ->  exp  e.  ( CC -cn-> CC ) )
19 ssel2 3092 . . . . 5  |-  ( ( ( 0 [,] U
)  C_  RR  /\  y  e.  ( 0 [,] U
) )  ->  y  e.  RR )
2019reefcld 11387 . . . 4  |-  ( ( ( 0 [,] U
)  C_  RR  /\  y  e.  ( 0 [,] U
) )  ->  ( exp `  y )  e.  RR )
215, 20sylan 281 . . 3  |-  ( ( ( U  e.  RR  /\  1  <  U )  /\  y  e.  ( 0 [,] U ) )  ->  ( exp `  y )  e.  RR )
22 ef0 11390 . . . . 5  |-  ( exp `  0 )  =  1
23 simpr 109 . . . . 5  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
1  <  U )
2422, 23eqbrtrid 3963 . . . 4  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( exp `  0
)  <  U )
25 peano2re 7910 . . . . . 6  |-  ( U  e.  RR  ->  ( U  +  1 )  e.  RR )
2625adantr 274 . . . . 5  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( U  +  1 )  e.  RR )
27 reefcl 11386 . . . . . 6  |-  ( U  e.  RR  ->  ( exp `  U )  e.  RR )
2827adantr 274 . . . . 5  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( exp `  U
)  e.  RR )
29 ltp1 8614 . . . . . 6  |-  ( U  e.  RR  ->  U  <  ( U  +  1 ) )
3029adantr 274 . . . . 5  |-  ( ( U  e.  RR  /\  1  <  U )  ->  U  <  ( U  + 
1 ) )
318recnd 7806 . . . . . . 7  |-  ( ( U  e.  RR  /\  1  <  U )  ->  U  e.  CC )
32 ax-1cn 7725 . . . . . . 7  |-  1  e.  CC
33 addcom 7911 . . . . . . 7  |-  ( ( U  e.  CC  /\  1  e.  CC )  ->  ( U  +  1 )  =  ( 1  +  U ) )
3431, 32, 33sylancl 409 . . . . . 6  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( U  +  1 )  =  ( 1  +  U ) )
358, 14elrpd 9493 . . . . . . 7  |-  ( ( U  e.  RR  /\  1  <  U )  ->  U  e.  RR+ )
36 efgt1p 11414 . . . . . . 7  |-  ( U  e.  RR+  ->  ( 1  +  U )  < 
( exp `  U
) )
3735, 36syl 14 . . . . . 6  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( 1  +  U
)  <  ( exp `  U ) )
3834, 37eqbrtrd 3950 . . . . 5  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( U  +  1 )  <  ( exp `  U ) )
398, 26, 28, 30, 38lttrd 7900 . . . 4  |-  ( ( U  e.  RR  /\  1  <  U )  ->  U  <  ( exp `  U
) )
4024, 39jca 304 . . 3  |-  ( ( U  e.  RR  /\  1  <  U )  -> 
( ( exp `  0
)  <  U  /\  U  <  ( exp `  U
) ) )
41 simplll 522 . . . . . . 7  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  U  e.  RR )
422, 41, 3sylancr 410 . . . . . 6  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  ( 0 [,] U )  C_  RR )
43 simplr 519 . . . . . 6  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  y  e.  ( 0 [,] U ) )
4442, 43sseldd 3098 . . . . 5  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  y  e.  RR )
45 simprl 520 . . . . . 6  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  z  e.  ( 0 [,] U ) )
4642, 45sseldd 3098 . . . . 5  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  z  e.  RR )
4744, 46jca 304 . . . 4  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  ( y  e.  RR  /\  z  e.  RR ) )
48 simprr 521 . . . 4  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  y  <  z
)
49 efltim 11416 . . . 4  |-  ( ( y  e.  RR  /\  z  e.  RR )  ->  ( y  <  z  ->  ( exp `  y
)  <  ( exp `  z ) ) )
5047, 48, 49sylc 62 . . 3  |-  ( ( ( ( U  e.  RR  /\  1  < 
U )  /\  y  e.  ( 0 [,] U
) )  /\  (
z  e.  ( 0 [,] U )  /\  y  <  z ) )  ->  ( exp `  y
)  <  ( exp `  z ) )
517, 8, 8, 14, 16, 18, 21, 40, 50ivthinc 12804 . 2  |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  (
0 (,) U ) ( exp `  x
)  =  U )
52 ssrexv 3162 . 2  |-  ( ( 0 (,) U ) 
C_  RR  ->  ( E. x  e.  ( 0 (,) U ) ( exp `  x )  =  U  ->  E. x  e.  RR  ( exp `  x
)  =  U ) )
536, 51, 52sylc 62 1  |-  ( ( U  e.  RR  /\  1  <  U )  ->  E. x  e.  RR  ( exp `  x )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   E.wrex 2417    C_ wss 3071   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632   1c1 7633    + caddc 7635    < clt 7812   RR+crp 9453   (,)cioo 9683   [,]cicc 9686   expce 11360   -cn->ccncf 12740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752  ax-pre-suploc 7753  ax-addf 7754  ax-mulf 7755
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-xneg 9571  df-xadd 9572  df-ioo 9687  df-ico 9689  df-icc 9690  df-fz 9803  df-fzo 9932  df-seqfrec 10231  df-exp 10305  df-fac 10484  df-bc 10506  df-ihash 10534  df-shft 10599  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060  df-sumdc 11135  df-ef 11366  df-rest 12136  df-topgen 12155  df-psmet 12170  df-xmet 12171  df-met 12172  df-bl 12173  df-mopn 12174  df-top 12179  df-topon 12192  df-bases 12224  df-ntr 12279  df-cn 12371  df-cnp 12372  df-tx 12436  df-cncf 12741  df-limced 12808  df-dvap 12809
This theorem is referenced by:  reeff1oleme  12876  reeff1o  12877
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