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Theorem xrmulc1cn 28725
Description: The operation multiplying an extended real number by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
Hypotheses
Ref Expression
xrmulc1cn.k  |-  J  =  (ordTop `  <_  )
xrmulc1cn.f  |-  F  =  ( x  e.  RR*  |->  ( x xe C ) )
xrmulc1cn.c  |-  ( ph  ->  C  e.  RR+ )
Assertion
Ref Expression
xrmulc1cn  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Distinct variable groups:    x, C    x, F    ph, x
Allowed substitution hint:    J( x)

Proof of Theorem xrmulc1cn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 letsr 16451 . . . 4  |-  <_  e.  TosetRel
21a1i 11 . . 3  |-  ( ph  ->  <_  e.  TosetRel  )
3 simpr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  x  e.  RR* )
4 xrmulc1cn.c . . . . . . . . 9  |-  ( ph  ->  C  e.  RR+ )
54adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR+ )
65rpxrd 11338 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR* )
73, 6xmulcld 11584 . . . . . 6  |-  ( (
ph  /\  x  e.  RR* )  ->  ( x xe C )  e.  RR* )
87ralrimiva 2837 . . . . 5  |-  ( ph  ->  A. x  e.  RR*  ( x xe C )  e.  RR* )
9 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  y  e.  RR* )
104adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR+ )
1110rpred 11337 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR )
1210rpne0d 11342 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  =/=  0 )
13 xreceu 28378 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  y )
149, 11, 12, 13syl3anc 1264 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  ( C xe x )  =  y )
15 eqcom 2429 . . . . . . . . 9  |-  ( y  =  ( x xe C )  <->  ( x xe C )  =  y )
16 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  x  e.  RR* )
176adantlr 719 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  C  e.  RR* )
18 xmulcom 11548 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
1916, 17, 18syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
2019eqeq1d 2422 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
( x xe C )  =  y  <-> 
( C xe x )  =  y ) )
2115, 20syl5bb 260 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
y  =  ( x xe C )  <-> 
( C xe x )  =  y ) )
2221reubidva 3010 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  ( E! x  e.  RR*  y  =  ( x xe C )  <->  E! x  e.  RR*  ( C xe x )  =  y ) )
2314, 22mpbird 235 . . . . . 6  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  y  =  ( x xe C ) )
2423ralrimiva 2837 . . . . 5  |-  ( ph  ->  A. y  e.  RR*  E! x  e.  RR*  y  =  ( x xe C ) )
25 xrmulc1cn.f . . . . . 6  |-  F  =  ( x  e.  RR*  |->  ( x xe C ) )
2625f1ompt 6051 . . . . 5  |-  ( F : RR* -1-1-onto-> RR*  <->  ( A. x  e.  RR*  ( x xe C )  e. 
RR*  /\  A. y  e.  RR*  E! x  e. 
RR*  y  =  ( x xe C ) ) )
278, 24, 26sylanbrc 668 . . . 4  |-  ( ph  ->  F : RR* -1-1-onto-> RR* )
28 simplr 760 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  x  e.  RR* )
29 simpr 462 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  y  e.  RR* )
304ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  C  e.  RR+ )
31 xlemul1 11572 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  C  e.  RR+ )  ->  ( x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
3228, 29, 30, 31syl3anc 1264 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
33 ovex 6325 . . . . . . . . 9  |-  ( x xe C )  e.  _V
3425fvmpt2 5965 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  (
x xe C )  e.  _V )  ->  ( F `  x
)  =  ( x xe C ) )
3528, 33, 34sylancl 666 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  x )  =  ( x xe C ) )
36 oveq1 6304 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
37 ovex 6325 . . . . . . . . . 10  |-  ( y xe C )  e.  _V
3836, 25, 37fvmpt 5956 . . . . . . . . 9  |-  ( y  e.  RR*  ->  ( F `
 y )  =  ( y xe C ) )
3938adantl 467 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  y )  =  ( y xe C ) )
4035, 39breq12d 4430 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
( F `  x
)  <_  ( F `  y )  <->  ( x xe C )  <_  ( y xe C ) ) )
4132, 40bitr4d 259 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
4241ralrimiva 2837 . . . . 5  |-  ( (
ph  /\  x  e.  RR* )  ->  A. y  e.  RR*  ( x  <_ 
y  <->  ( F `  x )  <_  ( F `  y )
) )
4342ralrimiva 2837 . . . 4  |-  ( ph  ->  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
44 df-isom 5602 . . . 4  |-  ( F 
Isom  <_  ,  <_  ( RR* ,  RR* )  <->  ( F : RR*
-1-1-onto-> RR* 
/\  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) ) )
4527, 43, 44sylanbrc 668 . . 3  |-  ( ph  ->  F  Isom  <_  ,  <_  (
RR* ,  RR* ) )
46 ledm 16448 . . . 4  |-  RR*  =  dom  <_
4746, 46ordthmeolem 20793 . . 3  |-  ( (  <_  e.  TosetRel  /\  <_  e.  TosetRel 
/\  F  Isom  <_  ,  <_  ( RR* ,  RR* ) )  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) ) )
482, 2, 45, 47syl3anc 1264 . 2  |-  ( ph  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
49 xrmulc1cn.k . . 3  |-  J  =  (ordTop `  <_  )
5049, 49oveq12i 6309 . 2  |-  ( J  Cn  J )  =  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )
5148, 50syl6eleqr 2519 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   A.wral 2773   E!wreu 2775   _Vcvv 3078   class class class wbr 4417    |-> cmpt 4476   -1-1-onto->wf1o 5592   ` cfv 5593    Isom wiso 5594  (class class class)co 6297   RRcr 9534   0cc0 9535   RR*cxr 9670    <_ cle 9672   RR+crp 11298   xecxmu 11404  ordTopcordt 15375    TosetRel ctsr 16423    Cn ccn 20217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-cnex 9591  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611  ax-pre-mulgt0 9612
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4477  df-mpt 4478  df-tr 4513  df-eprel 4757  df-id 4761  df-po 4767  df-so 4768  df-fr 4805  df-we 4807  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-ord 5437  df-on 5438  df-lim 5439  df-suc 5440  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6259  df-ov 6300  df-oprab 6301  df-mpt2 6302  df-om 6699  df-1st 6799  df-2nd 6800  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fi 7923  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677  df-sub 9858  df-neg 9859  df-div 10266  df-rp 11299  df-xneg 11405  df-xmul 11407  df-topgen 15320  df-ordt 15377  df-ps 16424  df-tsr 16425  df-top 19898  df-bases 19899  df-topon 19900  df-cn 20220
This theorem is referenced by:  xrge0mulc1cn  28736
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