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Theorem xrmulc1cn 28736
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 16473 . . . 4  |-  <_  e.  TosetRel
21a1i 11 . . 3  |-  ( ph  ->  <_  e.  TosetRel  )
3 simpr 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  x  e.  RR* )
4 xrmulc1cn.c . . . . . . . . 9  |-  ( ph  ->  C  e.  RR+ )
54adantr 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR+ )
65rpxrd 11342 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR* )
73, 6xmulcld 11588 . . . . . 6  |-  ( (
ph  /\  x  e.  RR* )  ->  ( x xe C )  e.  RR* )
87ralrimiva 2802 . . . . 5  |-  ( ph  ->  A. x  e.  RR*  ( x xe C )  e.  RR* )
9 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  y  e.  RR* )
104adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR+ )
1110rpred 11341 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR )
1210rpne0d 11346 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  =/=  0 )
13 xreceu 28391 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  y )
149, 11, 12, 13syl3anc 1268 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  ( C xe x )  =  y )
15 eqcom 2458 . . . . . . . . 9  |-  ( y  =  ( x xe C )  <->  ( x xe C )  =  y )
16 simpr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  x  e.  RR* )
176adantlr 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  C  e.  RR* )
18 xmulcom 11552 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
1916, 17, 18syl2anc 667 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
2019eqeq1d 2453 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
( x xe C )  =  y  <-> 
( C xe x )  =  y ) )
2115, 20syl5bb 261 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
y  =  ( x xe C )  <-> 
( C xe x )  =  y ) )
2221reubidva 2974 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  ( E! x  e.  RR*  y  =  ( x xe C )  <->  E! x  e.  RR*  ( C xe x )  =  y ) )
2314, 22mpbird 236 . . . . . 6  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  y  =  ( x xe C ) )
2423ralrimiva 2802 . . . . 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 6044 . . . . 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 670 . . . 4  |-  ( ph  ->  F : RR* -1-1-onto-> RR* )
28 simplr 762 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  x  e.  RR* )
29 simpr 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  y  e.  RR* )
304ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  C  e.  RR+ )
31 xlemul1 11576 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  C  e.  RR+ )  ->  ( x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
3228, 29, 30, 31syl3anc 1268 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
33 ovex 6318 . . . . . . . . 9  |-  ( x xe C )  e.  _V
3425fvmpt2 5957 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  (
x xe C )  e.  _V )  ->  ( F `  x
)  =  ( x xe C ) )
3528, 33, 34sylancl 668 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  x )  =  ( x xe C ) )
36 oveq1 6297 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
37 ovex 6318 . . . . . . . . . 10  |-  ( y xe C )  e.  _V
3836, 25, 37fvmpt 5948 . . . . . . . . 9  |-  ( y  e.  RR*  ->  ( F `
 y )  =  ( y xe C ) )
3938adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  y )  =  ( y xe C ) )
4035, 39breq12d 4415 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
( F `  x
)  <_  ( F `  y )  <->  ( x xe C )  <_  ( y xe C ) ) )
4132, 40bitr4d 260 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
4241ralrimiva 2802 . . . . 5  |-  ( (
ph  /\  x  e.  RR* )  ->  A. y  e.  RR*  ( x  <_ 
y  <->  ( F `  x )  <_  ( F `  y )
) )
4342ralrimiva 2802 . . . 4  |-  ( ph  ->  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
44 df-isom 5591 . . . 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 670 . . 3  |-  ( ph  ->  F  Isom  <_  ,  <_  (
RR* ,  RR* ) )
46 ledm 16470 . . . 4  |-  RR*  =  dom  <_
4746, 46ordthmeolem 20816 . . 3  |-  ( (  <_  e.  TosetRel  /\  <_  e.  TosetRel 
/\  F  Isom  <_  ,  <_  ( RR* ,  RR* ) )  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) ) )
482, 2, 45, 47syl3anc 1268 . 2  |-  ( ph  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
49 xrmulc1cn.k . . 3  |-  J  =  (ordTop `  <_  )
5049, 49oveq12i 6302 . 2  |-  ( J  Cn  J )  =  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )
5148, 50syl6eleqr 2540 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E!wreu 2739   _Vcvv 3045   class class class wbr 4402    |-> cmpt 4461   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583  (class class class)co 6290   RRcr 9538   0cc0 9539   RR*cxr 9674    <_ cle 9676   RR+crp 11302   xecxmu 11408  ordTopcordt 15397    TosetRel ctsr 16445    Cn ccn 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  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 7925  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-rp 11303  df-xneg 11409  df-xmul 11411  df-topgen 15342  df-ordt 15399  df-ps 16446  df-tsr 16447  df-top 19921  df-bases 19922  df-topon 19923  df-cn 20243
This theorem is referenced by:  xrge0mulc1cn  28747
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