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Theorem xrmulc1cn 24269
Description: The operation multiplying an extended real number by a non-negative constant is continuous. (Contributed by Thierry Arnoux, 5-Jul-2017.)
Hypotheses
Ref Expression
xrmulc1cn.k  |-  J  =  (ordTop `  <_  )
xrmulc1cn.f  |-  F  =  ( x  e.  RR*  |->  ( x x e 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 14627 . . . 4  |-  <_  e.  TosetRel
21a1i 11 . . 3  |-  ( ph  ->  <_  e.  TosetRel  )
3 simpr 448 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  x  e.  RR* )
4 xrmulc1cn.c . . . . . . . . 9  |-  ( ph  ->  C  e.  RR+ )
54adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR+ )
65rpxrd 10605 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR* )
73, 6xmulcld 10837 . . . . . 6  |-  ( (
ph  /\  x  e.  RR* )  ->  ( x x e C )  e. 
RR* )
87ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. x  e.  RR*  ( x x e C )  e.  RR* )
9 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  y  e.  RR* )
104adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR+ )
1110rpred 10604 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR )
1210rpne0d 10609 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  =/=  0 )
13 xreceu 24121 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C x e x )  =  y )
149, 11, 12, 13syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  ( C x e x )  =  y )
15 eqcom 2406 . . . . . . . . 9  |-  ( y  =  ( x x e C )  <->  ( x x e C )  =  y )
16 simpr 448 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  x  e.  RR* )
176adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  C  e.  RR* )
18 xmulcom 10801 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x x e C )  =  ( C x e x ) )
1916, 17, 18syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
x x e C )  =  ( C x e x ) )
2019eqeq1d 2412 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
( x x e C )  =  y  <-> 
( C x e x )  =  y ) )
2115, 20syl5bb 249 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
y  =  ( x x e C )  <-> 
( C x e x )  =  y ) )
2221reubidva 2851 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  ( E! x  e.  RR*  y  =  ( x x e C )  <->  E! x  e.  RR*  ( C x e x )  =  y ) )
2314, 22mpbird 224 . . . . . 6  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  y  =  ( x x e C ) )
2423ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. y  e.  RR*  E! x  e.  RR*  y  =  ( x x e C ) )
25 xrmulc1cn.f . . . . . 6  |-  F  =  ( x  e.  RR*  |->  ( x x e C ) )
2625f1ompt 5850 . . . . 5  |-  ( F : RR* -1-1-onto-> RR*  <->  ( A. x  e.  RR*  ( x x e C )  e. 
RR*  /\  A. y  e.  RR*  E! x  e. 
RR*  y  =  ( x x e C ) ) )
278, 24, 26sylanbrc 646 . . . 4  |-  ( ph  ->  F : RR* -1-1-onto-> RR* )
28 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  x  e.  RR* )
29 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  y  e.  RR* )
304ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  C  e.  RR+ )
31 xlemul1 10825 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  C  e.  RR+ )  ->  ( x  <_  y  <->  ( x x e C )  <_ 
( y x e C ) ) )
3228, 29, 30, 31syl3anc 1184 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x x e C )  <_ 
( y x e C ) ) )
33 ovex 6065 . . . . . . . . 9  |-  ( x x e C )  e.  _V
3425fvmpt2 5771 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  (
x x e C )  e.  _V )  ->  ( F `  x
)  =  ( x x e C ) )
3528, 33, 34sylancl 644 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  x )  =  ( x x e C ) )
36 oveq1 6047 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x x e C )  =  ( y x e C ) )
37 ovex 6065 . . . . . . . . . 10  |-  ( y x e C )  e.  _V
3836, 25, 37fvmpt 5765 . . . . . . . . 9  |-  ( y  e.  RR*  ->  ( F `
 y )  =  ( y x e C ) )
3938adantl 453 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  y )  =  ( y x e C ) )
4035, 39breq12d 4185 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
( F `  x
)  <_  ( F `  y )  <->  ( x x e C )  <_ 
( y x e C ) ) )
4132, 40bitr4d 248 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
4241ralrimiva 2749 . . . . 5  |-  ( (
ph  /\  x  e.  RR* )  ->  A. y  e.  RR*  ( x  <_ 
y  <->  ( F `  x )  <_  ( F `  y )
) )
4342ralrimiva 2749 . . . 4  |-  ( ph  ->  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
44 df-isom 5422 . . . 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 646 . . 3  |-  ( ph  ->  F  Isom  <_  ,  <_  (
RR* ,  RR* ) )
46 ledm 14624 . . . 4  |-  RR*  =  dom  <_
4746, 46ordthmeolem 17786 . . 3  |-  ( (  <_  e.  TosetRel  /\  <_  e.  TosetRel 
/\  F  Isom  <_  ,  <_  ( RR* ,  RR* ) )  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) ) )
482, 2, 45, 47syl3anc 1184 . 2  |-  ( ph  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
49 xrmulc1cn.k . . 3  |-  J  =  (ordTop `  <_  )
5049, 49oveq12i 6052 . 2  |-  ( J  Cn  J )  =  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )
5148, 50syl6eleqr 2495 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E!wreu 2668   _Vcvv 2916   class class class wbr 4172    e. cmpt 4226   -1-1-onto->wf1o 5412   ` cfv 5413    Isom wiso 5414  (class class class)co 6040   RRcr 8945   0cc0 8946   RR*cxr 9075    <_ cle 9077   RR+crp 10568   x ecxmu 10665  ordTopcordt 13676    TosetRel ctsr 14580    Cn ccn 17242
This theorem is referenced by:  xrge0mulc1cn  24280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-rp 10569  df-xneg 10666  df-xmul 10668  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245
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