Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrmulc1cn Structured version   Unicode version

Theorem xrmulc1cn 26214
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 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 15380 . . . 4  |-  <_  e.  TosetRel
21a1i 11 . . 3  |-  ( ph  ->  <_  e.  TosetRel  )
3 simpr 458 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  x  e.  RR* )
4 xrmulc1cn.c . . . . . . . . 9  |-  ( ph  ->  C  e.  RR+ )
54adantr 462 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR+ )
65rpxrd 11016 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR* )
73, 6xmulcld 11253 . . . . . 6  |-  ( (
ph  /\  x  e.  RR* )  ->  ( x xe C )  e.  RR* )
87ralrimiva 2789 . . . . 5  |-  ( ph  ->  A. x  e.  RR*  ( x xe C )  e.  RR* )
9 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  y  e.  RR* )
104adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR+ )
1110rpred 11015 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR )
1210rpne0d 11020 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  =/=  0 )
13 xreceu 25920 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  y )
149, 11, 12, 13syl3anc 1211 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  ( C xe x )  =  y )
15 eqcom 2435 . . . . . . . . 9  |-  ( y  =  ( x xe C )  <->  ( x xe C )  =  y )
16 simpr 458 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  x  e.  RR* )
176adantlr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  C  e.  RR* )
18 xmulcom 11217 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
1916, 17, 18syl2anc 654 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
2019eqeq1d 2441 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
( x xe C )  =  y  <-> 
( C xe x )  =  y ) )
2115, 20syl5bb 257 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
y  =  ( x xe C )  <-> 
( C xe x )  =  y ) )
2221reubidva 2894 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  ( E! x  e.  RR*  y  =  ( x xe C )  <->  E! x  e.  RR*  ( C xe x )  =  y ) )
2314, 22mpbird 232 . . . . . 6  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  y  =  ( x xe C ) )
2423ralrimiva 2789 . . . . 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 5853 . . . . 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 657 . . . 4  |-  ( ph  ->  F : RR* -1-1-onto-> RR* )
28 simplr 747 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  x  e.  RR* )
29 simpr 458 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  y  e.  RR* )
304ad2antrr 718 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  C  e.  RR+ )
31 xlemul1 11241 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  C  e.  RR+ )  ->  ( x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
3228, 29, 30, 31syl3anc 1211 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
33 ovex 6105 . . . . . . . . 9  |-  ( x xe C )  e.  _V
3425fvmpt2 5769 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  (
x xe C )  e.  _V )  ->  ( F `  x
)  =  ( x xe C ) )
3528, 33, 34sylancl 655 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  x )  =  ( x xe C ) )
36 oveq1 6087 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
37 ovex 6105 . . . . . . . . . 10  |-  ( y xe C )  e.  _V
3836, 25, 37fvmpt 5762 . . . . . . . . 9  |-  ( y  e.  RR*  ->  ( F `
 y )  =  ( y xe C ) )
3938adantl 463 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  y )  =  ( y xe C ) )
4035, 39breq12d 4293 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
( F `  x
)  <_  ( F `  y )  <->  ( x xe C )  <_  ( y xe C ) ) )
4132, 40bitr4d 256 . . . . . 6  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
4241ralrimiva 2789 . . . . 5  |-  ( (
ph  /\  x  e.  RR* )  ->  A. y  e.  RR*  ( x  <_ 
y  <->  ( F `  x )  <_  ( F `  y )
) )
4342ralrimiva 2789 . . . 4  |-  ( ph  ->  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
44 df-isom 5415 . . . 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 657 . . 3  |-  ( ph  ->  F  Isom  <_  ,  <_  (
RR* ,  RR* ) )
46 ledm 15377 . . . 4  |-  RR*  =  dom  <_
4746, 46ordthmeolem 19216 . . 3  |-  ( (  <_  e.  TosetRel  /\  <_  e.  TosetRel 
/\  F  Isom  <_  ,  <_  ( RR* ,  RR* ) )  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) ) )
482, 2, 45, 47syl3anc 1211 . 2  |-  ( ph  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
49 xrmulc1cn.k . . 3  |-  J  =  (ordTop `  <_  )
5049, 49oveq12i 6092 . 2  |-  ( J  Cn  J )  =  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )
5148, 50syl6eleqr 2524 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   E!wreu 2707   _Vcvv 2962   class class class wbr 4280    e. cmpt 4338   -1-1-onto->wf1o 5405   ` cfv 5406    Isom wiso 5407  (class class class)co 6080   RRcr 9269   0cc0 9270   RR*cxr 9405    <_ cle 9407   RR+crp 10979   xecxmu 11076  ordTopcordt 14420    TosetRel ctsr 15352    Cn ccn 18670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fi 7649  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-rp 10980  df-xneg 11077  df-xmul 11079  df-topgen 14365  df-ordt 14422  df-ps 15353  df-tsr 15354  df-top 18345  df-bases 18347  df-topon 18348  df-cn 18673
This theorem is referenced by:  xrge0mulc1cn  26225
  Copyright terms: Public domain W3C validator