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Theorem xrmulc1cn 26525
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 15519 . . . 4  |-  <_  e.  TosetRel
21a1i 11 . . 3  |-  ( ph  ->  <_  e.  TosetRel  )
3 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  x  e.  RR* )
4 xrmulc1cn.c . . . . . . . . 9  |-  ( ph  ->  C  e.  RR+ )
54adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR+ )
65rpxrd 11142 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR* )  ->  C  e.  RR* )
73, 6xmulcld 11379 . . . . . 6  |-  ( (
ph  /\  x  e.  RR* )  ->  ( x xe C )  e.  RR* )
87ralrimiva 2830 . . . . 5  |-  ( ph  ->  A. x  e.  RR*  ( x xe C )  e.  RR* )
9 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  y  e.  RR* )
104adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR+ )
1110rpred 11141 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  e.  RR )
1210rpne0d 11146 . . . . . . . 8  |-  ( (
ph  /\  y  e.  RR* )  ->  C  =/=  0 )
13 xreceu 26262 . . . . . . . 8  |-  ( ( y  e.  RR*  /\  C  e.  RR  /\  C  =/=  0 )  ->  E! x  e.  RR*  ( C xe x )  =  y )
149, 11, 12, 13syl3anc 1219 . . . . . . 7  |-  ( (
ph  /\  y  e.  RR* )  ->  E! x  e.  RR*  ( C xe x )  =  y )
15 eqcom 2463 . . . . . . . . 9  |-  ( y  =  ( x xe C )  <->  ( x xe C )  =  y )
16 simpr 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  x  e.  RR* )
176adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  C  e.  RR* )
18 xmulcom 11343 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
1916, 17, 18syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  y  e.  RR* )  /\  x  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
2019eqeq1d 2456 . . . . . . . . 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 3010 . . . . . . 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 2830 . . . . 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 5977 . . . . 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 664 . . . 4  |-  ( ph  ->  F : RR* -1-1-onto-> RR* )
28 simplr 754 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  x  e.  RR* )
29 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  y  e.  RR* )
304ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  C  e.  RR+ )
31 xlemul1 11367 . . . . . . . 8  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  C  e.  RR+ )  ->  ( x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
3228, 29, 30, 31syl3anc 1219 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  (
x  <_  y  <->  ( x xe C )  <_  ( y xe C ) ) )
33 ovex 6228 . . . . . . . . 9  |-  ( x xe C )  e.  _V
3425fvmpt2 5893 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  (
x xe C )  e.  _V )  ->  ( F `  x
)  =  ( x xe C ) )
3528, 33, 34sylancl 662 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  x )  =  ( x xe C ) )
36 oveq1 6210 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
37 ovex 6228 . . . . . . . . . 10  |-  ( y xe C )  e.  _V
3836, 25, 37fvmpt 5886 . . . . . . . . 9  |-  ( y  e.  RR*  ->  ( F `
 y )  =  ( y xe C ) )
3938adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  RR* )  /\  y  e.  RR* )  ->  ( F `  y )  =  ( y xe C ) )
4035, 39breq12d 4416 . . . . . . 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 2830 . . . . 5  |-  ( (
ph  /\  x  e.  RR* )  ->  A. y  e.  RR*  ( x  <_ 
y  <->  ( F `  x )  <_  ( F `  y )
) )
4342ralrimiva 2830 . . . 4  |-  ( ph  ->  A. x  e.  RR*  A. y  e.  RR*  (
x  <_  y  <->  ( F `  x )  <_  ( F `  y )
) )
44 df-isom 5538 . . . 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 664 . . 3  |-  ( ph  ->  F  Isom  <_  ,  <_  (
RR* ,  RR* ) )
46 ledm 15516 . . . 4  |-  RR*  =  dom  <_
4746, 46ordthmeolem 19509 . . 3  |-  ( (  <_  e.  TosetRel  /\  <_  e.  TosetRel 
/\  F  Isom  <_  ,  <_  ( RR* ,  RR* ) )  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) ) )
482, 2, 45, 47syl3anc 1219 . 2  |-  ( ph  ->  F  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
49 xrmulc1cn.k . . 3  |-  J  =  (ordTop `  <_  )
5049, 49oveq12i 6215 . 2  |-  ( J  Cn  J )  =  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )
5148, 50syl6eleqr 2553 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E!wreu 2801   _Vcvv 3078   class class class wbr 4403    |-> cmpt 4461   -1-1-onto->wf1o 5528   ` cfv 5529    Isom wiso 5530  (class class class)co 6203   RRcr 9395   0cc0 9396   RR*cxr 9531    <_ cle 9533   RR+crp 11105   xecxmu 11202  ordTopcordt 14559    TosetRel ctsr 15491    Cn ccn 18963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fi 7775  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-rp 11106  df-xneg 11203  df-xmul 11205  df-topgen 14504  df-ordt 14561  df-ps 15492  df-tsr 15493  df-top 18638  df-bases 18640  df-topon 18641  df-cn 18966
This theorem is referenced by:  xrge0mulc1cn  26536
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