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Theorem xrge0mulc1cn 28798
Description: The operation multiplying a nonnegative real numbers by a nonnegative constant is continuous. (Contributed by Thierry Arnoux, 6-Jul-2017.)
Hypotheses
Ref Expression
xrge0mulc1cn.k  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
xrge0mulc1cn.f  |-  F  =  ( x  e.  ( 0 [,] +oo )  |->  ( x xe C ) )
xrge0mulc1cn.c  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
Assertion
Ref Expression
xrge0mulc1cn  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Distinct variable group:    x, C
Allowed substitution hints:    ph( x)    F( x)    J( x)

Proof of Theorem xrge0mulc1cn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 xrge0mulc1cn.k . . . . . 6  |-  J  =  ( (ordTop `  <_  )t  ( 0 [,] +oo )
)
2 letopon 20276 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 11751 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
4 resttopon 20232 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
52, 3, 4mp2an 683 . . . . . 6  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
61, 5eqeltri 2536 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,] +oo ) ) )
8 0e0iccpnf 11778 . . . . 5  |-  0  e.  ( 0 [,] +oo )
98a1i 11 . . . 4  |-  ( C  =  0  ->  0  e.  ( 0 [,] +oo ) )
10 simpl 463 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  C  =  0 )
1110oveq2d 6336 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  ( x xe 0 ) )
12 simpr 467 . . . . . . . . . 10  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo )
)
133, 12sseldi 3442 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  RR* )
14 xmul01 11587 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x xe 0 )  =  0 )
1513, 14syl 17 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe 0 )  =  0 )
1611, 15eqtrd 2496 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  0 )
1716mpteq2dva 4505 . . . . . 6  |-  ( C  =  0  ->  (
x  e.  ( 0 [,] +oo )  |->  ( x xe C ) )  =  ( x  e.  ( 0 [,] +oo )  |->  0 ) )
18 xrge0mulc1cn.f . . . . . 6  |-  F  =  ( x  e.  ( 0 [,] +oo )  |->  ( x xe C ) )
19 fconstmpt 4900 . . . . . 6  |-  ( ( 0 [,] +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,] +oo )  |->  0 )
2017, 18, 193eqtr4g 2521 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,] +oo )  X. 
{ 0 } ) )
21 c0ex 9668 . . . . . 6  |-  0  e.  _V
2221fconst2 6150 . . . . 5  |-  ( F : ( 0 [,] +oo ) --> { 0 }  <-> 
F  =  ( ( 0 [,] +oo )  X.  { 0 } ) )
2320, 22sylibr 217 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] +oo ) --> { 0 } )
24 cnconst 20355 . . . 4  |-  ( ( ( J  e.  (TopOn `  ( 0 [,] +oo ) )  /\  J  e.  (TopOn `  ( 0 [,] +oo ) ) )  /\  ( 0  e.  ( 0 [,] +oo )  /\  F : ( 0 [,] +oo ) --> { 0 } ) )  ->  F  e.  ( J  Cn  J
) )
257, 7, 9, 23, 24syl22anc 1277 . . 3  |-  ( C  =  0  ->  F  e.  ( J  Cn  J
) )
2625adantl 472 . 2  |-  ( (
ph  /\  C  = 
0 )  ->  F  e.  ( J  Cn  J
) )
27 eqid 2462 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
28 oveq1 6327 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
2928cbvmptv 4511 . . . . . . . . 9  |-  ( x  e.  RR*  |->  ( x xe C ) )  =  ( y  e.  RR*  |->  ( y xe C ) )
30 id 22 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  e.  RR+ )
3127, 29, 30xrmulc1cn 28787 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
32 letopuni 20278 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3332cnrest 20356 . . . . . . . 8  |-  ( ( ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )  /\  ( 0 [,] +oo )  C_  RR* )  ->  ( (
x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop ` 
<_  ) ) )
3431, 3, 33sylancl 673 . . . . . . 7  |-  ( C  e.  RR+  ->  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop ` 
<_  ) ) )
35 resmpt 5176 . . . . . . . . 9  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  ( x  e.  ( 0 [,] +oo )  |->  ( x xe C ) ) )
363, 35ax-mp 5 . . . . . . . 8  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  ( x  e.  ( 0 [,] +oo )  |->  ( x xe C ) )
3736, 18eqtr4i 2487 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  F
381eqcomi 2471 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  J
3938oveq1i 6330 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4034, 37, 393eltr3g 2556 . . . . . 6  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (ordTop ` 
<_  ) ) )
412a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  (ordTop `  <_  )  e.  (TopOn `  RR* ) )
42 simpr 467 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo ) )
43 ioorp 11746 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  = 
RR+
44 ioossicc 11754 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  C_  ( 0 [,] +oo )
4543, 44eqsstr3i 3475 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,] +oo )
46 simpl 463 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  RR+ )
4745, 46sseldi 3442 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo ) )
48 ge0xmulcl 11782 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  e.  ( 0 [,] +oo ) )
4942, 47, 48syl2anc 671 . . . . . . . . 9  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  (
x xe C )  e.  ( 0 [,] +oo ) )
5049, 18fmptd 6074 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,] +oo )
--> ( 0 [,] +oo ) )
51 frn 5762 . . . . . . . 8  |-  ( F : ( 0 [,] +oo ) --> ( 0 [,] +oo )  ->  ran  F  C_  ( 0 [,] +oo ) )
5250, 51syl 17 . . . . . . 7  |-  ( C  e.  RR+  ->  ran  F  C_  ( 0 [,] +oo ) )
533a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  ( 0 [,] +oo )  C_  RR* )
54 cnrest2 20357 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ran  F  C_  ( 0 [,] +oo )  /\  ( 0 [,] +oo )  C_  RR* )  ->  ( F  e.  ( J  Cn  (ordTop `  <_  ) )  <-> 
F  e.  ( J  Cn  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) ) ) )
5541, 52, 53, 54syl3anc 1276 . . . . . 6  |-  ( C  e.  RR+  ->  ( F  e.  ( J  Cn  (ordTop `  <_  ) )  <->  F  e.  ( J  Cn  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) ) ) )
5640, 55mpbid 215 . . . . 5  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) ) )
571oveq2i 6331 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) )
5856, 57syl6eleqr 2551 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
5958, 43eleq2s 2558 . . 3  |-  ( C  e.  ( 0 (,) +oo )  ->  F  e.  ( J  Cn  J
) )
6059adantl 472 . 2  |-  ( (
ph  /\  C  e.  ( 0 (,) +oo ) )  ->  F  e.  ( J  Cn  J
) )
61 xrge0mulc1cn.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
62 0xr 9718 . . . 4  |-  0  e.  RR*
63 pnfxr 11446 . . . 4  |- +oo  e.  RR*
64 0ltpnf 11458 . . . 4  |-  0  < +oo
65 elicoelioo 28412 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  ->  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) ) )
6662, 63, 64, 65mp3an 1373 . . 3  |-  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) )
6761, 66sylib 201 . 2  |-  ( ph  ->  ( C  =  0  \/  C  e.  ( 0 (,) +oo )
) )
6826, 60, 67mpjaodan 800 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1455    e. wcel 1898    C_ wss 3416   {csn 3980   class class class wbr 4418    |-> cmpt 4477    X. cxp 4854   ran crn 4857    |` cres 4858   -->wf 5601   ` cfv 5605  (class class class)co 6320   0cc0 9570   +oocpnf 9703   RR*cxr 9705    < clt 9706    <_ cle 9707   RR+crp 11336   xecxmu 11442   (,)cioo 11669   [,)cico 11671   [,]cicc 11672   ↾t crest 15374  ordTopcordt 15452  TopOnctopon 19973    Cn ccn 20295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fi 7956  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-rp 11337  df-xneg 11443  df-xmul 11445  df-ioo 11673  df-ico 11675  df-icc 11676  df-rest 15376  df-topgen 15397  df-ordt 15454  df-ps 16501  df-tsr 16502  df-top 19976  df-bases 19977  df-topon 19978  df-cn 20298  df-cnp 20299
This theorem is referenced by:  esummulc1  28953
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