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Theorem xrge0mulc1cn 27559
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 19472 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 11603 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
4 resttopon 19428 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
52, 3, 4mp2an 672 . . . . . 6  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
61, 5eqeltri 2551 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,] +oo ) ) )
8 0e0iccpnf 11627 . . . . 5  |-  0  e.  ( 0 [,] +oo )
98a1i 11 . . . 4  |-  ( C  =  0  ->  0  e.  ( 0 [,] +oo ) )
10 simpl 457 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  C  =  0 )
1110oveq2d 6298 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  ( x xe 0 ) )
12 simpr 461 . . . . . . . . . 10  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo )
)
133, 12sseldi 3502 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  RR* )
14 xmul01 11455 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x xe 0 )  =  0 )
1513, 14syl 16 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe 0 )  =  0 )
1611, 15eqtrd 2508 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  0 )
1716mpteq2dva 4533 . . . . . 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 5042 . . . . . 6  |-  ( ( 0 [,] +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,] +oo )  |->  0 )
2017, 18, 193eqtr4g 2533 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,] +oo )  X. 
{ 0 } ) )
21 c0ex 9586 . . . . . 6  |-  0  e.  _V
2221fconst2 6115 . . . . 5  |-  ( F : ( 0 [,] +oo ) --> { 0 }  <-> 
F  =  ( ( 0 [,] +oo )  X.  { 0 } ) )
2320, 22sylibr 212 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] +oo ) --> { 0 } )
24 cnconst 19551 . . . 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 1229 . . 3  |-  ( C  =  0  ->  F  e.  ( J  Cn  J
) )
2625adantl 466 . 2  |-  ( (
ph  /\  C  = 
0 )  ->  F  e.  ( J  Cn  J
) )
27 eqid 2467 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
28 oveq1 6289 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
2928cbvmptv 4538 . . . . . . . . 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 27548 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
32 letopuni 19474 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3332cnrest 19552 . . . . . . . 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 662 . . . . . . 7  |-  ( C  e.  RR+  ->  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop ` 
<_  ) ) )
35 resmpt 5321 . . . . . . . . 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 2499 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  F
381eqcomi 2480 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  J
3938oveq1i 6292 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4034, 37, 393eltr3g 2571 . . . . . 6  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (ordTop ` 
<_  ) ) )
412a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  (ordTop `  <_  )  e.  (TopOn `  RR* ) )
42 simpr 461 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo ) )
43 ioorp 11598 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  = 
RR+
44 ioossicc 11606 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  C_  ( 0 [,] +oo )
4543, 44eqsstr3i 3535 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,] +oo )
46 simpl 457 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  RR+ )
4745, 46sseldi 3502 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo ) )
48 ge0xmulcl 11631 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  e.  ( 0 [,] +oo ) )
4942, 47, 48syl2anc 661 . . . . . . . . 9  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  (
x xe C )  e.  ( 0 [,] +oo ) )
5049, 18fmptd 6043 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,] +oo )
--> ( 0 [,] +oo ) )
51 frn 5735 . . . . . . . 8  |-  ( F : ( 0 [,] +oo ) --> ( 0 [,] +oo )  ->  ran  F  C_  ( 0 [,] +oo ) )
5250, 51syl 16 . . . . . . 7  |-  ( C  e.  RR+  ->  ran  F  C_  ( 0 [,] +oo ) )
533a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  ( 0 [,] +oo )  C_  RR* )
54 cnrest2 19553 . . . . . . 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 1228 . . . . . 6  |-  ( C  e.  RR+  ->  ( F  e.  ( J  Cn  (ordTop `  <_  ) )  <->  F  e.  ( J  Cn  ( (ordTop `  <_  )t  ( 0 [,] +oo ) ) ) ) )
5640, 55mpbid 210 . . . . 5  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) ) )
571oveq2i 6293 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) )
5856, 57syl6eleqr 2566 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
5958, 43eleq2s 2575 . . 3  |-  ( C  e.  ( 0 (,) +oo )  ->  F  e.  ( J  Cn  J
) )
6059adantl 466 . 2  |-  ( (
ph  /\  C  e.  ( 0 (,) +oo ) )  ->  F  e.  ( J  Cn  J
) )
61 xrge0mulc1cn.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
62 0xr 9636 . . . 4  |-  0  e.  RR*
63 pnfxr 11317 . . . 4  |- +oo  e.  RR*
64 0ltpnf 11328 . . . 4  |-  0  < +oo
65 elicoelioo 27257 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  ->  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) ) )
6662, 63, 64, 65mp3an 1324 . . 3  |-  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) )
6761, 66sylib 196 . 2  |-  ( ph  ->  ( C  =  0  \/  C  e.  ( 0 (,) +oo )
) )
6826, 60, 67mpjaodan 784 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000    |` cres 5001   -->wf 5582   ` cfv 5586  (class class class)co 6282   0cc0 9488   +oocpnf 9621   RR*cxr 9623    < clt 9624    <_ cle 9625   RR+crp 11216   xecxmu 11313   (,)cioo 11525   [,)cico 11527   [,]cicc 11528   ↾t crest 14672  ordTopcordt 14750  TopOnctopon 19162    Cn ccn 19491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fi 7867  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-rp 11217  df-xneg 11314  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-rest 14674  df-topgen 14695  df-ordt 14752  df-ps 15683  df-tsr 15684  df-top 19166  df-bases 19168  df-topon 19169  df-cn 19494  df-cnp 19495
This theorem is referenced by:  esummulc1  27727
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