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Theorem xrge0mulc1cn 24319
Description: The operation multiplying a non-negative real numbers by a non-negative 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 x e 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 17261 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 10985 . . . . . . 7  |-  ( 0 [,]  +oo )  C_  RR*
4 resttopon 17217 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,]  +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) )  e.  (TopOn `  ( 0 [,]  +oo ) ) )
52, 3, 4mp2an 654 . . . . . 6  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  e.  (TopOn `  ( 0 [,]  +oo ) )
61, 5eqeltri 2505 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,]  +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,]  +oo ) ) )
8 0xr 9123 . . . . . 6  |-  0  e.  RR*
9 pnfxr 10705 . . . . . 6  |-  +oo  e.  RR*
10 pnfge 10719 . . . . . . 7  |-  ( 0  e.  RR*  ->  0  <_  +oo )
118, 10ax-mp 8 . . . . . 6  |-  0  <_  +oo
12 lbicc2 11005 . . . . . 6  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <_  +oo )  ->  0  e.  ( 0 [,]  +oo ) )
138, 9, 11, 12mp3an 1279 . . . . 5  |-  0  e.  ( 0 [,]  +oo )
1413a1i 11 . . . 4  |-  ( C  =  0  ->  0  e.  ( 0 [,]  +oo ) )
15 simpl 444 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  =  0 )
1615oveq2d 6089 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x x e C )  =  ( x x e 0 ) )
17 simpr 448 . . . . . . . . . 10  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  ( 0 [,]  +oo ) )
183, 17sseldi 3338 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  RR* )
19 xmul01 10838 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x x e 0 )  =  0 )
2018, 19syl 16 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x x e 0 )  =  0 )
2116, 20eqtrd 2467 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x x e C )  =  0 )
2221mpteq2dva 4287 . . . . . 6  |-  ( C  =  0  ->  (
x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) )  =  ( x  e.  ( 0 [,] 
+oo )  |->  0 ) )
23 xrge0mulc1cn.f . . . . . 6  |-  F  =  ( x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) )
24 fconstmpt 4913 . . . . . 6  |-  ( ( 0 [,]  +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,]  +oo )  |->  0 )
2522, 23, 243eqtr4g 2492 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,]  +oo )  X.  {
0 } ) )
26 c0ex 9077 . . . . . 6  |-  0  e.  _V
2726fconst2 5940 . . . . 5  |-  ( F : ( 0 [,] 
+oo ) --> { 0 }  <->  F  =  (
( 0 [,]  +oo )  X.  { 0 } ) )
2825, 27sylibr 204 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] 
+oo ) --> { 0 } )
29 cnconst 17340 . . . 4  |-  ( ( ( J  e.  (TopOn `  ( 0 [,]  +oo ) )  /\  J  e.  (TopOn `  ( 0 [,]  +oo ) ) )  /\  ( 0  e.  ( 0 [,]  +oo )  /\  F : ( 0 [,]  +oo ) --> { 0 } ) )  ->  F  e.  ( J  Cn  J
) )
307, 7, 14, 28, 29syl22anc 1185 . . 3  |-  ( C  =  0  ->  F  e.  ( J  Cn  J
) )
3130adantl 453 . 2  |-  ( (
ph  /\  C  = 
0 )  ->  F  e.  ( J  Cn  J
) )
32 eqid 2435 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
33 oveq1 6080 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x x e C )  =  ( y x e C ) )
3433cbvmptv 4292 . . . . . . . . 9  |-  ( x  e.  RR*  |->  ( x x e C ) )  =  ( y  e.  RR*  |->  ( y x e C ) )
35 id 20 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  e.  RR+ )
3632, 34, 35xrmulc1cn 24308 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x x e C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
37 letopuni 17263 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3837cnrest 17341 . . . . . . . 8  |-  ( ( ( x  e.  RR*  |->  ( x x e C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop `  <_  ) )  /\  ( 0 [,]  +oo )  C_  RR* )  ->  ( ( x  e. 
RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo )
)  Cn  (ordTop `  <_  ) ) )
3936, 3, 38sylancl 644 . . . . . . 7  |-  ( C  e.  RR+  ->  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )  Cn  (ordTop ` 
<_  ) ) )
40 resmpt 5183 . . . . . . . . 9  |-  ( ( 0 [,]  +oo )  C_ 
RR*  ->  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  =  ( x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) ) )
413, 40ax-mp 8 . . . . . . . 8  |-  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  =  ( x  e.  ( 0 [,]  +oo )  |->  ( x x e C ) )
4241, 23eqtr4i 2458 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  =  F
431eqcomi 2439 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  J
4443oveq1i 6083 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4539, 42, 443eltr3g 2517 . . . . . 6  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (ordTop ` 
<_  ) ) )
462a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  (ordTop `  <_  )  e.  (TopOn `  RR* ) )
47 simpr 448 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  ( 0 [,]  +oo ) )
48 ioorp 10980 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
49 ioossicc 10988 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  C_  (
0 [,]  +oo )
5048, 49eqsstr3i 3371 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,]  +oo )
51 simpl 444 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  RR+ )
5250, 51sseldi 3338 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  ( 0 [,]  +oo ) )
53 ge0xmulcl 11004 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  (
x x e C )  e.  ( 0 [,]  +oo ) )
5447, 52, 53syl2anc 643 . . . . . . . . 9  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  (
x x e C )  e.  ( 0 [,]  +oo ) )
5554, 23fmptd 5885 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,]  +oo )
--> ( 0 [,]  +oo ) )
56 frn 5589 . . . . . . . 8  |-  ( F : ( 0 [,] 
+oo ) --> ( 0 [,]  +oo )  ->  ran  F 
C_  ( 0 [,] 
+oo ) )
5755, 56syl 16 . . . . . . 7  |-  ( C  e.  RR+  ->  ran  F  C_  ( 0 [,]  +oo ) )
583a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  ( 0 [,]  +oo )  C_  RR* )
59 cnrest2 17342 . . . . . . 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 ) ) ) ) )
6046, 57, 58, 59syl3anc 1184 . . . . . 6  |-  ( C  e.  RR+  ->  ( F  e.  ( J  Cn  (ordTop `  <_  ) )  <->  F  e.  ( J  Cn  ( (ordTop `  <_  )t  ( 0 [,]  +oo ) ) ) ) )
6145, 60mpbid 202 . . . . 5  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) ) )
621oveq2i 6084 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) )
6361, 62syl6eleqr 2526 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
6463, 48eleq2s 2527 . . 3  |-  ( C  e.  ( 0 (,) 
+oo )  ->  F  e.  ( J  Cn  J
) )
6564adantl 453 . 2  |-  ( (
ph  /\  C  e.  ( 0 (,)  +oo ) )  ->  F  e.  ( J  Cn  J
) )
66 xrge0mulc1cn.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,)  +oo ) )
67 0re 9083 . . . . 5  |-  0  e.  RR
68 ltpnf 10713 . . . . 5  |-  ( 0  e.  RR  ->  0  <  +oo )
6967, 68ax-mp 8 . . . 4  |-  0  <  +oo
70 elicoelioo 24133 . . . 4  |-  ( ( 0  e.  RR*  /\  +oo  e.  RR*  /\  0  <  +oo )  ->  ( C  e.  ( 0 [,) 
+oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,)  +oo ) ) ) )
718, 9, 69, 70mp3an 1279 . . 3  |-  ( C  e.  ( 0 [,) 
+oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,)  +oo ) ) )
7266, 71sylib 189 . 2  |-  ( ph  ->  ( C  =  0  \/  C  e.  ( 0 (,)  +oo )
) )
7331, 65, 72mpjaodan 762 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312   {csn 3806   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   ran crn 4871    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113   RR+crp 10604   x ecxmu 10701   (,)cioo 10908   [,)cico 10910   [,]cicc 10911   ↾t crest 13640  ordTopcordt 13713  TopOnctopon 16951    Cn ccn 17280
This theorem is referenced by:  esummulc1  24463
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-rp 10605  df-xneg 10702  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-rest 13642  df-topgen 13659  df-ordt 13717  df-ps 14621  df-tsr 14622  df-top 16955  df-bases 16957  df-topon 16958  df-cn 17283  df-cnp 17284
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