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Theorem xrge0mulc1cn 26386
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 18824 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 11393 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
4 resttopon 18780 . . . . . . 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 2513 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,] +oo ) ) )
8 0e0iccpnf 11411 . . . . 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 6122 . . . . . . . 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 3369 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  RR* )
14 xmul01 11245 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x xe 0 )  =  0 )
1513, 14syl 16 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe 0 )  =  0 )
1611, 15eqtrd 2475 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  0 )
1716mpteq2dva 4393 . . . . . 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 4897 . . . . . 6  |-  ( ( 0 [,] +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,] +oo )  |->  0 )
2017, 18, 193eqtr4g 2500 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,] +oo )  X. 
{ 0 } ) )
21 c0ex 9395 . . . . . 6  |-  0  e.  _V
2221fconst2 5949 . . . . 5  |-  ( F : ( 0 [,] +oo ) --> { 0 }  <-> 
F  =  ( ( 0 [,] +oo )  X.  { 0 } ) )
2320, 22sylibr 212 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] +oo ) --> { 0 } )
24 cnconst 18903 . . . 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 1219 . . 3  |-  ( C  =  0  ->  F  e.  ( J  Cn  J
) )
2625adantl 466 . 2  |-  ( (
ph  /\  C  = 
0 )  ->  F  e.  ( J  Cn  J
) )
27 eqid 2443 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
28 oveq1 6113 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
2928cbvmptv 4398 . . . . . . . . 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 26375 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
32 letopuni 18826 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3332cnrest 18904 . . . . . . . 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 5171 . . . . . . . . 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 2466 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  F
381eqcomi 2447 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  J
3938oveq1i 6116 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4034, 37, 393eltr3g 2525 . . . . . 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 11388 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  = 
RR+
44 ioossicc 11396 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  C_  ( 0 [,] +oo )
4543, 44eqsstr3i 3402 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,] +oo )
46 simpl 457 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  RR+ )
4745, 46sseldi 3369 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo ) )
48 ge0xmulcl 11415 . . . . . . . . . 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 5882 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,] +oo )
--> ( 0 [,] +oo ) )
51 frn 5580 . . . . . . . 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 18905 . . . . . . 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 1218 . . . . . 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 6117 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) )
5856, 57syl6eleqr 2534 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
5958, 43eleq2s 2535 . . 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 9445 . . . 4  |-  0  e.  RR*
63 pnfxr 11107 . . . 4  |- +oo  e.  RR*
64 0ltpnf 11118 . . . 4  |-  0  < +oo
65 elicoelioo 26083 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  ->  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) ) )
6662, 63, 64, 65mp3an 1314 . . 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 1369    e. wcel 1756    C_ wss 3343   {csn 3892   class class class wbr 4307    e. cmpt 4365    X. cxp 4853   ran crn 4856    |` cres 4857   -->wf 5429   ` cfv 5433  (class class class)co 6106   0cc0 9297   +oocpnf 9430   RR*cxr 9432    < clt 9433    <_ cle 9434   RR+crp 11006   xecxmu 11103   (,)cioo 11315   [,)cico 11317   [,]cicc 11318   ↾t crest 14374  ordTopcordt 14452  TopOnctopon 18514    Cn ccn 18843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fi 7676  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-rp 11007  df-xneg 11104  df-xmul 11106  df-ioo 11319  df-ico 11321  df-icc 11322  df-rest 14376  df-topgen 14397  df-ordt 14454  df-ps 15385  df-tsr 15386  df-top 18518  df-bases 18520  df-topon 18521  df-cn 18846  df-cnp 18847
This theorem is referenced by:  esummulc1  26545
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