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Theorem xrge0mulc1cn 26307
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 18768 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 11374 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
4 resttopon 18724 . . . . . . 7  |-  ( ( (ordTop `  <_  )  e.  (TopOn `  RR* )  /\  ( 0 [,] +oo )  C_  RR* )  ->  (
(ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  (
0 [,] +oo )
) )
52, 3, 4mp2an 667 . . . . . 6  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  e.  (TopOn `  ( 0 [,] +oo ) )
61, 5eqeltri 2511 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,] +oo ) ) )
8 0e0iccpnf 11392 . . . . 5  |-  0  e.  ( 0 [,] +oo )
98a1i 11 . . . 4  |-  ( C  =  0  ->  0  e.  ( 0 [,] +oo ) )
10 simpl 454 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  C  =  0 )
1110oveq2d 6106 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  ( x xe 0 ) )
12 simpr 458 . . . . . . . . . 10  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo )
)
133, 12sseldi 3351 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  RR* )
14 xmul01 11226 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x xe 0 )  =  0 )
1513, 14syl 16 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe 0 )  =  0 )
1611, 15eqtrd 2473 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  0 )
1716mpteq2dva 4375 . . . . . 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 4878 . . . . . 6  |-  ( ( 0 [,] +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,] +oo )  |->  0 )
2017, 18, 193eqtr4g 2498 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,] +oo )  X. 
{ 0 } ) )
21 c0ex 9376 . . . . . 6  |-  0  e.  _V
2221fconst2 5931 . . . . 5  |-  ( F : ( 0 [,] +oo ) --> { 0 }  <-> 
F  =  ( ( 0 [,] +oo )  X.  { 0 } ) )
2320, 22sylibr 212 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] +oo ) --> { 0 } )
24 cnconst 18847 . . . 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 1214 . . 3  |-  ( C  =  0  ->  F  e.  ( J  Cn  J
) )
2625adantl 463 . 2  |-  ( (
ph  /\  C  = 
0 )  ->  F  e.  ( J  Cn  J
) )
27 eqid 2441 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
28 oveq1 6097 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
2928cbvmptv 4380 . . . . . . . . 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 26296 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
32 letopuni 18770 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3332cnrest 18848 . . . . . . . 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 657 . . . . . . 7  |-  ( C  e.  RR+  ->  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  e.  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop ` 
<_  ) ) )
35 resmpt 5153 . . . . . . . . 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 2464 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  F
381eqcomi 2445 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  J
3938oveq1i 6100 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4034, 37, 393eltr3g 2523 . . . . . 6  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  (ordTop ` 
<_  ) ) )
412a1i 11 . . . . . . 7  |-  ( C  e.  RR+  ->  (ordTop `  <_  )  e.  (TopOn `  RR* ) )
42 simpr 458 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  ( 0 [,] +oo ) )
43 ioorp 11369 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  = 
RR+
44 ioossicc 11377 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  C_  ( 0 [,] +oo )
4543, 44eqsstr3i 3384 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,] +oo )
46 simpl 454 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  RR+ )
4745, 46sseldi 3351 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo ) )
48 ge0xmulcl 11396 . . . . . . . . . 10  |-  ( ( x  e.  ( 0 [,] +oo )  /\  C  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  e.  ( 0 [,] +oo ) )
4942, 47, 48syl2anc 656 . . . . . . . . 9  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  (
x xe C )  e.  ( 0 [,] +oo ) )
5049, 18fmptd 5864 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,] +oo )
--> ( 0 [,] +oo ) )
51 frn 5562 . . . . . . . 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 18849 . . . . . . 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 1213 . . . . . 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 6101 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) )
5856, 57syl6eleqr 2532 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
5958, 43eleq2s 2533 . . 3  |-  ( C  e.  ( 0 (,) +oo )  ->  F  e.  ( J  Cn  J
) )
6059adantl 463 . 2  |-  ( (
ph  /\  C  e.  ( 0 (,) +oo ) )  ->  F  e.  ( J  Cn  J
) )
61 xrge0mulc1cn.c . . 3  |-  ( ph  ->  C  e.  ( 0 [,) +oo ) )
62 0xr 9426 . . . 4  |-  0  e.  RR*
63 pnfxr 11088 . . . 4  |- +oo  e.  RR*
64 0ltpnf 11099 . . . 4  |-  0  < +oo
65 elicoelioo 26001 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  ->  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) ) )
6662, 63, 64, 65mp3an 1309 . . 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 779 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1364    e. wcel 1761    C_ wss 3325   {csn 3874   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837    |` cres 4838   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278   +oocpnf 9411   RR*cxr 9413    < clt 9414    <_ cle 9415   RR+crp 10987   xecxmu 11084   (,)cioo 11296   [,)cico 11298   [,]cicc 11299   ↾t crest 14355  ordTopcordt 14433  TopOnctopon 18458    Cn ccn 18787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fi 7657  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-rp 10988  df-xneg 11085  df-xmul 11087  df-ioo 11300  df-ico 11302  df-icc 11303  df-rest 14357  df-topgen 14378  df-ordt 14435  df-ps 15366  df-tsr 15367  df-top 18462  df-bases 18464  df-topon 18465  df-cn 18790  df-cnp 18791
This theorem is referenced by:  esummulc1  26466
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