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Theorem xrge0mulc1cn 27796
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 19579 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 11616 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
4 resttopon 19535 . . . . . . 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 2527 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,] +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,] +oo ) ) )
8 0e0iccpnf 11640 . . . . 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 6297 . . . . . . . 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 3487 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  x  e.  RR* )
14 xmul01 11468 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x xe 0 )  =  0 )
1513, 14syl 16 . . . . . . . 8  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe 0 )  =  0 )
1611, 15eqtrd 2484 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,] +oo ) )  ->  ( x xe C )  =  0 )
1716mpteq2dva 4523 . . . . . 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 5033 . . . . . 6  |-  ( ( 0 [,] +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,] +oo )  |->  0 )
2017, 18, 193eqtr4g 2509 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,] +oo )  X. 
{ 0 } ) )
21 c0ex 9593 . . . . . 6  |-  0  e.  _V
2221fconst2 6112 . . . . 5  |-  ( F : ( 0 [,] +oo ) --> { 0 }  <-> 
F  =  ( ( 0 [,] +oo )  X.  { 0 } ) )
2320, 22sylibr 212 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] +oo ) --> { 0 } )
24 cnconst 19658 . . . 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 1230 . . 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 6288 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x xe C )  =  ( y xe C ) )
2928cbvmptv 4528 . . . . . . . . 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 27785 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x xe C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
32 letopuni 19581 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3332cnrest 19659 . . . . . . . 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 5313 . . . . . . . . 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 2475 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x xe C ) )  |`  ( 0 [,] +oo ) )  =  F
381eqcomi 2456 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  =  J
3938oveq1i 6291 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,] +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4034, 37, 393eltr3g 2547 . . . . . 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 11611 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  = 
RR+
44 ioossicc 11619 . . . . . . . . . . . 12  |-  ( 0 (,) +oo )  C_  ( 0 [,] +oo )
4543, 44eqsstr3i 3520 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,] +oo )
46 simpl 457 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  RR+ )
4745, 46sseldi 3487 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,] +oo ) )  ->  C  e.  ( 0 [,] +oo ) )
48 ge0xmulcl 11644 . . . . . . . . . 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 6040 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,] +oo )
--> ( 0 [,] +oo ) )
51 frn 5727 . . . . . . . 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 19660 . . . . . . 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 1229 . . . . . 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 6292 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,] +oo ) ) )
5856, 57syl6eleqr 2542 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
5958, 43eleq2s 2551 . . 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 9643 . . . 4  |-  0  e.  RR*
63 pnfxr 11330 . . . 4  |- +oo  e.  RR*
64 0ltpnf 11341 . . . 4  |-  0  < +oo
65 elicoelioo 27461 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  0  < +oo )  ->  ( C  e.  ( 0 [,) +oo )  <->  ( C  =  0  \/  C  e.  ( 0 (,) +oo ) ) ) )
6662, 63, 64, 65mp3an 1325 . . 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 786 1  |-  ( ph  ->  F  e.  ( J  Cn  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    C_ wss 3461   {csn 4014   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   ran crn 4990    |` cres 4991   -->wf 5574   ` cfv 5578  (class class class)co 6281   0cc0 9495   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632   RR+crp 11229   xecxmu 11326   (,)cioo 11538   [,)cico 11540   [,]cicc 11541   ↾t crest 14695  ordTopcordt 14773  TopOnctopon 19268    Cn ccn 19598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fi 7873  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-rp 11230  df-xneg 11327  df-xmul 11329  df-ioo 11542  df-ico 11544  df-icc 11545  df-rest 14697  df-topgen 14718  df-ordt 14775  df-ps 15704  df-tsr 15705  df-top 19272  df-bases 19274  df-topon 19275  df-cn 19601  df-cnp 19602
This theorem is referenced by:  esummulc1  27960
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