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Theorem xrge0mulc1cn 24280
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 17223 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 10949 . . . . . . 7  |-  ( 0 [,]  +oo )  C_  RR*
4 resttopon 17179 . . . . . . 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 2474 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,]  +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,]  +oo ) ) )
8 0xr 9087 . . . . . 6  |-  0  e.  RR*
9 pnfxr 10669 . . . . . 6  |-  +oo  e.  RR*
10 pnfge 10683 . . . . . . 7  |-  ( 0  e.  RR*  ->  0  <_  +oo )
118, 10ax-mp 8 . . . . . 6  |-  0  <_  +oo
12 lbicc2 10969 . . . . . 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 6056 . . . . . . . 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 3306 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  RR* )
19 xmul01 10802 . . . . . . . . 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 2436 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x x e C )  =  0 )
2221mpteq2dva 4255 . . . . . 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 4880 . . . . . 6  |-  ( ( 0 [,]  +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,]  +oo )  |->  0 )
2522, 23, 243eqtr4g 2461 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,]  +oo )  X.  {
0 } ) )
26 c0ex 9041 . . . . . 6  |-  0  e.  _V
2726fconst2 5907 . . . . 5  |-  ( F : ( 0 [,] 
+oo ) --> { 0 }  <->  F  =  (
( 0 [,]  +oo )  X.  { 0 } ) )
2825, 27sylibr 204 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] 
+oo ) --> { 0 } )
29 cnconst 17302 . . . 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 2404 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
33 oveq1 6047 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x x e C )  =  ( y x e C ) )
3433cbvmptv 4260 . . . . . . . . 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 24269 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x x e C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
37 letopuni 17225 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3837cnrest 17303 . . . . . . . 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 5150 . . . . . . . . 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 2427 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  =  F
431eqcomi 2408 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  J
4443oveq1i 6050 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4539, 42, 443eltr3g 2486 . . . . . 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 10944 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
49 ioossicc 10952 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  C_  (
0 [,]  +oo )
5048, 49eqsstr3i 3339 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,]  +oo )
51 simpl 444 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  RR+ )
5250, 51sseldi 3306 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  ( 0 [,]  +oo ) )
53 ge0xmulcl 10968 . . . . . . . . . 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 5852 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,]  +oo )
--> ( 0 [,]  +oo ) )
56 frn 5556 . . . . . . . 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 17304 . . . . . . 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 6051 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) )
6361, 62syl6eleqr 2495 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
6463, 48eleq2s 2496 . . 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 9047 . . . . 5  |-  0  e.  RR
68 ltpnf 10677 . . . . 5  |-  ( 0  e.  RR  ->  0  <  +oo )
6967, 68ax-mp 8 . . . 4  |-  0  <  +oo
70 elicoelioo 24094 . . . 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 1649    e. wcel 1721    C_ wss 3280   {csn 3774   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   ran crn 4838    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946    +oocpnf 9073   RR*cxr 9075    < clt 9076    <_ cle 9077   RR+crp 10568   x ecxmu 10665   (,)cioo 10872   [,)cico 10874   [,]cicc 10875   ↾t crest 13603  ordTopcordt 13676  TopOnctopon 16914    Cn ccn 17242
This theorem is referenced by:  esummulc1  24424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-rp 10569  df-xneg 10666  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-rest 13605  df-topgen 13622  df-ordt 13680  df-ps 14584  df-tsr 14585  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245  df-cnp 17246
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