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Theorem xrge0mulc1cn 24132
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 17192 . . . . . . 7  |-  (ordTop `  <_  )  e.  (TopOn `  RR* )
3 iccssxr 10926 . . . . . . 7  |-  ( 0 [,]  +oo )  C_  RR*
4 resttopon 17148 . . . . . . 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 2458 . . . . 5  |-  J  e.  (TopOn `  ( 0 [,]  +oo ) )
76a1i 11 . . . 4  |-  ( C  =  0  ->  J  e.  (TopOn `  ( 0 [,]  +oo ) ) )
8 0xr 9065 . . . . . 6  |-  0  e.  RR*
9 pnfxr 10646 . . . . . 6  |-  +oo  e.  RR*
10 pnfge 10660 . . . . . . 7  |-  ( 0  e.  RR*  ->  0  <_  +oo )
118, 10ax-mp 8 . . . . . 6  |-  0  <_  +oo
12 lbicc2 10946 . . . . . 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 6037 . . . . . . . 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 3290 . . . . . . . . 9  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  ->  x  e.  RR* )
19 xmul01 10779 . . . . . . . . 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 2420 . . . . . . 7  |-  ( ( C  =  0  /\  x  e.  ( 0 [,]  +oo ) )  -> 
( x x e C )  =  0 )
2221mpteq2dva 4237 . . . . . 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 4862 . . . . . 6  |-  ( ( 0 [,]  +oo )  X.  { 0 } )  =  ( x  e.  ( 0 [,]  +oo )  |->  0 )
2522, 23, 243eqtr4g 2445 . . . . 5  |-  ( C  =  0  ->  F  =  ( ( 0 [,]  +oo )  X.  {
0 } ) )
26 c0ex 9019 . . . . . 6  |-  0  e.  _V
2726fconst2 5888 . . . . 5  |-  ( F : ( 0 [,] 
+oo ) --> { 0 }  <->  F  =  (
( 0 [,]  +oo )  X.  { 0 } ) )
2825, 27sylibr 204 . . . 4  |-  ( C  =  0  ->  F : ( 0 [,] 
+oo ) --> { 0 } )
29 cnconst 17271 . . . 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 2388 . . . . . . . . 9  |-  (ordTop `  <_  )  =  (ordTop `  <_  )
33 oveq1 6028 . . . . . . . . . 10  |-  ( x  =  y  ->  (
x x e C )  =  ( y x e C ) )
3433cbvmptv 4242 . . . . . . . . 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 24121 . . . . . . . 8  |-  ( C  e.  RR+  ->  ( x  e.  RR*  |->  ( x x e C ) )  e.  ( (ordTop `  <_  )  Cn  (ordTop ` 
<_  ) ) )
37 letopuni 17194 . . . . . . . . 9  |-  RR*  =  U. (ordTop `  <_  )
3837cnrest 17272 . . . . . . . 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 5132 . . . . . . . . 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 2411 . . . . . . 7  |-  ( ( x  e.  RR*  |->  ( x x e C ) )  |`  ( 0 [,]  +oo ) )  =  F
431eqcomi 2392 . . . . . . . 8  |-  ( (ordTop `  <_  )t  ( 0 [,] 
+oo ) )  =  J
4443oveq1i 6031 . . . . . . 7  |-  ( ( (ordTop `  <_  )t  ( 0 [,]  +oo ) )  Cn  (ordTop `  <_  ) )  =  ( J  Cn  (ordTop `  <_  ) )
4539, 42, 443eltr3g 2470 . . . . . 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 10921 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  =  RR+
49 ioossicc 10929 . . . . . . . . . . . 12  |-  ( 0 (,)  +oo )  C_  (
0 [,]  +oo )
5048, 49eqsstr3i 3323 . . . . . . . . . . 11  |-  RR+  C_  (
0 [,]  +oo )
51 simpl 444 . . . . . . . . . . 11  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  RR+ )
5250, 51sseldi 3290 . . . . . . . . . 10  |-  ( ( C  e.  RR+  /\  x  e.  ( 0 [,]  +oo ) )  ->  C  e.  ( 0 [,]  +oo ) )
53 ge0xmulcl 10945 . . . . . . . . . 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 5833 . . . . . . . 8  |-  ( C  e.  RR+  ->  F :
( 0 [,]  +oo )
--> ( 0 [,]  +oo ) )
56 frn 5538 . . . . . . . 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 17273 . . . . . . 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 6032 . . . . 5  |-  ( J  Cn  J )  =  ( J  Cn  (
(ordTop `  <_  )t  ( 0 [,]  +oo ) ) )
6361, 62syl6eleqr 2479 . . . 4  |-  ( C  e.  RR+  ->  F  e.  ( J  Cn  J
) )
6463, 48eleq2s 2480 . . 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 9025 . . . . 5  |-  0  e.  RR
68 ltpnf 10654 . . . . 5  |-  ( 0  e.  RR  ->  0  <  +oo )
6967, 68ax-mp 8 . . . 4  |-  0  <  +oo
70 elicoelioo 23978 . . . 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 1717    C_ wss 3264   {csn 3758   class class class wbr 4154    e. cmpt 4208    X. cxp 4817   ran crn 4820    |` cres 4821   -->wf 5391   ` cfv 5395  (class class class)co 6021   RRcr 8923   0cc0 8924    +oocpnf 9051   RR*cxr 9053    < clt 9054    <_ cle 9055   RR+crp 10545   x ecxmu 10642   (,)cioo 10849   [,)cico 10851   [,]cicc 10852   ↾t crest 13576  ordTopcordt 13649  TopOnctopon 16883    Cn ccn 17211
This theorem is referenced by:  esummulc1  24268
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-rp 10546  df-xneg 10643  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-rest 13578  df-topgen 13595  df-ordt 13653  df-ps 14557  df-tsr 14558  df-top 16887  df-bases 16889  df-topon 16890  df-cn 17214  df-cnp 17215
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