Theorem xrge0mulc1cn 28798

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.

Step	Hyp	Ref	Expression
1		xrge0mulc1cn.k	. . . . . 6 |- J = ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) )
2		letopon 20276	. . . . . . 7 |- (ordTop ` <_ ) e. (TopOn ` RR* )
3		iccssxr 11751	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
4		resttopon 20232	. . . . . . 7 |- ( ( (ordTop ` <_ ) e. (TopOn ` RR* ) /\ ( 0 [,] +oo ) C_ RR* ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) ) )
5	2, 3, 4	mp2an 683	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) )
6	1, 5	eqeltri 2536	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 11	. . . 4 |- ( C = 0 -> J e. (TopOn ` ( 0 [,] +oo ) ) )
8		0e0iccpnf 11778	. . . . 5 |- 0 e. ( 0 [,] +oo )
9	8	a1i 11	. . . 4 |- ( C = 0 -> 0 e. ( 0 [,] +oo ) )
10		simpl 463	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> C = 0 )
11	10	oveq2d 6336	. . . . . . . 8 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = ( x xe 0 ) )
12		simpr 467	. . . . . . . . . 10 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
13	3, 12	sseldi 3442	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
14		xmul01 11587	. . . . . . . . 9 |- ( x e. RR* -> ( x xe 0 ) = 0 )
15	13, 14	syl 17	. . . . . . . 8 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe 0 ) = 0 )
16	11, 15	eqtrd 2496	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = 0 )
17	16	mpteq2dva 4505	. . . . . 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 4900	. . . . . 6 |- ( ( 0 [,] +oo ) X. { 0 } ) = ( x e. ( 0 [,] +oo ) |-> 0 )
20	17, 18, 19	3eqtr4g 2521	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
21		c0ex 9668	. . . . . 6 |- 0 e. _V
22	21	fconst2 6150	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
23	20, 22	sylibr 217	. . . 4 |- ( C = 0 -> F : ( 0 [,] +oo ) --> { 0 } )
24		cnconst 20355	. . . 4 |- ( ( ( J e. (TopOn ` ( 0 [,] +oo ) ) /\ J e. (TopOn ` ( 0 [,] +oo ) ) ) /\ ( 0 e. ( 0 [,] +oo ) /\ F : ( 0 [,] +oo ) --> { 0 } ) ) -> F e. ( J Cn J ) )
25	7, 7, 9, 23, 24	syl22anc 1277	. . 3 |- ( C = 0 -> F e. ( J Cn J ) )
26	25	adantl 472	. 2 |- ( ( ph /\ C = 0 ) -> F e. ( J Cn J ) )
27		eqid 2462	. . . . . . . . 9 |- (ordTop ` <_ ) = (ordTop ` <_ )
28		oveq1 6327	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
29	28	cbvmptv 4511	. . . . . . . . 9 |- ( x e. RR* |-> ( x xe C ) ) = ( y e. RR* |-> ( y xe C ) )
30		id 22	. . . . . . . . 9 |- ( C e. RR+ -> C e. RR+ )
31	27, 29, 30	xrmulc1cn 28787	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x xe C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
32		letopuni 20278	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
33	32	cnrest 20356	. . . . . . . 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 ` <_ ) ) )
34	31, 3, 33	sylancl 673	. . . . . . 7 |- ( C e. RR+ -> ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) e. ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) )
35		resmpt 5176	. . . . . . . . 9 |- ( ( 0 [,] +oo ) C_ RR* -> ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = ( x e. ( 0 [,] +oo ) |-> ( x xe C ) ) )
36	3, 35	ax-mp 5	. . . . . . . 8 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = ( x e. ( 0 [,] +oo ) |-> ( x xe C ) )
37	36, 18	eqtr4i 2487	. . . . . . 7 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = F
38	1	eqcomi 2471	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
39	38	oveq1i 6330	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
40	34, 37, 39	3eltr3g 2556	. . . . . 6 |- ( C e. RR+ -> F e. ( J Cn (ordTop ` <_ ) ) )
41	2	a1i 11	. . . . . . 7 |- ( C e. RR+ -> (ordTop ` <_ ) e. (TopOn ` RR* ) )
42		simpr 467	. . . . . . . . . 10 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
43		ioorp 11746	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
44		ioossicc 11754	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) C_ ( 0 [,] +oo )
45	43, 44	eqsstr3i 3475	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
46		simpl 463	. . . . . . . . . . 11 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. RR+ )
47	45, 46	sseldi 3442	. . . . . . . . . 10 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. ( 0 [,] +oo ) )
48		ge0xmulcl 11782	. . . . . . . . . 10 |- ( ( x e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( x xe C ) e. ( 0 [,] +oo ) )
49	42, 47, 48	syl2anc 671	. . . . . . . . 9 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) e. ( 0 [,] +oo ) )
50	49, 18	fmptd 6074	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
51		frn 5762	. . . . . . . 8 |- ( F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) -> ran F C_ ( 0 [,] +oo ) )
52	50, 51	syl 17	. . . . . . 7 |- ( C e. RR+ -> ran F C_ ( 0 [,] +oo ) )
53	3	a1i 11	. . . . . . 7 |- ( C e. RR+ -> ( 0 [,] +oo ) C_ RR* )
54		cnrest2 20357	. . . . . . 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 ) ) ) ) )
55	41, 52, 53, 54	syl3anc 1276	. . . . . 6 |- ( C e. RR+ -> ( F e. ( J Cn (ordTop ` <_ ) ) <-> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) ) )
56	40, 55	mpbid 215	. . . . 5 |- ( C e. RR+ -> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) )
57	1	oveq2i 6331	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
58	56, 57	syl6eleqr 2551	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
59	58, 43	eleq2s 2558	. . 3 |- ( C e. ( 0 (,) +oo ) -> F e. ( J Cn J ) )
60	59	adantl 472	. 2 |- ( ( ph /\ C e. ( 0 (,) +oo ) ) -> F e. ( J Cn J ) )
61		xrge0mulc1cn.c	. . 3 |- ( ph -> C e. ( 0 [,) +oo ) )
62		0xr 9718	. . . 4 |- 0 e. RR*
63		pnfxr 11446	. . . 4 |- +oo e. RR*
64		0ltpnf 11458	. . . 4 |- 0 < +oo
65		elicoelioo 28412	. . . 4 |- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 < +oo ) -> ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) ) )
66	62, 63, 64, 65	mp3an 1373	. . 3 |- ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
67	61, 66	sylib 201	. 2 |- ( ph -> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
68	26, 60, 67	mpjaodan 800	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   C_ wss 3416   {csn 3980   class class class wbr 4418   |-> cmpt 4477   X. cxp 4854   ran crn 4857   |` cres 4858   -->wf 5601   ` cfv 5605  (class class class)co 6320   0cc0 9570   +oocpnf 9703   RR*cxr 9705   < clt 9706   <_ cle 9707   RR+crp 11336   xecxmu 11442   (,)cioo 11669   [,)cico 11671   [,]cicc 11672   ↾_t crest 15374  ordTopcordt 15452  TopOnctopon 19973   Cn ccn 20295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-fi 7956  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-rp 11337  df-xneg 11443  df-xmul 11445  df-ioo 11673  df-ico 11675  df-icc 11676  df-rest 15376  df-topgen 15397  df-ordt 15454  df-ps 16501  df-tsr 16502  df-top 19976  df-bases 19977  df-topon 19978  df-cn 20298  df-cnp 20299

This theorem is referenced by:  esummulc1  28953