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.

Step	Hyp	Ref	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 ) ) )
5	2, 3, 4	mp2an 672	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) )
6	1, 5	eqeltri 2527	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 11	. . . 4 |- ( C = 0 -> J e. (TopOn ` ( 0 [,] +oo ) ) )
8		0e0iccpnf 11640	. . . . 5 |- 0 e. ( 0 [,] +oo )
9	8	a1i 11	. . . 4 |- ( C = 0 -> 0 e. ( 0 [,] +oo ) )
10		simpl 457	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> C = 0 )
11	10	oveq2d 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 ) )
13	3, 12	sseldi 3487	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
14		xmul01 11468	. . . . . . . . 9 |- ( x e. RR* -> ( x xe 0 ) = 0 )
15	13, 14	syl 16	. . . . . . . 8 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe 0 ) = 0 )
16	11, 15	eqtrd 2484	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = 0 )
17	16	mpteq2dva 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 )
20	17, 18, 19	3eqtr4g 2509	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
21		c0ex 9593	. . . . . 6 |- 0 e. _V
22	21	fconst2 6112	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
23	20, 22	sylibr 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 ) )
25	7, 7, 9, 23, 24	syl22anc 1230	. . 3 |- ( C = 0 -> F e. ( J Cn J ) )
26	25	adantl 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 ) )
29	28	cbvmptv 4528	. . . . . . . . 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 27785	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x xe C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
32		letopuni 19581	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
33	32	cnrest 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 ` <_ ) ) )
34	31, 3, 33	sylancl 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 ) ) )
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 2475	. . . . . . 7 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = F
38	1	eqcomi 2456	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
39	38	oveq1i 6291	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
40	34, 37, 39	3eltr3g 2547	. . . . . 6 |- ( C e. RR+ -> F e. ( J Cn (ordTop ` <_ ) ) )
41	2	a1i 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 )
45	43, 44	eqsstr3i 3520	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
46		simpl 457	. . . . . . . . . . 11 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. RR+ )
47	45, 46	sseldi 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 ) )
49	42, 47, 48	syl2anc 661	. . . . . . . . 9 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) e. ( 0 [,] +oo ) )
50	49, 18	fmptd 6040	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
51		frn 5727	. . . . . . . 8 |- ( F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) -> ran F C_ ( 0 [,] +oo ) )
52	50, 51	syl 16	. . . . . . 7 |- ( C e. RR+ -> ran F C_ ( 0 [,] +oo ) )
53	3	a1i 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 ) ) ) ) )
55	41, 52, 53, 54	syl3anc 1229	. . . . . 6 |- ( C e. RR+ -> ( F e. ( J Cn (ordTop ` <_ ) ) <-> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) ) )
56	40, 55	mpbid 210	. . . . 5 |- ( C e. RR+ -> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) )
57	1	oveq2i 6292	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
58	56, 57	syl6eleqr 2542	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
59	58, 43	eleq2s 2551	. . 3 |- ( C e. ( 0 (,) +oo ) -> F e. ( J Cn J ) )
60	59	adantl 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 ) ) ) )
66	62, 63, 64, 65	mp3an 1325	. . 3 |- ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
67	61, 66	sylib 196	. 2 |- ( ph -> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
68	26, 60, 67	mpjaodan 786	1 |- ( ph -> F e. ( J Cn J ) )

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