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.

Step	Hyp	Ref	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 ) ) )
5	2, 3, 4	mp2an 654	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) )
6	1, 5	eqeltri 2474	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 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 )
11	8, 10	ax-mp 8	. . . . . 6 |- 0 <_ +oo
12		lbicc2 10969	. . . . . 6 |- ( ( 0 e. RR* /\ +oo e. RR* /\ 0 <_ +oo ) -> 0 e. ( 0 [,] +oo ) )
13	8, 9, 11, 12	mp3an 1279	. . . . 5 |- 0 e. ( 0 [,] +oo )
14	13	a1i 11	. . . 4 |- ( C = 0 -> 0 e. ( 0 [,] +oo ) )
15		simpl 444	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> C = 0 )
16	15	oveq2d 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 ) )
18	3, 17	sseldi 3306	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
19		xmul01 10802	. . . . . . . . 9 |- ( x e. RR* -> ( x x e 0 ) = 0 )
20	18, 19	syl 16	. . . . . . . 8 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x x e 0 ) = 0 )
21	16, 20	eqtrd 2436	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x x e C ) = 0 )
22	21	mpteq2dva 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 )
25	22, 23, 24	3eqtr4g 2461	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
26		c0ex 9041	. . . . . 6 |- 0 e. _V
27	26	fconst2 5907	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
28	25, 27	sylibr 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 ) )
30	7, 7, 14, 28, 29	syl22anc 1185	. . 3 |- ( C = 0 -> F e. ( J Cn J ) )
31	30	adantl 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 ) )
34	33	cbvmptv 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+ )
36	32, 34, 35	xrmulc1cn 24269	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x x e C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
37		letopuni 17225	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
38	37	cnrest 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 ` <_ ) ) )
39	36, 3, 38	sylancl 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 ) ) )
41	3, 40	ax-mp 8	. . . . . . . 8 |- ( ( x e. RR* |-> ( x x e C ) ) |` ( 0 [,] +oo ) ) = ( x e. ( 0 [,] +oo ) |-> ( x x e C ) )
42	41, 23	eqtr4i 2427	. . . . . . 7 |- ( ( x e. RR* |-> ( x x e C ) ) |` ( 0 [,] +oo ) ) = F
43	1	eqcomi 2408	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
44	43	oveq1i 6050	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
45	39, 42, 44	3eltr3g 2486	. . . . . 6 |- ( C e. RR+ -> F e. ( J Cn (ordTop ` <_ ) ) )
46	2	a1i 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 )
50	48, 49	eqsstr3i 3339	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
51		simpl 444	. . . . . . . . . . 11 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. RR+ )
52	50, 51	sseldi 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 ) )
54	47, 52, 53	syl2anc 643	. . . . . . . . 9 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> ( x x e C ) e. ( 0 [,] +oo ) )
55	54, 23	fmptd 5852	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
56		frn 5556	. . . . . . . 8 |- ( F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) -> ran F C_ ( 0 [,] +oo ) )
57	55, 56	syl 16	. . . . . . 7 |- ( C e. RR+ -> ran F C_ ( 0 [,] +oo ) )
58	3	a1i 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 ) ) ) ) )
60	46, 57, 58, 59	syl3anc 1184	. . . . . 6 |- ( C e. RR+ -> ( F e. ( J Cn (ordTop ` <_ ) ) <-> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) ) )
61	45, 60	mpbid 202	. . . . 5 |- ( C e. RR+ -> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) )
62	1	oveq2i 6051	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
63	61, 62	syl6eleqr 2495	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
64	63, 48	eleq2s 2496	. . 3 |- ( C e. ( 0 (,) +oo ) -> F e. ( J Cn J ) )
65	64	adantl 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 )
69	67, 68	ax-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 ) ) ) )
71	8, 9, 69, 70	mp3an 1279	. . 3 |- ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
72	66, 71	sylib 189	. 2 |- ( ph -> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
73	31, 65, 72	mpjaodan 762	1 |- ( ph -> F e. ( J Cn J ) )

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