Theorem xrge0mulc1cn 28389

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 20001	. . . . . . 7 |- (ordTop ` <_ ) e. (TopOn ` RR* )
3		iccssxr 11663	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
4		resttopon 19957	. . . . . . 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 2488	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 11	. . . 4 |- ( C = 0 -> J e. (TopOn ` ( 0 [,] +oo ) ) )
8		0e0iccpnf 11687	. . . . 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 6296	. . . . . . . 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 3442	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
14		xmul01 11514	. . . . . . . . 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 2445	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = 0 )
17	16	mpteq2dva 4483	. . . . . 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 4869	. . . . . 6 |- ( ( 0 [,] +oo ) X. { 0 } ) = ( x e. ( 0 [,] +oo ) |-> 0 )
20	17, 18, 19	3eqtr4g 2470	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
21		c0ex 9622	. . . . . 6 |- 0 e. _V
22	21	fconst2 6110	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
23	20, 22	sylibr 214	. . . 4 |- ( C = 0 -> F : ( 0 [,] +oo ) --> { 0 } )
24		cnconst 20080	. . . 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 1233	. . 3 |- ( C = 0 -> F e. ( J Cn J ) )
26	25	adantl 466	. 2 |- ( ( ph /\ C = 0 ) -> F e. ( J Cn J ) )
27		eqid 2404	. . . . . . . . 9 |- (ordTop ` <_ ) = (ordTop ` <_ )
28		oveq1 6287	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
29	28	cbvmptv 4489	. . . . . . . . 9 |- ( x e. RR* |-> ( x xe C ) ) = ( y e. RR* |-> ( y xe C ) )
30		id 23	. . . . . . . . 9 |- ( C e. RR+ -> C e. RR+ )
31	27, 29, 30	xrmulc1cn 28378	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x xe C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
32		letopuni 20003	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
33	32	cnrest 20081	. . . . . . . 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 5145	. . . . . . . . 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 2436	. . . . . . 7 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = F
38	1	eqcomi 2417	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
39	38	oveq1i 6290	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
40	34, 37, 39	3eltr3g 2508	. . . . . 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 11658	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
44		ioossicc 11666	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) C_ ( 0 [,] +oo )
45	43, 44	eqsstr3i 3475	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
46		simpl 457	. . . . . . . . . . 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 11691	. . . . . . . . . 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 6035	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
51		frn 5722	. . . . . . . 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 20082	. . . . . . 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 1232	. . . . . 6 |- ( C e. RR+ -> ( F e. ( J Cn (ordTop ` <_ ) ) <-> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) ) )
56	40, 55	mpbid 212	. . . . 5 |- ( C e. RR+ -> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) )
57	1	oveq2i 6291	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
58	56, 57	syl6eleqr 2503	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
59	58, 43	eleq2s 2512	. . 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 9672	. . . 4 |- 0 e. RR*
63		pnfxr 11376	. . . 4 |- +oo e. RR*
64		0ltpnf 11387	. . . 4 |- 0 < +oo
65		elicoelioo 28050	. . . 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 1328	. . 3 |- ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
67	61, 66	sylib 198	. 2 |- ( ph -> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
68	26, 60, 67	mpjaodan 789	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 186   \/ wo 368   /\ wa 369   = wceq 1407   e. wcel 1844   C_ wss 3416   {csn 3974   class class class wbr 4397   |-> cmpt 4455   X. cxp 4823   ran crn 4826   |` cres 4827   -->wf 5567   ` cfv 5571  (class class class)co 6280   0cc0 9524   +oocpnf 9657   RR*cxr 9659   < clt 9660   <_ cle 9661   RR+crp 11267   xecxmu 11372   (,)cioo 11584   [,)cico 11586   [,]cicc 11587   ↾_t crest 15037  ordTopcordt 15115  TopOnctopon 19689   Cn ccn 20020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-fi 7907  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-div 10250  df-rp 11268  df-xneg 11373  df-xmul 11375  df-ioo 11588  df-ico 11590  df-icc 11591  df-rest 15039  df-topgen 15060  df-ordt 15117  df-ps 16156  df-tsr 16157  df-top 19693  df-bases 19695  df-topon 19696  df-cn 20023  df-cnp 20024

This theorem is referenced by:  esummulc1  28541