Theorem xrge0mulc1cn 29987

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 21009	. . . . . . 7 |- (ordTop ` <_ ) e. (TopOn ` RR* )
3		iccssxr 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
4		resttopon 20965	. . . . . . 7 |- ( ( (ordTop ` <_ ) e. (TopOn ` RR* ) /\ ( 0 [,] +oo ) C_ RR* ) -> ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) ) )
5	2, 3, 4	mp2an 708	. . . . . 6 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) e. (TopOn ` ( 0 [,] +oo ) )
6	1, 5	eqeltri 2697	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 11	. . . 4 |- ( C = 0 -> J e. (TopOn ` ( 0 [,] +oo ) ) )
8		0e0iccpnf 12283	. . . . 5 |- 0 e. ( 0 [,] +oo )
9	8	a1i 11	. . . 4 |- ( C = 0 -> 0 e. ( 0 [,] +oo ) )
10		simpl 473	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> C = 0 )
11	10	oveq2d 6666	. . . . . . . 8 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = ( x xe 0 ) )
12		simpr 477	. . . . . . . . . 10 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
13	3, 12	sseldi 3601	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
14		xmul01 12097	. . . . . . . . 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 2656	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = 0 )
17	16	mpteq2dva 4744	. . . . . 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 5163	. . . . . 6 |- ( ( 0 [,] +oo ) X. { 0 } ) = ( x e. ( 0 [,] +oo ) |-> 0 )
20	17, 18, 19	3eqtr4g 2681	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
21		c0ex 10034	. . . . . 6 |- 0 e. _V
22	21	fconst2 6470	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
23	20, 22	sylibr 224	. . . 4 |- ( C = 0 -> F : ( 0 [,] +oo ) --> { 0 } )
24		cnconst 21088	. . . 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 1327	. . 3 |- ( C = 0 -> F e. ( J Cn J ) )
26	25	adantl 482	. 2 |- ( ( ph /\ C = 0 ) -> F e. ( J Cn J ) )
27		eqid 2622	. . . . . . . . 9 |- (ordTop ` <_ ) = (ordTop ` <_ )
28		oveq1 6657	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
29	28	cbvmptv 4750	. . . . . . . . 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 29976	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x xe C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
32		letopuni 21011	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
33	32	cnrest 21089	. . . . . . . 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 694	. . . . . . 7 |- ( C e. RR+ -> ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) e. ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) )
35		resmpt 5449	. . . . . . . . 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 2647	. . . . . . 7 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = F
38	1	eqcomi 2631	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
39	38	oveq1i 6660	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
40	34, 37, 39	3eltr3g 2717	. . . . . 6 |- ( C e. RR+ -> F e. ( J Cn (ordTop ` <_ ) ) )
41	2	a1i 11	. . . . . . 7 |- ( C e. RR+ -> (ordTop ` <_ ) e. (TopOn ` RR* ) )
42		simpr 477	. . . . . . . . . 10 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> x e. ( 0 [,] +oo ) )
43		ioorp 12251	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
44		ioossicc 12259	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) C_ ( 0 [,] +oo )
45	43, 44	eqsstr3i 3636	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
46		simpl 473	. . . . . . . . . . 11 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. RR+ )
47	45, 46	sseldi 3601	. . . . . . . . . 10 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. ( 0 [,] +oo ) )
48		ge0xmulcl 12287	. . . . . . . . . 10 |- ( ( x e. ( 0 [,] +oo ) /\ C e. ( 0 [,] +oo ) ) -> ( x xe C ) e. ( 0 [,] +oo ) )
49	42, 47, 48	syl2anc 693	. . . . . . . . 9 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) e. ( 0 [,] +oo ) )
50	49, 18	fmptd 6385	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
51		frn 6053	. . . . . . . 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 21090	. . . . . . 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 1326	. . . . . 6 |- ( C e. RR+ -> ( F e. ( J Cn (ordTop ` <_ ) ) <-> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) ) )
56	40, 55	mpbid 222	. . . . 5 |- ( C e. RR+ -> F e. ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) ) )
57	1	oveq2i 6661	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
58	56, 57	syl6eleqr 2712	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
59	58, 43	eleq2s 2719	. . 3 |- ( C e. ( 0 (,) +oo ) -> F e. ( J Cn J ) )
60	59	adantl 482	. 2 |- ( ( ph /\ C e. ( 0 (,) +oo ) ) -> F e. ( J Cn J ) )
61		xrge0mulc1cn.c	. . 3 |- ( ph -> C e. ( 0 [,) +oo ) )
62		0xr 10086	. . . 4 |- 0 e. RR*
63		pnfxr 10092	. . . 4 |- +oo e. RR*
64		0ltpnf 11956	. . . 4 |- 0 < +oo
65		elicoelioo 29540	. . . 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 1424	. . 3 |- ( C e. ( 0 [,) +oo ) <-> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
67	61, 66	sylib 208	. 2 |- ( ph -> ( C = 0 \/ C e. ( 0 (,) +oo ) ) )
68	26, 60, 67	mpjaodan 827	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   xecxmu 11945   (,)cioo 12175   [,)cico 12177   [,]cicc 12178   ↾_t crest 16081  ordTopcordt 16159  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833  df-xneg 11946  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-rest 16083  df-topgen 16104  df-ordt 16161  df-ps 17200  df-tsr 17201  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-cnp 21032

This theorem is referenced by:  esummulc1  30143