Theorem xrge0mulc1cn 26386

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 18824	. . . . . . 7 |- (ordTop ` <_ ) e. (TopOn ` RR* )
3		iccssxr 11393	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
4		resttopon 18780	. . . . . . 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 2513	. . . . 5 |- J e. (TopOn ` ( 0 [,] +oo ) )
7	6	a1i 11	. . . 4 |- ( C = 0 -> J e. (TopOn ` ( 0 [,] +oo ) ) )
8		0e0iccpnf 11411	. . . . 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 6122	. . . . . . . 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 3369	. . . . . . . . 9 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> x e. RR* )
14		xmul01 11245	. . . . . . . . 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 2475	. . . . . . 7 |- ( ( C = 0 /\ x e. ( 0 [,] +oo ) ) -> ( x xe C ) = 0 )
17	16	mpteq2dva 4393	. . . . . 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 4897	. . . . . 6 |- ( ( 0 [,] +oo ) X. { 0 } ) = ( x e. ( 0 [,] +oo ) |-> 0 )
20	17, 18, 19	3eqtr4g 2500	. . . . 5 |- ( C = 0 -> F = ( ( 0 [,] +oo ) X. { 0 } ) )
21		c0ex 9395	. . . . . 6 |- 0 e. _V
22	21	fconst2 5949	. . . . 5 |- ( F : ( 0 [,] +oo ) --> { 0 } <-> F = ( ( 0 [,] +oo ) X. { 0 } ) )
23	20, 22	sylibr 212	. . . 4 |- ( C = 0 -> F : ( 0 [,] +oo ) --> { 0 } )
24		cnconst 18903	. . . 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 1219	. . 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 6113	. . . . . . . . . 10 |- ( x = y -> ( x xe C ) = ( y xe C ) )
29	28	cbvmptv 4398	. . . . . . . . 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 26375	. . . . . . . 8 |- ( C e. RR+ -> ( x e. RR* |-> ( x xe C ) ) e. ( (ordTop ` <_ ) Cn (ordTop ` <_ ) ) )
32		letopuni 18826	. . . . . . . . 9 |- RR* = U. (ordTop ` <_ )
33	32	cnrest 18904	. . . . . . . 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 5171	. . . . . . . . 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 2466	. . . . . . 7 |- ( ( x e. RR* |-> ( x xe C ) ) |` ( 0 [,] +oo ) ) = F
38	1	eqcomi 2447	. . . . . . . 8 |- ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) = J
39	38	oveq1i 6116	. . . . . . 7 |- ( ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) Cn (ordTop ` <_ ) ) = ( J Cn (ordTop ` <_ ) )
40	34, 37, 39	3eltr3g 2525	. . . . . 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 11388	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) = RR+
44		ioossicc 11396	. . . . . . . . . . . 12 |- ( 0 (,) +oo ) C_ ( 0 [,] +oo )
45	43, 44	eqsstr3i 3402	. . . . . . . . . . 11 |- RR+ C_ ( 0 [,] +oo )
46		simpl 457	. . . . . . . . . . 11 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. RR+ )
47	45, 46	sseldi 3369	. . . . . . . . . 10 |- ( ( C e. RR+ /\ x e. ( 0 [,] +oo ) ) -> C e. ( 0 [,] +oo ) )
48		ge0xmulcl 11415	. . . . . . . . . 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 5882	. . . . . . . 8 |- ( C e. RR+ -> F : ( 0 [,] +oo ) --> ( 0 [,] +oo ) )
51		frn 5580	. . . . . . . 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 18905	. . . . . . 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 1218	. . . . . 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 6117	. . . . 5 |- ( J Cn J ) = ( J Cn ( (ordTop ` <_ ) ↾t ( 0 [,] +oo ) ) )
58	56, 57	syl6eleqr 2534	. . . 4 |- ( C e. RR+ -> F e. ( J Cn J ) )
59	58, 43	eleq2s 2535	. . 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 9445	. . . 4 |- 0 e. RR*
63		pnfxr 11107	. . . 4 |- +oo e. RR*
64		0ltpnf 11118	. . . 4 |- 0 < +oo
65		elicoelioo 26083	. . . 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 1314	. . 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 784	1 |- ( ph -> F e. ( J Cn J ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   C_ wss 3343   {csn 3892   class class class wbr 4307   e. cmpt 4365   X. cxp 4853   ran crn 4856   |` cres 4857   -->wf 5429   ` cfv 5433  (class class class)co 6106   0cc0 9297   +oocpnf 9430   RR*cxr 9432   < clt 9433   <_ cle 9434   RR+crp 11006   xecxmu 11103   (,)cioo 11315   [,)cico 11317   [,]cicc 11318   ↾_t crest 14374  ordTopcordt 14452  TopOnctopon 18514   Cn ccn 18843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-map 7231  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fi 7676  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-rp 11007  df-xneg 11104  df-xmul 11106  df-ioo 11319  df-ico 11321  df-icc 11322  df-rest 14376  df-topgen 14397  df-ordt 14454  df-ps 15385  df-tsr 15386  df-top 18518  df-bases 18520  df-topon 18521  df-cn 18846  df-cnp 18847

This theorem is referenced by:  esummulc1  26545