xrge0mulc1cn - Mathbox for Thierry Arnoux

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Theorem xrge0mulc1cn 32909

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[,]+∞))
xrge0mulc1cn.f	⊢ 𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·_e 𝐶))
xrge0mulc1cn.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))

**Assertion**
Ref	Expression
xrge0mulc1cn	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽))

Distinct variable group: 𝑥,𝐶 Allowed substitution hints: 𝜑(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[,]+∞))
2		letopon 22700	. . . . . . 7 ⊢ (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*)
3		iccssxr 13403	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
4		resttopon 22656	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ (0[,]+∞) ⊆ ℝ^) → ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞)))
5	2, 3, 4	mp2an 690	. . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) ∈ (TopOn‘(0[,]+∞))
6	1, 5	eqeltri 2829	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘(0[,]+∞))
7	6	a1i 11	. . . 4 ⊢ (𝐶 = 0 → 𝐽 ∈ (TopOn‘(0[,]+∞)))
8		0e0iccpnf 13432	. . . . 5 ⊢ 0 ∈ (0[,]+∞)
9	8	a1i 11	. . . 4 ⊢ (𝐶 = 0 → 0 ∈ (0[,]+∞))
10		simpl 483	. . . . . . . . 9 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → 𝐶 = 0)
11	10	oveq2d 7421	. . . . . . . 8 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → (𝑥 ·_e 𝐶) = (𝑥 ·_e 0))
12		simpr 485	. . . . . . . . . 10 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → 𝑥 ∈ (0[,]+∞))
13	3, 12	sselid 3979	. . . . . . . . 9 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → 𝑥 ∈ ℝ^*)
14		xmul01 13242	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* → (𝑥 ·_e 0) = 0)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → (𝑥 ·_e 0) = 0)
16	11, 15	eqtrd 2772	. . . . . . 7 ⊢ ((𝐶 = 0 ∧ 𝑥 ∈ (0[,]+∞)) → (𝑥 ·_e 𝐶) = 0)
17	16	mpteq2dva 5247	. . . . . 6 ⊢ (𝐶 = 0 → (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·_e 𝐶)) = (𝑥 ∈ (0[,]+∞) ↦ 0))
18		xrge0mulc1cn.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·_e 𝐶))
19		fconstmpt 5736	. . . . . 6 ⊢ ((0[,]+∞) × {0}) = (𝑥 ∈ (0[,]+∞) ↦ 0)
20	17, 18, 19	3eqtr4g 2797	. . . . 5 ⊢ (𝐶 = 0 → 𝐹 = ((0[,]+∞) × {0}))
21		c0ex 11204	. . . . . 6 ⊢ 0 ∈ V
22	21	fconst2 7202	. . . . 5 ⊢ (𝐹:(0[,]+∞)⟶{0} ↔ 𝐹 = ((0[,]+∞) × {0}))
23	20, 22	sylibr 233	. . . 4 ⊢ (𝐶 = 0 → 𝐹:(0[,]+∞)⟶{0})
24		cnconst 22779	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘(0[,]+∞)) ∧ 𝐽 ∈ (TopOn‘(0[,]+∞))) ∧ (0 ∈ (0[,]+∞) ∧ 𝐹:(0[,]+∞)⟶{0})) → 𝐹 ∈ (𝐽 Cn 𝐽))
25	7, 7, 9, 23, 24	syl22anc 837	. . 3 ⊢ (𝐶 = 0 → 𝐹 ∈ (𝐽 Cn 𝐽))
26	25	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐶 = 0) → 𝐹 ∈ (𝐽 Cn 𝐽))
27		eqid 2732	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ )
28		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ·_e 𝐶) = (𝑦 ·_e 𝐶))
29	28	cbvmptv 5260	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) = (𝑦 ∈ ℝ^* ↦ (𝑦 ·_e 𝐶))
30		id 22	. . . . . . . . 9 ⊢ (𝐶 ∈ ℝ⁺ → 𝐶 ∈ ℝ⁺)
31	27, 29, 30	xrmulc1cn 32898	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → (𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ∈ ((ordTop‘ ≤ ) Cn (ordTop‘ ≤ )))
32		letopuni 22702	. . . . . . . . 9 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
33	32	cnrest 22780	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ∈ ((ordTop‘ ≤ ) Cn (ordTop‘ ≤ )) ∧ (0[,]+∞) ⊆ ℝ^) → ((𝑥 ∈ ℝ^ ↦ (𝑥 ·_e 𝐶)) ↾ (0[,]+∞)) ∈ (((ordTop‘ ≤ ) ↾_t (0[,]+∞)) Cn (ordTop‘ ≤ )))
34	31, 3, 33	sylancl 586	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → ((𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ↾ (0[,]+∞)) ∈ (((ordTop‘ ≤ ) ↾_t (0[,]+∞)) Cn (ordTop‘ ≤ )))
35		resmpt 6035	. . . . . . . . 9 ⊢ ((0[,]+∞) ⊆ ℝ^* → ((𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ↾ (0[,]+∞)) = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·_e 𝐶)))
36	3, 35	ax-mp 5	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ↾ (0[,]+∞)) = (𝑥 ∈ (0[,]+∞) ↦ (𝑥 ·_e 𝐶))
37	36, 18	eqtr4i 2763	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ^* ↦ (𝑥 ·_e 𝐶)) ↾ (0[,]+∞)) = 𝐹
38	1	eqcomi 2741	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = 𝐽
39	38	oveq1i 7415	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ↾_t (0[,]+∞)) Cn (ordTop‘ ≤ )) = (𝐽 Cn (ordTop‘ ≤ ))
40	34, 37, 39	3eltr3g 2849	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → 𝐹 ∈ (𝐽 Cn (ordTop‘ ≤ )))
41	2	a1i 11	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → (ordTop‘ ≤ ) ∈ (TopOn‘ℝ^*))
42		simpr 485	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑥 ∈ (0[,]+∞)) → 𝑥 ∈ (0[,]+∞))
43		ioorp 13398	. . . . . . . . . . . 12 ⊢ (0(,)+∞) = ℝ⁺
44		ioossicc 13406	. . . . . . . . . . . 12 ⊢ (0(,)+∞) ⊆ (0[,]+∞)
45	43, 44	eqsstrri 4016	. . . . . . . . . . 11 ⊢ ℝ⁺ ⊆ (0[,]+∞)
46		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑥 ∈ (0[,]+∞)) → 𝐶 ∈ ℝ⁺)
47	45, 46	sselid 3979	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑥 ∈ (0[,]+∞)) → 𝐶 ∈ (0[,]+∞))
48		ge0xmulcl 13436	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]+∞) ∧ 𝐶 ∈ (0[,]+∞)) → (𝑥 ·_e 𝐶) ∈ (0[,]+∞))
49	42, 47, 48	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ⁺ ∧ 𝑥 ∈ (0[,]+∞)) → (𝑥 ·_e 𝐶) ∈ (0[,]+∞))
50	49, 18	fmptd 7110	. . . . . . . 8 ⊢ (𝐶 ∈ ℝ⁺ → 𝐹:(0[,]+∞)⟶(0[,]+∞))
51	50	frnd 6722	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → ran 𝐹 ⊆ (0[,]+∞))
52	3	a1i 11	. . . . . . 7 ⊢ (𝐶 ∈ ℝ⁺ → (0[,]+∞) ⊆ ℝ^*)
53		cnrest2 22781	. . . . . . 7 ⊢ (((ordTop‘ ≤ ) ∈ (TopOn‘ℝ^) ∧ ran 𝐹 ⊆ (0[,]+∞) ∧ (0[,]+∞) ⊆ ℝ^) → (𝐹 ∈ (𝐽 Cn (ordTop‘ ≤ )) ↔ 𝐹 ∈ (𝐽 Cn ((ordTop‘ ≤ ) ↾_t (0[,]+∞)))))
54	41, 51, 52, 53	syl3anc 1371	. . . . . 6 ⊢ (𝐶 ∈ ℝ⁺ → (𝐹 ∈ (𝐽 Cn (ordTop‘ ≤ )) ↔ 𝐹 ∈ (𝐽 Cn ((ordTop‘ ≤ ) ↾_t (0[,]+∞)))))
55	40, 54	mpbid 231	. . . . 5 ⊢ (𝐶 ∈ ℝ⁺ → 𝐹 ∈ (𝐽 Cn ((ordTop‘ ≤ ) ↾_t (0[,]+∞))))
56	1	oveq2i 7416	. . . . 5 ⊢ (𝐽 Cn 𝐽) = (𝐽 Cn ((ordTop‘ ≤ ) ↾_t (0[,]+∞)))
57	55, 56	eleqtrrdi 2844	. . . 4 ⊢ (𝐶 ∈ ℝ⁺ → 𝐹 ∈ (𝐽 Cn 𝐽))
58	57, 43	eleq2s 2851	. . 3 ⊢ (𝐶 ∈ (0(,)+∞) → 𝐹 ∈ (𝐽 Cn 𝐽))
59	58	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (0(,)+∞)) → 𝐹 ∈ (𝐽 Cn 𝐽))
60		xrge0mulc1cn.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
61		0xr 11257	. . . 4 ⊢ 0 ∈ ℝ^*
62		pnfxr 11264	. . . 4 ⊢ +∞ ∈ ℝ^*
63		0ltpnf 13098	. . . 4 ⊢ 0 < +∞
64		elicoelioo 31976	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 0 < +∞) → (𝐶 ∈ (0[,)+∞) ↔ (𝐶 = 0 ∨ 𝐶 ∈ (0(,)+∞))))
65	61, 62, 63, 64	mp3an 1461	. . 3 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 = 0 ∨ 𝐶 ∈ (0(,)+∞)))
66	60, 65	sylib 217	. 2 ⊢ (𝜑 → (𝐶 = 0 ∨ 𝐶 ∈ (0(,)+∞)))
67	26, 59, 66	mpjaodan 957	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℝ⁺crp 12970 ·_e cxmu 13087 (,)cioo 13320 [,)cico 13322 [,]cicc 13323 ↾_t crest 17362 ordTopcordt 17441 TopOnctopon 22403 Cn ccn 22719

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-rp 12971 df-xneg 13088 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-rest 17364 df-topgen 17385 df-ordt 17443 df-ps 18515 df-tsr 18516 df-top 22387 df-topon 22404 df-bases 22440 df-cn 22722 df-cnp 22723

This theorem is referenced by: esummulc1 33067

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