lebnumlem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lebnumlem2		Structured version Visualization version GIF version

Theorem lebnumlem2 25121

Description: Lemma for lebnum 25123. As a finite sum of point-to-set distance functions, which are continuous by metdscn 25014, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.)

**Hypotheses**
Ref	Expression
lebnum.j	⊢ 𝐽 = (MetOpen‘𝐷)
lebnum.d	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
lebnum.c	⊢ (𝜑 → 𝐽 ∈ Comp)
lebnum.s	⊢ (𝜑 → 𝑈 ⊆ 𝐽)
lebnum.u	⊢ (𝜑 → 𝑋 = ∪ 𝑈)
lebnumlem1.u	⊢ (𝜑 → 𝑈 ∈ Fin)
lebnumlem1.n	⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
lebnumlem1.f	⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
lebnumlem2.k	⊢ 𝐾 = (topGen‘ran (,))

**Assertion**
Ref	Expression
lebnumlem2	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Distinct variable groups: 𝑦,𝑘,𝑧,𝐷 𝑘,𝐽,𝑦,𝑧 𝑈,𝑘,𝑦,𝑧 𝜑,𝑘,𝑦,𝑧 𝑘,𝑋,𝑦,𝑧 Allowed substitution hints: 𝐹(𝑦,𝑧,𝑘) 𝐾(𝑦,𝑧,𝑘)

**Proof of Theorem lebnumlem2**
Step	Hyp	Ref	Expression
1		lebnumlem1.f	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < ))
2		eqid 2763	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
3		lebnum.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24491	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		lebnum.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
7	6	mopntopon 24596	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 18	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lebnumlem1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
10	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
11		difssd 4091	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
12	5	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
13	12, 7	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
14		lebnum.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
15	14	sselda 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
16		toponss 23084	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝐽) → 𝑘 ⊆ 𝑋)
17	13, 15, 16	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2851	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 321	. . . . . . . . . 10 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 2973	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
24		pssdifn0 4323	. . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
25	17, 23, 24	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
26		eqid 2763	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
27	26, 6, 2	metdscn2 25015	. . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑘) ≠ ∅) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
28	10, 11, 25, 27	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
29	2, 8, 9, 28	fsumcn 25029	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
30	1, 29	eqeltrid 2867	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
31	2	cnfldtopon 24939	. . . . 5 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
32	31	a1i 11	. . . 4 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
33		lebnum.c	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Comp)
34		lebnum.u	. . . . . . 7 ⊢ (𝜑 → 𝑋 = ∪ 𝑈)
35	6, 3, 33, 14, 34, 9, 18, 1	lebnumlem1 25120	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
36	35	frnd 6714	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ⁺)
37		rpssre 13019	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
38	36, 37	sstrdi 3949	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
39		ax-resscn 11152	. . . . 5 ⊢ ℝ ⊆ ℂ
40	39	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ)
41		cnrest2 23443	. . . 4 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran 𝐹 ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
42	32, 38, 40, 41	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
43	30, 42	mpbid 235	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44		lebnumlem2.k	. . . 4 ⊢ 𝐾 = (topGen‘ran (,))
45		tgioo4 24962	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
46	44, 45	eqtri 2786	. . 3 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47	46	oveq2i 7421	. 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
48	43, 47	eleqtrrdi 2874	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∖ cdif 3902 ⊆ wss 3905 ∅c0 4286 ∪ cuni 4872 ↦ cmpt 5192 ran crn 5662 ‘cfv 6536 (class class class)co 7410 Fincfn 8939 infcinf 9397 ℂcc 11093 ℝcr 11094 ℝ^*cxr 11237 < clt 11238 ℝ⁺crp 13011 (,)cioo 13367 Σcsu 15733 ↾_t crest 17468 TopOpenctopn 17469 topGenctg 17485 ∞Metcxmet 21507 Metcmet 21508 MetOpencmopn 21512 ℂ_fldccnfld 21522 TopOnctopon 23067 Cn ccn 23381 Compccmp 23543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-ec 8692 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-ntr 23177 df-cls 23178 df-cn 23384 df-cnp 23385 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479

This theorem is referenced by: lebnumlem3 25122

W3C validator