lebnumlem2 - Metamath Proof Explorer

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Theorem lebnumlem2 24031

Description: Lemma for lebnum 24033. As a finite sum of point-to-set distance functions, which are continuous by metdscn 23925, 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 2738	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
3		lebnum.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 23395	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
6		lebnum.j	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
7	6	mopntopon 23500	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		lebnumlem1.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ Fin)
10	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋))
11		difssd 4063	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋)
12	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (∞Met‘𝑋))
13	12, 7	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐽 ∈ (TopOn‘𝑋))
14		lebnum.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝐽)
15	14	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
16		toponss 21984	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝐽) → 𝑘 ⊆ 𝑋)
17	13, 15, 16	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋)
18		lebnumlem1.n	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)
19		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈))
20	19	notbid 317	. . . . . . . . . 10 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈))
21	18, 20	syl5ibrcom 246	. . . . . . . . 9 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈))
22	21	necon2ad 2957	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
24		pssdifn0 4296	. . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
25	17, 23, 24	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
26		eqid 2738	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
27	26, 6, 2	metdscn2 23926	. . . . . 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 1369	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
29	2, 8, 9, 28	fsumcn 23939	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
30	1, 29	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
31	2	cnfldtopon 23852	. . . . 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 24030	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
36	35	frnd 6592	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ⁺)
37		rpssre 12666	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
38	36, 37	sstrdi 3929	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
39		ax-resscn 10859	. . . . 5 ⊢ ℝ ⊆ ℂ
40	39	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ)
41		cnrest2 22345	. . . 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 1369	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
43	30, 42	mpbid 231	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
44		lebnumlem2.k	. . . 4 ⊢ 𝐾 = (topGen‘ran (,))
45	2	tgioo2 23872	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
46	44, 45	eqtri 2766	. . 3 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47	46	oveq2i 7266	. 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
48	43, 47	eleqtrrdi 2850	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 ∪ cuni 4836 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 infcinf 9130 ℂcc 10800 ℝcr 10801 ℝ^*cxr 10939 < clt 10940 ℝ⁺crp 12659 (,)cioo 13008 Σcsu 15325 ↾_t crest 17048 TopOpenctopn 17049 topGenctg 17065 ∞Metcxmet 20495 Metcmet 20496 MetOpencmopn 20500 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 Compccmp 22445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-ec 8458 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-sum 15326 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-cn 22286 df-cnp 22287 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383

This theorem is referenced by: lebnumlem3 24032

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