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Theorem lebnumlem2 25090

Description: Lemma for lebnum 25092. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24983, 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 2769	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
3		lebnum.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		metxmet 24460	. . . . . . 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 24565	. . . . . 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 4099	. . . . . 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 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽)
16		toponss 23053	. . . . . . . 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 2857	. . . . . . . . . . 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 2979	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋))
23	22	imp 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋)
24		pssdifn0 4331	. . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅)
25	17, 23, 24	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅)
26		eqid 2769	. . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^, < ))
27	26, 6, 2	metdscn2 24984	. . . . . 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 1396	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
29	2, 8, 9, 28	fsumcn 24998	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ^*, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
30	1, 29	eqeltrid 2873	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
31	2	cnfldtopon 24908	. . . . 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 25089	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ⁺)
36	35	frnd 6715	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ⁺)
37		rpssre 13024	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
38	36, 37	sstrdi 3957	. . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
39		ax-resscn 11157	. . . . 5 ⊢ ℝ ⊆ ℂ
40	39	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ)
41		cnrest2 23412	. . . 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 1396	. . 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 24931	. . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
46	44, 45	eqtri 2792	. . 3 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
47	46	oveq2i 7422	. 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
48	43, 47	eleqtrrdi 2880	1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3910 ⊆ wss 3913 ∅c0 4294 ∪ cuni 4876 ↦ cmpt 5196 ran crn 5663 ‘cfv 6537 (class class class)co 7411 Fincfn 8943 infcinf 9401 ℂcc 11098 ℝcr 11099 ℝ^*cxr 11242 < clt 11243 ℝ⁺crp 13016 (,)cioo 13372 Σcsu 15737 ↾_t crest 17473 TopOpenctopn 17474 topGenctg 17490 ∞Metcxmet 21476 Metcmet 21477 MetOpencmopn 21481 ℂ_fldccnfld 21491 TopOnctopon 23036 Cn ccn 23350 Compccmp 23512

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-ec 8696 df-map 8826 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-cn 23353 df-cnp 23354 df-tx 23688 df-hmeo 23881 df-xms 24446 df-ms 24447 df-tms 24448

This theorem is referenced by: lebnumlem3 25091

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