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Mirrors > Home > MPE Home > Th. List > lebnumlem2 | Structured version Visualization version GIF version |
Description: Lemma for lebnum 24462. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24354, the function 𝐹 is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by AV, 30-Sep-2020.) |
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 (,)) |
Ref | Expression |
---|---|
lebnumlem2 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lebnumlem1.f | . . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) | |
2 | eqid 2733 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
3 | lebnum.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
4 | metxmet 23822 | . . . . . . 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 23927 | . . . . . 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 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋)) |
11 | difssd 4131 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋) | |
12 | 5 | adantr 482 | . . . . . . . . 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 3981 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽) |
16 | toponss 22411 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝐽) → 𝑘 ⊆ 𝑋) | |
17 | 13, 15, 16 | syl2anc 585 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋) |
18 | lebnumlem1.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
19 | eleq1 2822 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈)) | |
20 | 19 | notbid 318 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈)) |
21 | 18, 20 | syl5ibrcom 246 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈)) |
22 | 21 | necon2ad 2956 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋)) |
23 | 22 | imp 408 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋) |
24 | pssdifn0 4364 | . . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅) | |
25 | 17, 23, 24 | syl2anc 585 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅) |
26 | eqid 2733 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) | |
27 | 26, 6, 2 | metdscn2 24355 | . . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝑋 ∖ 𝑘) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑘) ≠ ∅) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
28 | 10, 11, 25, 27 | syl3anc 1372 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
29 | 2, 8, 9, 28 | fsumcn 24368 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
30 | 1, 29 | eqeltrid 2838 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
31 | 2 | cnfldtopon 24281 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (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 24459 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+) |
36 | 35 | frnd 6722 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ+) |
37 | rpssre 12977 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
38 | 36, 37 | sstrdi 3993 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
39 | ax-resscn 11163 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ) |
41 | cnrest2 22772 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran 𝐹 ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))) | |
42 | 32, 38, 40, 41 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))) |
43 | 30, 42 | mpbid 231 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))) |
44 | lebnumlem2.k | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
45 | 2 | tgioo2 24301 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
46 | 44, 45 | eqtri 2761 | . . 3 ⊢ 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ) |
47 | 46 | oveq2i 7415 | . 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) |
48 | 43, 47 | eleqtrrdi 2845 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 ↦ cmpt 5230 ran crn 5676 ‘cfv 6540 (class class class)co 7404 Fincfn 8935 infcinf 9432 ℂcc 11104 ℝcr 11105 ℝ*cxr 11243 < clt 11244 ℝ+crp 12970 (,)cioo 13320 Σcsu 15628 ↾t crest 17362 TopOpenctopn 17363 topGenctg 17379 ∞Metcxmet 20914 Metcmet 20915 MetOpencmopn 20919 ℂfldccnfld 20929 TopOnctopon 22394 Cn ccn 22710 Compccmp 22872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 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 ax-pre-sup 11184 ax-addf 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-tp 4632 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-se 5631 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-pred 6297 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 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-ec 8701 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 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-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-cn 22713 df-cnp 22714 df-tx 23048 df-hmeo 23241 df-xms 23808 df-ms 23809 df-tms 23810 |
This theorem is referenced by: lebnumlem3 24461 |
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