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Mirrors > Home > MPE Home > Th. List > lebnumlem2 | Structured version Visualization version GIF version |
Description: Lemma for lebnum 23569. As a finite sum of point-to-set distance functions, which are continuous by metdscn 23461, 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 2798 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
3 | lebnum.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
4 | metxmet 22941 | . . . . . . 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 23046 | . . . . . 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 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝐷 ∈ (Met‘𝑋)) |
11 | difssd 4060 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ⊆ 𝑋) | |
12 | 5 | adantr 484 | . . . . . . . . 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 3915 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽) |
16 | toponss 21532 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝐽) → 𝑘 ⊆ 𝑋) | |
17 | 13, 15, 16 | syl2anc 587 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋) |
18 | lebnumlem1.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
19 | eleq1 2877 | . . . . . . . . . . 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 3002 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋)) |
23 | 22 | imp 410 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋) |
24 | pssdifn0 4279 | . . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅) | |
25 | 17, 23, 24 | syl2anc 587 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅) |
26 | eqid 2798 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) | |
27 | 26, 6, 2 | metdscn2 23462 | . . . . . 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 1368 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
29 | 2, 8, 9, 28 | fsumcn 23475 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
30 | 1, 29 | eqeltrid 2894 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
31 | 2 | cnfldtopon 23388 | . . . . 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 23566 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+) |
36 | 35 | frnd 6494 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ+) |
37 | rpssre 12384 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
38 | 36, 37 | sstrdi 3927 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
39 | ax-resscn 10583 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ) |
41 | cnrest2 21891 | . . . 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 1368 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))) |
43 | 30, 42 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))) |
44 | lebnumlem2.k | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
45 | 2 | tgioo2 23408 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
46 | 44, 45 | eqtri 2821 | . . 3 ⊢ 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ) |
47 | 46 | oveq2i 7146 | . 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) |
48 | 43, 47 | eleqtrrdi 2901 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 infcinf 8889 ℂcc 10524 ℝcr 10525 ℝ*cxr 10663 < clt 10664 ℝ+crp 12377 (,)cioo 12726 Σcsu 15034 ↾t crest 16686 TopOpenctopn 16687 topGenctg 16703 ∞Metcxmet 20076 Metcmet 20077 MetOpencmopn 20081 ℂfldccnfld 20091 TopOnctopon 21515 Cn ccn 21829 Compccmp 21991 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 |
This theorem is referenced by: lebnumlem3 23568 |
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