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Mirrors > Home > MPE Home > Th. List > lebnumlem2 | Structured version Visualization version GIF version |
Description: Lemma for lebnum 25010. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24892, 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 2735 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
3 | lebnum.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
4 | metxmet 24360 | . . . . . . 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 24465 | . . . . . 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 4147 | . . . . . 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 3995 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽) |
16 | toponss 22949 | . . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ∈ 𝐽) → 𝑘 ⊆ 𝑋) | |
17 | 13, 15, 16 | syl2anc 584 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ⊆ 𝑋) |
18 | lebnumlem1.n | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) | |
19 | eleq1 2827 | . . . . . . . . . . 11 ⊢ (𝑘 = 𝑋 → (𝑘 ∈ 𝑈 ↔ 𝑋 ∈ 𝑈)) | |
20 | 19 | notbid 318 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑋 → (¬ 𝑘 ∈ 𝑈 ↔ ¬ 𝑋 ∈ 𝑈)) |
21 | 18, 20 | syl5ibrcom 247 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 = 𝑋 → ¬ 𝑘 ∈ 𝑈)) |
22 | 21 | necon2ad 2953 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋)) |
23 | 22 | imp 406 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋) |
24 | pssdifn0 4374 | . . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅) | |
25 | 17, 23, 24 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅) |
26 | eqid 2735 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) | |
27 | 26, 6, 2 | metdscn2 24893 | . . . . . 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 1370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
29 | 2, 8, 9, 28 | fsumcn 24908 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
30 | 1, 29 | eqeltrid 2843 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
31 | 2 | cnfldtopon 24819 | . . . . 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 25007 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+) |
36 | 35 | frnd 6745 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ+) |
37 | rpssre 13040 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
38 | 36, 37 | sstrdi 4008 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
39 | ax-resscn 11210 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ) |
41 | cnrest2 23310 | . . . 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 1370 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))) |
43 | 30, 42 | mpbid 232 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))) |
44 | lebnumlem2.k | . . . 4 ⊢ 𝐾 = (topGen‘ran (,)) | |
45 | 2 | tgioo2 24839 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
46 | 44, 45 | eqtri 2763 | . . 3 ⊢ 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ) |
47 | 46 | oveq2i 7442 | . 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 ∪ cuni 4912 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 infcinf 9479 ℂcc 11151 ℝcr 11152 ℝ*cxr 11292 < clt 11293 ℝ+crp 13032 (,)cioo 13384 Σcsu 15719 ↾t crest 17467 TopOpenctopn 17468 topGenctg 17484 ∞Metcxmet 21367 Metcmet 21368 MetOpencmopn 21372 ℂfldccnfld 21382 TopOnctopon 22932 Cn ccn 23248 Compccmp 23410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 |
This theorem is referenced by: lebnumlem3 25009 |
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