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| Mirrors > Home > MPE Home > Th. List > lebnumlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for lebnum 24996. As a finite sum of point-to-set distance functions, which are continuous by metdscn 24878, 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 2737 | . . . . 5 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 3 | lebnum.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) | |
| 4 | metxmet 24344 | . . . . . . 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 24449 | . . . . . 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 4137 | . . . . . 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 3983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ∈ 𝐽) |
| 16 | toponss 22933 | . . . . . . . 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 2829 | . . . . . . . . . . 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 2955 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑈 → 𝑘 ≠ 𝑋)) |
| 23 | 22 | imp 406 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑘 ≠ 𝑋) |
| 24 | pssdifn0 4368 | . . . . . . 7 ⊢ ((𝑘 ⊆ 𝑋 ∧ 𝑘 ≠ 𝑋) → (𝑋 ∖ 𝑘) ≠ ∅) | |
| 25 | 17, 23, 24 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑋 ∖ 𝑘) ≠ ∅) |
| 26 | eqid 2737 | . . . . . . 7 ⊢ (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) = (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) | |
| 27 | 26, 6, 2 | metdscn2 24879 | . . . . . 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 1373 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → (𝑦 ∈ 𝑋 ↦ inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
| 29 | 2, 8, 9, 28 | fsumcn 24894 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝑈 inf(ran (𝑧 ∈ (𝑋 ∖ 𝑘) ↦ (𝑦𝐷𝑧)), ℝ*, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
| 30 | 1, 29 | eqeltrid 2845 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld))) |
| 31 | 2 | cnfldtopon 24803 | . . . . 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 24993 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ+) |
| 36 | 35 | frnd 6744 | . . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ+) |
| 37 | rpssre 13042 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
| 38 | 36, 37 | sstrdi 3996 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
| 39 | ax-resscn 11212 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
| 40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ) |
| 41 | cnrest2 23294 | . . . 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 1373 | . . 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 | tgioo4 24826 | . . . 4 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | |
| 46 | 44, 45 | eqtri 2765 | . . 3 ⊢ 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ) |
| 47 | 46 | oveq2i 7442 | . 2 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) |
| 48 | 43, 47 | eleqtrrdi 2852 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 ∪ cuni 4907 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 infcinf 9481 ℂcc 11153 ℝcr 11154 ℝ*cxr 11294 < clt 11295 ℝ+crp 13034 (,)cioo 13387 Σcsu 15722 ↾t crest 17465 TopOpenctopn 17466 topGenctg 17482 ∞Metcxmet 21349 Metcmet 21350 MetOpencmopn 21354 ℂfldccnfld 21364 TopOnctopon 22916 Cn ccn 23232 Compccmp 23394 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-cn 23235 df-cnp 23236 df-tx 23570 df-hmeo 23763 df-xms 24330 df-ms 24331 df-tms 24332 |
| This theorem is referenced by: lebnumlem3 24995 |
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