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Mirrors > Home > MPE Home > Th. List > lnocni | Structured version Visualization version GIF version |
Description: If a linear operator is continuous at any point, it is continuous everywhere. Theorem 2.7-9(b) of [Kreyszig] p. 97. (Contributed by NM, 18-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
blocni.8 | ⊢ 𝐶 = (IndMet‘𝑈) |
blocni.d | ⊢ 𝐷 = (IndMet‘𝑊) |
blocni.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
blocni.k | ⊢ 𝐾 = (MetOpen‘𝐷) |
blocni.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
blocni.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
blocni.u | ⊢ 𝑈 ∈ NrmCVec |
blocni.w | ⊢ 𝑊 ∈ NrmCVec |
blocni.l | ⊢ 𝑇 ∈ 𝐿 |
lnocni.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
Ref | Expression |
---|---|
lnocni | ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blocni.8 | . . 3 ⊢ 𝐶 = (IndMet‘𝑈) | |
2 | blocni.d | . . 3 ⊢ 𝐷 = (IndMet‘𝑊) | |
3 | blocni.j | . . 3 ⊢ 𝐽 = (MetOpen‘𝐶) | |
4 | blocni.k | . . 3 ⊢ 𝐾 = (MetOpen‘𝐷) | |
5 | blocni.4 | . . 3 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
6 | blocni.5 | . . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
7 | blocni.u | . . 3 ⊢ 𝑈 ∈ NrmCVec | |
8 | blocni.w | . . 3 ⊢ 𝑊 ∈ NrmCVec | |
9 | blocni.l | . . 3 ⊢ 𝑇 ∈ 𝐿 | |
10 | lnocni.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | blocnilem 28214 | . 2 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | blocni 28215 | . 2 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) |
13 | 11, 12 | sylibr 226 | 1 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 MetOpencmopn 20096 Cn ccn 21399 CnP ccnp 21400 NrmCVeccnv 27994 BaseSetcba 27996 IndMetcims 28001 LnOp clno 28150 BLnOp cblo 28152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-top 21069 df-topon 21086 df-bases 21121 df-cn 21402 df-cnp 21403 df-grpo 27903 df-gid 27904 df-ginv 27905 df-gdiv 27906 df-ablo 27955 df-vc 27969 df-nv 28002 df-va 28005 df-ba 28006 df-sm 28007 df-0v 28008 df-vs 28009 df-nmcv 28010 df-ims 28011 df-lno 28154 df-nmoo 28155 df-blo 28156 df-0o 28157 |
This theorem is referenced by: (None) |
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