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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 30732 | . 2 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ 𝐵) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | blocni 30733 | . 2 ⊢ (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵) |
13 | 11, 12 | sylibr 233 | 1 ⊢ ((𝑃 ∈ 𝑋 ∧ 𝑇 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑇 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6544 (class class class)co 7414 MetOpencmopn 21327 Cn ccn 23214 CnP ccnp 23215 NrmCVeccnv 30512 BaseSetcba 30514 IndMetcims 30519 LnOp clno 30668 BLnOp cblo 30670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 ax-addf 11226 ax-mulf 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-z 12603 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-topgen 17451 df-psmet 21329 df-xmet 21330 df-met 21331 df-bl 21332 df-mopn 21333 df-top 22882 df-topon 22899 df-bases 22935 df-cn 23217 df-cnp 23218 df-grpo 30421 df-gid 30422 df-ginv 30423 df-gdiv 30424 df-ablo 30473 df-vc 30487 df-nv 30520 df-va 30523 df-ba 30524 df-sm 30525 df-0v 30526 df-vs 30527 df-nmcv 30528 df-ims 30529 df-lno 30672 df-nmoo 30673 df-blo 30674 df-0o 30675 |
This theorem is referenced by: (None) |
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