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Mirrors > Home > MPE Home > Th. List > blocn2 | Structured version Visualization version GIF version |
Description: A bounded linear operator is continuous. (Contributed by NM, 25-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
blocn.8 | ⊢ 𝐶 = (IndMet‘𝑈) |
blocn.d | ⊢ 𝐷 = (IndMet‘𝑊) |
blocn.j | ⊢ 𝐽 = (MetOpen‘𝐶) |
blocn.k | ⊢ 𝐾 = (MetOpen‘𝐷) |
blocn.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
blocn.u | ⊢ 𝑈 ∈ NrmCVec |
blocn.w | ⊢ 𝑊 ∈ NrmCVec |
Ref | Expression |
---|---|
blocn2 | ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blocn.u | . . 3 ⊢ 𝑈 ∈ NrmCVec | |
2 | blocn.w | . . 3 ⊢ 𝑊 ∈ NrmCVec | |
3 | eqid 2738 | . . . 4 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊) | |
4 | blocn.5 | . . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
5 | 3, 4 | bloln 28719 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊)) |
6 | 1, 2, 5 | mp3an12 1452 | . 2 ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝑈 LnOp 𝑊)) |
7 | blocn.8 | . . . 4 ⊢ 𝐶 = (IndMet‘𝑈) | |
8 | blocn.d | . . . 4 ⊢ 𝐷 = (IndMet‘𝑊) | |
9 | blocn.j | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐶) | |
10 | blocn.k | . . . 4 ⊢ 𝐾 = (MetOpen‘𝐷) | |
11 | 7, 8, 9, 10, 4, 1, 2, 3 | blocn 28742 | . . 3 ⊢ (𝑇 ∈ (𝑈 LnOp 𝑊) → (𝑇 ∈ (𝐽 Cn 𝐾) ↔ 𝑇 ∈ 𝐵)) |
12 | 11 | biimprd 251 | . 2 ⊢ (𝑇 ∈ (𝑈 LnOp 𝑊) → (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾))) |
13 | 6, 12 | mpcom 38 | 1 ⊢ (𝑇 ∈ 𝐵 → 𝑇 ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 MetOpencmopn 20207 Cn ccn 21975 NrmCVeccnv 28519 IndMetcims 28526 LnOp clno 28675 BLnOp cblo 28677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-topgen 16820 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-top 21645 df-topon 21662 df-bases 21697 df-cn 21978 df-cnp 21979 df-grpo 28428 df-gid 28429 df-ginv 28430 df-gdiv 28431 df-ablo 28480 df-vc 28494 df-nv 28527 df-va 28530 df-ba 28531 df-sm 28532 df-0v 28533 df-vs 28534 df-nmcv 28535 df-ims 28536 df-lno 28679 df-nmoo 28680 df-blo 28681 df-0o 28682 |
This theorem is referenced by: ubthlem1 28805 ubthlem2 28806 |
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