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Mirrors > Home > MPE Home > Th. List > isblo2 | Structured version Visualization version GIF version |
Description: The predicate "is a bounded linear operator." (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bloval.3 | ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) |
bloval.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
bloval.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
Ref | Expression |
---|---|
isblo2 | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bloval.3 | . . 3 ⊢ 𝑁 = (𝑈 normOpOLD 𝑊) | |
2 | bloval.4 | . . 3 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
3 | bloval.5 | . . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
4 | 1, 2, 3 | isblo 28553 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) |
5 | eqid 2821 | . . . . . 6 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
6 | eqid 2821 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
7 | 5, 6, 2 | lnof 28526 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) |
8 | 5, 6, 1 | nmoreltpnf 28540 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇:(BaseSet‘𝑈)⟶(BaseSet‘𝑊)) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞)) |
9 | 7, 8 | syld3an3 1405 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞)) |
10 | 9 | 3expa 1114 | . . 3 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐿) → ((𝑁‘𝑇) ∈ ℝ ↔ (𝑁‘𝑇) < +∞)) |
11 | 10 | pm5.32da 581 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → ((𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ) ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) < +∞))) |
12 | 4, 11 | bitr4d 284 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ (𝑁‘𝑇) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 +∞cpnf 10666 < clt 10669 NrmCVeccnv 28355 BaseSetcba 28357 LnOp clno 28511 normOpOLD cnmoo 28512 BLnOp cblo 28513 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-lno 28515 df-nmoo 28516 df-blo 28517 |
This theorem is referenced by: 0blo 28563 |
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