Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bloln | Structured version Visualization version GIF version |
Description: A bounded operator is a linear operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bloln.4 | ⊢ 𝐿 = (𝑈 LnOp 𝑊) |
bloln.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
Ref | Expression |
---|---|
bloln | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2824 | . . . 4 ⊢ (𝑈 normOpOLD 𝑊) = (𝑈 normOpOLD 𝑊) | |
2 | bloln.4 | . . . 4 ⊢ 𝐿 = (𝑈 LnOp 𝑊) | |
3 | bloln.5 | . . . 4 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
4 | 1, 2, 3 | isblo 28562 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑇 ∈ 𝐵 ↔ (𝑇 ∈ 𝐿 ∧ ((𝑈 normOpOLD 𝑊)‘𝑇) < +∞))) |
5 | 4 | simprbda 501 | . 2 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿) |
6 | 5 | 3impa 1106 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 +∞cpnf 10675 < clt 10678 NrmCVeccnv 28364 LnOp clno 28520 normOpOLD cnmoo 28521 BLnOp cblo 28522 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-blo 28526 |
This theorem is referenced by: blof 28565 nmblolbii 28579 isblo3i 28581 blometi 28583 blocn2 28588 ubthlem2 28651 |
Copyright terms: Public domain | W3C validator |