Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > blof | Structured version Visualization version GIF version |
Description: A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
blof.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
blof.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
blof.5 | ⊢ 𝐵 = (𝑈 BLnOp 𝑊) |
Ref | Expression |
---|---|
blof | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊) | |
2 | blof.5 | . . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
3 | 1, 2 | bloln 28819 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇 ∈ (𝑈 LnOp 𝑊)) |
4 | blof.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
5 | blof.2 | . . 3 ⊢ 𝑌 = (BaseSet‘𝑊) | |
6 | 4, 5, 1 | lnof 28790 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) → 𝑇:𝑋⟶𝑌) |
7 | 3, 6 | syld3an3 1411 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 NrmCVeccnv 28619 BaseSetcba 28621 LnOp clno 28775 BLnOp cblo 28777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-lno 28779 df-blo 28781 |
This theorem is referenced by: nmblore 28821 nmblolbii 28834 blometi 28838 ubthlem3 28907 htthlem 28952 |
Copyright terms: Public domain | W3C validator |