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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 2823 | . . 3 ⊢ (𝑈 LnOp 𝑊) = (𝑈 LnOp 𝑊) | |
2 | blof.5 | . . 3 ⊢ 𝐵 = (𝑈 BLnOp 𝑊) | |
3 | 1, 2 | bloln 28563 | . 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 28534 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ (𝑈 LnOp 𝑊)) → 𝑇:𝑋⟶𝑌) |
7 | 3, 6 | syld3an3 1405 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec ∧ 𝑇 ∈ 𝐵) → 𝑇:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 NrmCVeccnv 28363 BaseSetcba 28365 LnOp clno 28519 BLnOp cblo 28521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-lno 28523 df-blo 28525 |
This theorem is referenced by: nmblore 28565 nmblolbii 28578 blometi 28582 ubthlem3 28651 htthlem 28696 |
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