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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflcl | Structured version Visualization version GIF version |
Description: A linear functional value is a scalar. (Contributed by NM, 15-Apr-2014.) |
Ref | Expression |
---|---|
lflf.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lflf.k | ⊢ 𝐾 = (Base‘𝐷) |
lflf.v | ⊢ 𝑉 = (Base‘𝑊) |
lflf.f | ⊢ 𝐹 = (LFnl‘𝑊) |
Ref | Expression |
---|---|
lflcl | ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflf.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
2 | lflf.k | . . . 4 ⊢ 𝐾 = (Base‘𝐷) | |
3 | lflf.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lflf.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | 1, 2, 3, 4 | lflf 36359 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
6 | 5 | 3adant3 1129 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝐺:𝑉⟶𝐾) |
7 | simp3 1135 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | 6, 7 | ffvelrnd 6829 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⟶wf 6320 ‘cfv 6324 Basecbs 16475 Scalarcsca 16560 LFnlclfn 36353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-lfl 36354 |
This theorem is referenced by: lfl0 36361 lfladd 36362 lflsub 36363 lflmul 36364 lfl1 36366 lfladdcl 36367 lflnegcl 36371 lflvscl 36373 lkrsc 36393 eqlkr 36395 eqlkr3 36397 lkrlsp 36398 ldualvsubval 36453 dochkr1 38774 dochkr1OLDN 38775 lcfl7lem 38795 lclkrlem2m 38815 lclkrlem2o 38817 lclkrlem2p 38818 lcfrlem1 38838 lcfrlem2 38839 lcfrlem3 38840 lcfrlem29 38867 lcfrlem31 38869 lcfrlem33 38871 lcdvbasecl 38892 |
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