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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 38665 | . . 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 | ffvelcdmd 7094 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⟶wf 6545 ‘cfv 6549 Basecbs 17183 Scalarcsca 17239 LFnlclfn 38659 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-map 8847 df-lfl 38660 |
This theorem is referenced by: lfl0 38667 lfladd 38668 lflsub 38669 lflmul 38670 lfl1 38672 lfladdcl 38673 lflnegcl 38677 lflvscl 38679 lkrsc 38699 eqlkr 38701 eqlkr3 38703 lkrlsp 38704 ldualvsubval 38759 dochkr1 41081 dochkr1OLDN 41082 lcfl7lem 41102 lclkrlem2m 41122 lclkrlem2o 41124 lclkrlem2p 41125 lcfrlem1 41145 lcfrlem2 41146 lcfrlem3 41147 lcfrlem29 41174 lcfrlem31 41176 lcfrlem33 41178 lcdvbasecl 41199 |
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