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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 36231 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
6 | 5 | 3adant3 1128 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝐺:𝑉⟶𝐾) |
7 | simp3 1134 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | 6, 7 | ffvelrnd 6838 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⟶wf 6337 ‘cfv 6341 Basecbs 16466 Scalarcsca 16551 LFnlclfn 36225 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-map 8394 df-lfl 36226 |
This theorem is referenced by: lfl0 36233 lfladd 36234 lflsub 36235 lflmul 36236 lfl1 36238 lfladdcl 36239 lflnegcl 36243 lflvscl 36245 lkrsc 36265 eqlkr 36267 eqlkr3 36269 lkrlsp 36270 ldualvsubval 36325 dochkr1 38646 dochkr1OLDN 38647 lcfl7lem 38667 lclkrlem2m 38687 lclkrlem2o 38689 lclkrlem2p 38690 lcfrlem1 38710 lcfrlem2 38711 lcfrlem3 38712 lcfrlem29 38739 lcfrlem31 38741 lcfrlem33 38743 lcdvbasecl 38764 |
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