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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 35138 | . . 3 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
6 | 5 | 3adant3 1168 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝐺:𝑉⟶𝐾) |
7 | simp3 1174 | . 2 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉) | |
8 | 6, 7 | ffvelrnd 6609 | 1 ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⟶wf 6119 ‘cfv 6123 Basecbs 16222 Scalarcsca 16308 LFnlclfn 35132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-map 8124 df-lfl 35133 |
This theorem is referenced by: lfl0 35140 lfladd 35141 lflsub 35142 lflmul 35143 lfl1 35145 lfladdcl 35146 lflnegcl 35150 lflvscl 35152 lkrsc 35172 eqlkr 35174 eqlkr3 35176 lkrlsp 35177 ldualvsubval 35232 dochkr1 37553 dochkr1OLDN 37554 lcfl7lem 37574 lclkrlem2m 37594 lclkrlem2o 37596 lclkrlem2p 37597 lcfrlem1 37617 lcfrlem2 37618 lcfrlem3 37619 lcfrlem29 37646 lcfrlem31 37648 lcfrlem33 37650 lcdvbasecl 37671 |
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