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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 36214 | . . 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 6852 | 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 6351 ‘cfv 6355 Basecbs 16483 Scalarcsca 16568 LFnlclfn 36208 |
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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-lfl 36209 |
This theorem is referenced by: lfl0 36216 lfladd 36217 lflsub 36218 lflmul 36219 lfl1 36221 lfladdcl 36222 lflnegcl 36226 lflvscl 36228 lkrsc 36248 eqlkr 36250 eqlkr3 36252 lkrlsp 36253 ldualvsubval 36308 dochkr1 38629 dochkr1OLDN 38630 lcfl7lem 38650 lclkrlem2m 38670 lclkrlem2o 38672 lclkrlem2p 38673 lcfrlem1 38693 lcfrlem2 38694 lcfrlem3 38695 lcfrlem29 38722 lcfrlem31 38724 lcfrlem33 38726 lcdvbasecl 38747 |
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