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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfl0sc | Structured version Visualization version GIF version |
Description: The (right vector space) scalar product of a functional with zero is the zero functional. Note that the first occurrence of (𝑉 × { 0 }) represents the zero scalar, and the second is the zero functional. (Contributed by NM, 7-Oct-2014.) |
Ref | Expression |
---|---|
lfl0sc.v | ⊢ 𝑉 = (Base‘𝑊) |
lfl0sc.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lfl0sc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lfl0sc.k | ⊢ 𝐾 = (Base‘𝐷) |
lfl0sc.t | ⊢ · = (.r‘𝐷) |
lfl0sc.o | ⊢ 0 = (0g‘𝐷) |
lfl0sc.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lfl0sc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lfl0sc | ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × { 0 })) = (𝑉 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfl0sc.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6448 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lfl0sc.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lfl0sc.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
6 | lfl0sc.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
7 | lfl0sc.k | . . . 4 ⊢ 𝐾 = (Base‘𝐷) | |
8 | lfl0sc.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 35139 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 581 | . 2 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | 6 | lmodring 19228 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring) |
12 | 4, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐷 ∈ Ring) |
13 | lfl0sc.o | . . . 4 ⊢ 0 = (0g‘𝐷) | |
14 | 7, 13 | ring0cl 18924 | . . 3 ⊢ (𝐷 ∈ Ring → 0 ∈ 𝐾) |
15 | 12, 14 | syl 17 | . 2 ⊢ (𝜑 → 0 ∈ 𝐾) |
16 | lfl0sc.t | . . . 4 ⊢ · = (.r‘𝐷) | |
17 | 7, 16, 13 | ringrz 18943 | . . 3 ⊢ ((𝐷 ∈ Ring ∧ 𝑘 ∈ 𝐾) → (𝑘 · 0 ) = 0 ) |
18 | 12, 17 | sylan 577 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → (𝑘 · 0 ) = 0 ) |
19 | 3, 10, 15, 15, 18 | caofid1 7188 | 1 ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × { 0 })) = (𝑉 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3415 {csn 4398 × cxp 5341 ⟶wf 6120 ‘cfv 6124 (class class class)co 6906 ∘𝑓 cof 7156 Basecbs 16223 .rcmulr 16307 Scalarcsca 16309 0gc0g 16454 Ringcrg 18902 LModclmod 19220 LFnlclfn 35133 |
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 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-of 7158 df-om 7328 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-plusg 16319 df-0g 16456 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-mgp 18845 df-ring 18904 df-lmod 19222 df-lfl 35134 |
This theorem is referenced by: lkrscss 35174 lfl1dim 35197 lfl1dim2N 35198 |
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