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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvaddval | Structured version Visualization version GIF version |
Description: The value of the value of vector addition in the dual of a vector space. (Contributed by NM, 7-Jan-2015.) |
Ref | Expression |
---|---|
ldualvaddval.v | ⊢ 𝑉 = (Base‘𝑊) |
ldualvaddval.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvaddval.a | ⊢ + = (+g‘𝑅) |
ldualvaddval.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvaddval.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvaddval.p | ⊢ ✚ = (+g‘𝐷) |
ldualvaddval.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvaddval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvaddval.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
ldualvaddval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
ldualvaddval | ⊢ (𝜑 → ((𝐺 ✚ 𝐻)‘𝑋) = ((𝐺‘𝑋) + (𝐻‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvaddval.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | ldualvaddval.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | ldualvaddval.a | . . . 4 ⊢ + = (+g‘𝑅) | |
4 | ldualvaddval.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | ldualvaddval.p | . . . 4 ⊢ ✚ = (+g‘𝐷) | |
6 | ldualvaddval.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | ldualvaddval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
8 | ldualvaddval.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | ldualvadd 36880 | . . 3 ⊢ (𝜑 → (𝐺 ✚ 𝐻) = (𝐺 ∘f + 𝐻)) |
10 | 9 | fveq1d 6719 | . 2 ⊢ (𝜑 → ((𝐺 ✚ 𝐻)‘𝑋) = ((𝐺 ∘f + 𝐻)‘𝑋)) |
11 | ldualvaddval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
12 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
13 | ldualvaddval.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
14 | 2, 12, 13, 1 | lflf 36814 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶(Base‘𝑅)) |
15 | 14 | ffnd 6546 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺 Fn 𝑉) |
16 | 6, 7, 15 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝑉) |
17 | 2, 12, 13, 1 | lflf 36814 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻:𝑉⟶(Base‘𝑅)) |
18 | 17 | ffnd 6546 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → 𝐻 Fn 𝑉) |
19 | 6, 8, 18 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐻 Fn 𝑉) |
20 | 13 | fvexi 6731 | . . . . 5 ⊢ 𝑉 ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑉 ∈ V) |
22 | inidm 4133 | . . . 4 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
23 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) = (𝐺‘𝑋)) | |
24 | eqidd 2738 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝐻‘𝑋) = (𝐻‘𝑋)) | |
25 | 16, 19, 21, 21, 22, 23, 24 | ofval 7479 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → ((𝐺 ∘f + 𝐻)‘𝑋) = ((𝐺‘𝑋) + (𝐻‘𝑋))) |
26 | 11, 25 | mpdan 687 | . 2 ⊢ (𝜑 → ((𝐺 ∘f + 𝐻)‘𝑋) = ((𝐺‘𝑋) + (𝐻‘𝑋))) |
27 | 10, 26 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐺 ✚ 𝐻)‘𝑋) = ((𝐺‘𝑋) + (𝐻‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 Fn wfn 6375 ‘cfv 6380 (class class class)co 7213 ∘f cof 7467 Basecbs 16760 +gcplusg 16802 Scalarcsca 16805 LModclmod 19899 LFnlclfn 36808 LDualcld 36874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-sca 16818 df-vsca 16819 df-lfl 36809 df-ldual 36875 |
This theorem is referenced by: ldualvsubval 36908 lkrin 36915 lcdvaddval 39349 |
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