Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdvaddval | Structured version Visualization version GIF version |
Description: The value of the value of vector addition in the closed kernel vector space dual. (Contributed by NM, 10-Jun-2015.) |
Ref | Expression |
---|---|
lcdvaddval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdvaddval.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdvaddval.v | ⊢ 𝑉 = (Base‘𝑈) |
lcdvaddval.r | ⊢ 𝑅 = (Scalar‘𝑈) |
lcdvaddval.a | ⊢ + = (+g‘𝑅) |
lcdvaddval.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdvaddval.d | ⊢ 𝐷 = (Base‘𝐶) |
lcdvaddval.p | ⊢ ✚ = (+g‘𝐶) |
lcdvaddval.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcdvaddval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
lcdvaddval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
lcdvaddval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
lcdvaddval | ⊢ (𝜑 → ((𝐹 ✚ 𝐺)‘𝑋) = ((𝐹‘𝑋) + (𝐺‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdvaddval.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcdvaddval.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | eqid 2821 | . . . . 5 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
4 | eqid 2821 | . . . . 5 ⊢ (+g‘(LDual‘𝑈)) = (+g‘(LDual‘𝑈)) | |
5 | lcdvaddval.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
6 | lcdvaddval.p | . . . . 5 ⊢ ✚ = (+g‘𝐶) | |
7 | lcdvaddval.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | lcdvadd 38732 | . . . 4 ⊢ (𝜑 → ✚ = (+g‘(LDual‘𝑈))) |
9 | 8 | oveqd 7172 | . . 3 ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹(+g‘(LDual‘𝑈))𝐺)) |
10 | 9 | fveq1d 6671 | . 2 ⊢ (𝜑 → ((𝐹 ✚ 𝐺)‘𝑋) = ((𝐹(+g‘(LDual‘𝑈))𝐺)‘𝑋)) |
11 | lcdvaddval.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
12 | lcdvaddval.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
13 | lcdvaddval.a | . . 3 ⊢ + = (+g‘𝑅) | |
14 | eqid 2821 | . . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
15 | 1, 2, 7 | dvhlmod 38245 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
16 | lcdvaddval.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
17 | lcdvaddval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | 1, 5, 16, 2, 14, 7, 17 | lcdvbaselfl 38730 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (LFnl‘𝑈)) |
19 | lcdvaddval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐷) | |
20 | 1, 5, 16, 2, 14, 7, 19 | lcdvbaselfl 38730 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (LFnl‘𝑈)) |
21 | lcdvaddval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
22 | 11, 12, 13, 14, 3, 4, 15, 18, 20, 21 | ldualvaddval 36266 | . 2 ⊢ (𝜑 → ((𝐹(+g‘(LDual‘𝑈))𝐺)‘𝑋) = ((𝐹‘𝑋) + (𝐺‘𝑋))) |
23 | 10, 22 | eqtrd 2856 | 1 ⊢ (𝜑 → ((𝐹 ✚ 𝐺)‘𝑋) = ((𝐹‘𝑋) + (𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Scalarcsca 16567 LFnlclfn 36192 LDualcld 36258 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 LCDualclcd 38721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lvec 19874 df-lfl 36193 df-ldual 36259 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tendo 37890 df-edring 37892 df-dvech 38214 df-lcdual 38722 |
This theorem is referenced by: lcdvsubval 38753 hdmaplna2 39045 |
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