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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsub | Structured version Visualization version GIF version |
Description: The value of vector subtraction in the dual of a vector space. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
ldualvsub.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsub.n | ⊢ 𝑁 = (invg‘𝑅) |
ldualvsub.u | ⊢ 1 = (1r‘𝑅) |
ldualvsub.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsub.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsub.p | ⊢ + = (+g‘𝐷) |
ldualvsub.t | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsub.m | ⊢ − = (-g‘𝐷) |
ldualvsub.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsub.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvsub.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsub | ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsub.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
2 | ldualvsub.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | 1, 2 | lduallmod 36288 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
4 | ldualvsub.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | ldualvsub.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | 4, 1, 5, 2, 6 | ldualelvbase 36262 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
8 | ldualvsub.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 4, 1, 5, 2, 8 | ldualelvbase 36262 | . . 3 ⊢ (𝜑 → 𝐻 ∈ (Base‘𝐷)) |
10 | ldualvsub.p | . . . 4 ⊢ + = (+g‘𝐷) | |
11 | ldualvsub.m | . . . 4 ⊢ − = (-g‘𝐷) | |
12 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
13 | ldualvsub.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
14 | eqid 2821 | . . . 4 ⊢ (invg‘(Scalar‘𝐷)) = (invg‘(Scalar‘𝐷)) | |
15 | eqid 2821 | . . . 4 ⊢ (1r‘(Scalar‘𝐷)) = (1r‘(Scalar‘𝐷)) | |
16 | 5, 10, 11, 12, 13, 14, 15 | lmodvsubval2 19688 | . . 3 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷) ∧ 𝐻 ∈ (Base‘𝐷)) → (𝐺 − 𝐻) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
17 | 3, 7, 9, 16 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
18 | ldualvsub.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑊) | |
19 | eqid 2821 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
20 | 18, 19, 1, 12, 2 | ldualsca 36267 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝐷) = (oppr‘𝑅)) |
21 | 20 | fveq2d 6673 | . . . . . 6 ⊢ (𝜑 → (invg‘(Scalar‘𝐷)) = (invg‘(oppr‘𝑅))) |
22 | ldualvsub.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
23 | 19, 22 | opprneg 19384 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
24 | 21, 23 | syl6reqr 2875 | . . . . 5 ⊢ (𝜑 → 𝑁 = (invg‘(Scalar‘𝐷))) |
25 | 20 | fveq2d 6673 | . . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝐷)) = (1r‘(oppr‘𝑅))) |
26 | ldualvsub.u | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
27 | 19, 26 | oppr1 19383 | . . . . . 6 ⊢ 1 = (1r‘(oppr‘𝑅)) |
28 | 25, 27 | syl6reqr 2875 | . . . . 5 ⊢ (𝜑 → 1 = (1r‘(Scalar‘𝐷))) |
29 | 24, 28 | fveq12d 6676 | . . . 4 ⊢ (𝜑 → (𝑁‘ 1 ) = ((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷)))) |
30 | 29 | oveq1d 7170 | . . 3 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝐻) = (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻)) |
31 | 30 | oveq2d 7171 | . 2 ⊢ (𝜑 → (𝐺 + ((𝑁‘ 1 ) · 𝐻)) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
32 | 17, 31 | eqtr4d 2859 | 1 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Scalarcsca 16567 ·𝑠 cvsca 16568 invgcminusg 18103 -gcsg 18104 1rcur 19250 opprcoppr 19371 LModclmod 19633 LFnlclfn 36192 LDualcld 36258 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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-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-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-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-lmod 19635 df-lfl 36193 df-ldual 36259 |
This theorem is referenced by: ldualvsubcl 36291 lcfrlem2 38678 |
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