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Mathbox for Norm Megill |
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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 34758 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
4 | ldualvsub.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | eqid 2651 | . . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷) | |
6 | ldualvsub.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | 4, 1, 5, 2, 6 | ldualelvbase 34732 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷)) |
8 | ldualvsub.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 4, 1, 5, 2, 8 | ldualelvbase 34732 | . . 3 ⊢ (𝜑 → 𝐻 ∈ (Base‘𝐷)) |
10 | ldualvsub.p | . . . 4 ⊢ + = (+g‘𝐷) | |
11 | ldualvsub.m | . . . 4 ⊢ − = (-g‘𝐷) | |
12 | eqid 2651 | . . . 4 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷) | |
13 | ldualvsub.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
14 | eqid 2651 | . . . 4 ⊢ (invg‘(Scalar‘𝐷)) = (invg‘(Scalar‘𝐷)) | |
15 | eqid 2651 | . . . 4 ⊢ (1r‘(Scalar‘𝐷)) = (1r‘(Scalar‘𝐷)) | |
16 | 5, 10, 11, 12, 13, 14, 15 | lmodvsubval2 18966 | . . 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 1366 | . 2 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
18 | ldualvsub.r | . . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑊) | |
19 | eqid 2651 | . . . . . . . 8 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
20 | 18, 19, 1, 12, 2 | ldualsca 34737 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝐷) = (oppr‘𝑅)) |
21 | 20 | fveq2d 6233 | . . . . . 6 ⊢ (𝜑 → (invg‘(Scalar‘𝐷)) = (invg‘(oppr‘𝑅))) |
22 | ldualvsub.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑅) | |
23 | 19, 22 | opprneg 18681 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑅)) |
24 | 21, 23 | syl6reqr 2704 | . . . . 5 ⊢ (𝜑 → 𝑁 = (invg‘(Scalar‘𝐷))) |
25 | 20 | fveq2d 6233 | . . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝐷)) = (1r‘(oppr‘𝑅))) |
26 | ldualvsub.u | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
27 | 19, 26 | oppr1 18680 | . . . . . 6 ⊢ 1 = (1r‘(oppr‘𝑅)) |
28 | 25, 27 | syl6reqr 2704 | . . . . 5 ⊢ (𝜑 → 1 = (1r‘(Scalar‘𝐷))) |
29 | 24, 28 | fveq12d 6235 | . . . 4 ⊢ (𝜑 → (𝑁‘ 1 ) = ((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷)))) |
30 | 29 | oveq1d 6705 | . . 3 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝐻) = (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻)) |
31 | 30 | oveq2d 6706 | . 2 ⊢ (𝜑 → (𝐺 + ((𝑁‘ 1 ) · 𝐻)) = (𝐺 + (((invg‘(Scalar‘𝐷))‘(1r‘(Scalar‘𝐷))) · 𝐻))) |
32 | 17, 31 | eqtr4d 2688 | 1 ⊢ (𝜑 → (𝐺 − 𝐻) = (𝐺 + ((𝑁‘ 1 ) · 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 Scalarcsca 15991 ·𝑠 cvsca 15992 invgcminusg 17470 -gcsg 17471 1rcur 18547 opprcoppr 18668 LModclmod 18911 LFnlclfn 34662 LDualcld 34728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-lmod 18913 df-lfl 34663 df-ldual 34729 |
This theorem is referenced by: ldualvsubcl 34761 lcfrlem2 37149 |
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