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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcdvsub | Structured version Visualization version GIF version |
Description: The value of vector subtraction in the closed kernel dual space. (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
lcdvsub.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lcdvsub.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lcdvsub.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcdvsub.n | ⊢ 𝑁 = (invg‘𝑆) |
lcdvsub.e | ⊢ 1 = (1r‘𝑆) |
lcdvsub.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
lcdvsub.v | ⊢ 𝑉 = (Base‘𝐶) |
lcdvsub.p | ⊢ + = (+g‘𝐶) |
lcdvsub.t | ⊢ · = ( ·𝑠 ‘𝐶) |
lcdvsub.m | ⊢ − = (-g‘𝐶) |
lcdvsub.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lcdvsub.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
lcdvsub.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
Ref | Expression |
---|---|
lcdvsub | ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹 + ((𝑁‘ 1 ) · 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcdvsub.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lcdvsub.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | lcdvsub.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 37666 | . . 3 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lcdvsub.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
6 | lcdvsub.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
7 | lcdvsub.v | . . . 4 ⊢ 𝑉 = (Base‘𝐶) | |
8 | lcdvsub.p | . . . 4 ⊢ + = (+g‘𝐶) | |
9 | lcdvsub.m | . . . 4 ⊢ − = (-g‘𝐶) | |
10 | eqid 2824 | . . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
11 | lcdvsub.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐶) | |
12 | eqid 2824 | . . . 4 ⊢ (invg‘(Scalar‘𝐶)) = (invg‘(Scalar‘𝐶)) | |
13 | eqid 2824 | . . . 4 ⊢ (1r‘(Scalar‘𝐶)) = (1r‘(Scalar‘𝐶)) | |
14 | 7, 8, 9, 10, 11, 12, 13 | lmodvsubval2 19273 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉) → (𝐹 − 𝐺) = (𝐹 + (((invg‘(Scalar‘𝐶))‘(1r‘(Scalar‘𝐶))) · 𝐺))) |
15 | 4, 5, 6, 14 | syl3anc 1496 | . 2 ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹 + (((invg‘(Scalar‘𝐶))‘(1r‘(Scalar‘𝐶))) · 𝐺))) |
16 | lcdvsub.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
17 | lcdvsub.s | . . . . . . . 8 ⊢ 𝑆 = (Scalar‘𝑈) | |
18 | eqid 2824 | . . . . . . . 8 ⊢ (oppr‘𝑆) = (oppr‘𝑆) | |
19 | 1, 16, 17, 18, 2, 10, 3 | lcdsca 37673 | . . . . . . 7 ⊢ (𝜑 → (Scalar‘𝐶) = (oppr‘𝑆)) |
20 | 19 | fveq2d 6436 | . . . . . 6 ⊢ (𝜑 → (invg‘(Scalar‘𝐶)) = (invg‘(oppr‘𝑆))) |
21 | lcdvsub.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝑆) | |
22 | 18, 21 | opprneg 18988 | . . . . . 6 ⊢ 𝑁 = (invg‘(oppr‘𝑆)) |
23 | 20, 22 | syl6reqr 2879 | . . . . 5 ⊢ (𝜑 → 𝑁 = (invg‘(Scalar‘𝐶))) |
24 | 19 | fveq2d 6436 | . . . . . 6 ⊢ (𝜑 → (1r‘(Scalar‘𝐶)) = (1r‘(oppr‘𝑆))) |
25 | lcdvsub.e | . . . . . . 7 ⊢ 1 = (1r‘𝑆) | |
26 | 18, 25 | oppr1 18987 | . . . . . 6 ⊢ 1 = (1r‘(oppr‘𝑆)) |
27 | 24, 26 | syl6reqr 2879 | . . . . 5 ⊢ (𝜑 → 1 = (1r‘(Scalar‘𝐶))) |
28 | 23, 27 | fveq12d 6439 | . . . 4 ⊢ (𝜑 → (𝑁‘ 1 ) = ((invg‘(Scalar‘𝐶))‘(1r‘(Scalar‘𝐶)))) |
29 | 28 | oveq1d 6919 | . . 3 ⊢ (𝜑 → ((𝑁‘ 1 ) · 𝐺) = (((invg‘(Scalar‘𝐶))‘(1r‘(Scalar‘𝐶))) · 𝐺)) |
30 | 29 | oveq2d 6920 | . 2 ⊢ (𝜑 → (𝐹 + ((𝑁‘ 1 ) · 𝐺)) = (𝐹 + (((invg‘(Scalar‘𝐶))‘(1r‘(Scalar‘𝐶))) · 𝐺))) |
31 | 15, 30 | eqtr4d 2863 | 1 ⊢ (𝜑 → (𝐹 − 𝐺) = (𝐹 + ((𝑁‘ 1 ) · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 +gcplusg 16304 Scalarcsca 16307 ·𝑠 cvsca 16308 invgcminusg 17776 -gcsg 17777 1rcur 18854 opprcoppr 18975 LModclmod 19218 HLchlt 35424 LHypclh 36058 DVecHcdvh 37152 LCDualclcd 37660 |
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 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-tpos 7616 df-undef 7663 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-sca 16320 df-vsca 16321 df-0g 16454 df-mre 16598 df-mrc 16599 df-acs 16601 df-proset 17280 df-poset 17298 df-plt 17310 df-lub 17326 df-glb 17327 df-join 17328 df-meet 17329 df-p0 17391 df-p1 17392 df-lat 17398 df-clat 17460 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-grp 17778 df-minusg 17779 df-sbg 17780 df-subg 17941 df-cntz 18099 df-oppg 18125 df-lsm 18401 df-cmn 18547 df-abl 18548 df-mgp 18843 df-ur 18855 df-ring 18902 df-oppr 18976 df-dvdsr 18994 df-unit 18995 df-invr 19025 df-dvr 19036 df-drng 19104 df-lmod 19220 df-lss 19288 df-lsp 19330 df-lvec 19461 df-lsatoms 35050 df-lshyp 35051 df-lcv 35093 df-lfl 35132 df-lkr 35160 df-ldual 35198 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tgrp 36817 df-tendo 36829 df-edring 36831 df-dveca 37077 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 df-doch 37422 df-djh 37469 df-lcdual 37661 |
This theorem is referenced by: mapdpglem30 37776 |
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