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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvadd | Structured version Visualization version GIF version |
Description: Vector addition in the dual of a vector space. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualvadd.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvadd.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvadd.a | ⊢ + = (+g‘𝑅) |
ldualvadd.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvadd.p | ⊢ ✚ = (+g‘𝐷) |
ldualvadd.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
ldualvadd.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
ldualvadd.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvadd | ⊢ (𝜑 → (𝐺 ✚ 𝐻) = (𝐺 ∘𝑓 + 𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvadd.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | ldualvadd.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | ldualvadd.a | . . . 4 ⊢ + = (+g‘𝑅) | |
4 | ldualvadd.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | ldualvadd.p | . . . 4 ⊢ ✚ = (+g‘𝐷) | |
6 | ldualvadd.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
7 | eqid 2778 | . . . 4 ⊢ ( ∘𝑓 + ↾ (𝐹 × 𝐹)) = ( ∘𝑓 + ↾ (𝐹 × 𝐹)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ldualfvadd 35284 | . . 3 ⊢ (𝜑 → ✚ = ( ∘𝑓 + ↾ (𝐹 × 𝐹))) |
9 | 8 | oveqd 6939 | . 2 ⊢ (𝜑 → (𝐺 ✚ 𝐻) = (𝐺( ∘𝑓 + ↾ (𝐹 × 𝐹))𝐻)) |
10 | ldualvadd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | ldualvadd.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
12 | 10, 11 | ofmresval 7187 | . 2 ⊢ (𝜑 → (𝐺( ∘𝑓 + ↾ (𝐹 × 𝐹))𝐻) = (𝐺 ∘𝑓 + 𝐻)) |
13 | 9, 12 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝐺 ✚ 𝐻) = (𝐺 ∘𝑓 + 𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 × cxp 5353 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 +gcplusg 16338 Scalarcsca 16341 LFnlclfn 35213 LDualcld 35279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-sca 16354 df-vsca 16355 df-ldual 35280 |
This theorem is referenced by: ldualvaddcl 35286 ldualvaddval 35287 ldualvaddcom 35296 ldualvsdi1 35299 ldualvsdi2 35300 ldualgrplem 35301 ldual0v 35306 |
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