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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualfvadd | 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) |
ldualfvadd.q | ⊢ ⨣ = ( ∘f + ↾ (𝐹 × 𝐹)) |
Ref | Expression |
---|---|
ldualfvadd | ⊢ (𝜑 → ✚ = ⨣ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | ldualvadd.a | . . . 4 ⊢ + = (+g‘𝑅) | |
3 | ldualfvadd.q | . . . 4 ⊢ ⨣ = ( ∘f + ↾ (𝐹 × 𝐹)) | |
4 | ldualvadd.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
5 | ldualvadd.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
6 | ldualvadd.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
7 | eqid 2735 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
8 | eqid 2735 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
9 | eqid 2735 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | eqid 2735 | . . . 4 ⊢ (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘}))) = (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘}))) | |
11 | ldualvadd.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 39107 | . . 3 ⊢ (𝜑 → 𝐷 = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉})) |
13 | 12 | fveq2d 6911 | . 2 ⊢ (𝜑 → (+g‘𝐷) = (+g‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉}))) |
14 | ldualvadd.p | . 2 ⊢ ✚ = (+g‘𝐷) | |
15 | 4 | fvexi 6921 | . . . . 5 ⊢ 𝐹 ∈ V |
16 | id 22 | . . . . . 6 ⊢ (𝐹 ∈ V → 𝐹 ∈ V) | |
17 | 16, 16 | ofmresex 8009 | . . . . 5 ⊢ (𝐹 ∈ V → ( ∘f + ↾ (𝐹 × 𝐹)) ∈ V) |
18 | 15, 17 | ax-mp 5 | . . . 4 ⊢ ( ∘f + ↾ (𝐹 × 𝐹)) ∈ V |
19 | 3, 18 | eqeltri 2835 | . . 3 ⊢ ⨣ ∈ V |
20 | eqid 2735 | . . . 4 ⊢ ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉}) = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉}) | |
21 | 20 | lmodplusg 17373 | . . 3 ⊢ ( ⨣ ∈ V → ⨣ = (+g‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉}))) |
22 | 19, 21 | ax-mp 5 | . 2 ⊢ ⨣ = (+g‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ⨣ 〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ (Base‘𝑅), 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑘})))〉})) |
23 | 13, 14, 22 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → ✚ = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 {csn 4631 {ctp 4635 〈cop 4637 × cxp 5687 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ∘f cof 7695 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 opprcoppr 20350 LFnlclfn 39039 LDualcld 39105 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ldual 39106 |
This theorem is referenced by: ldualvadd 39111 |
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