| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualfvs | Structured version Visualization version GIF version | ||
| Description: Scalar product operation for the dual of a vector space. (Contributed by NM, 18-Oct-2014.) |
| Ref | Expression |
|---|---|
| ldualfvs.f | ⊢ 𝐹 = (LFnl‘𝑊) |
| ldualfvs.v | ⊢ 𝑉 = (Base‘𝑊) |
| ldualfvs.r | ⊢ 𝑅 = (Scalar‘𝑊) |
| ldualfvs.k | ⊢ 𝐾 = (Base‘𝑅) |
| ldualfvs.t | ⊢ × = (.r‘𝑅) |
| ldualfvs.d | ⊢ 𝐷 = (LDual‘𝑊) |
| ldualfvs.s | ⊢ ∙ = ( ·𝑠 ‘𝐷) |
| ldualfvs.w | ⊢ (𝜑 → 𝑊 ∈ 𝑌) |
| ldualfvs.m | ⊢ · = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) |
| Ref | Expression |
|---|---|
| ldualfvs | ⊢ (𝜑 → ∙ = · ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ldualfvs.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
| 2 | eqid 2737 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 3 | eqid 2737 | . . . 4 ⊢ ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹)) = ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹)) | |
| 4 | ldualfvs.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
| 5 | ldualfvs.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
| 6 | ldualfvs.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
| 7 | ldualfvs.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
| 8 | ldualfvs.t | . . . 4 ⊢ × = (.r‘𝑅) | |
| 9 | eqid 2737 | . . . 4 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
| 10 | eqid 2737 | . . . 4 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) | |
| 11 | ldualfvs.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑌) | |
| 12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 39126 | . . 3 ⊢ (𝜑 → 𝐷 = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
| 13 | 12 | fveq2d 6910 | . 2 ⊢ (𝜑 → ( ·𝑠 ‘𝐷) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}))) |
| 14 | ldualfvs.s | . 2 ⊢ ∙ = ( ·𝑠 ‘𝐷) | |
| 15 | ldualfvs.m | . . 3 ⊢ · = (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) | |
| 16 | 7 | fvexi 6920 | . . . . 5 ⊢ 𝐾 ∈ V |
| 17 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝐹 ∈ V |
| 18 | 16, 17 | mpoex 8104 | . . . 4 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) ∈ V |
| 19 | eqid 2737 | . . . . 5 ⊢ ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}) = ({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}) | |
| 20 | 19 | lmodvsca 17373 | . . . 4 ⊢ ((𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) ∈ V → (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉}))) |
| 21 | 18, 20 | ax-mp 5 | . . 3 ⊢ (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘}))) = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
| 22 | 15, 21 | eqtri 2765 | . 2 ⊢ · = ( ·𝑠 ‘({〈(Base‘ndx), 𝐹〉, 〈(+g‘ndx), ( ∘f (+g‘𝑅) ↾ (𝐹 × 𝐹))〉, 〈(Scalar‘ndx), (oppr‘𝑅)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑘 ∈ 𝐾, 𝑓 ∈ 𝐹 ↦ (𝑓 ∘f × (𝑉 × {𝑘})))〉})) |
| 23 | 13, 14, 22 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → ∙ = · ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 {csn 4626 {ctp 4630 〈cop 4632 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ∘f cof 7695 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 opprcoppr 20333 LFnlclfn 39058 LDualcld 39124 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-sca 17313 df-vsca 17314 df-ldual 39125 |
| This theorem is referenced by: ldualvs 39138 |
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