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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvafvadd | Structured version Visualization version GIF version |
Description: The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.) |
Ref | Expression |
---|---|
dvafvadd.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvafvadd.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvafvadd.u | ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) |
dvafvadd.v | ⊢ + = (+g‘𝑈) |
Ref | Expression |
---|---|
dvafvadd | ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → + = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvafvadd.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvafvadd.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | eqid 2760 | . . . 4 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
4 | eqid 2760 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
5 | dvafvadd.u | . . . 4 ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dvaset 36795 | . . 3 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉})) |
7 | 6 | fveq2d 6356 | . 2 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → (+g‘𝑈) = (+g‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}))) |
8 | dvafvadd.v | . 2 ⊢ + = (+g‘𝑈) | |
9 | fvex 6362 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V | |
10 | 2, 9 | eqeltri 2835 | . . . 4 ⊢ 𝑇 ∈ V |
11 | 10, 10 | mpt2ex 7415 | . . 3 ⊢ (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) ∈ V |
12 | eqid 2760 | . . . 4 ⊢ ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}) = ({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}) | |
13 | 12 | lmodplusg 16221 | . . 3 ⊢ ((𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) ∈ V → (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (+g‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉}))) |
14 | 11, 13 | ax-mp 5 | . 2 ⊢ (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)) = (+g‘({〈(Base‘ndx), 𝑇〉, 〈(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑓 ∈ 𝑇 ↦ (𝑠‘𝑓))〉})) |
15 | 7, 8, 14 | 3eqtr4g 2819 | 1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → + = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∪ cun 3713 {csn 4321 {ctp 4325 〈cop 4327 ∘ ccom 5270 ‘cfv 6049 ↦ cmpt2 6815 ndxcnx 16056 Basecbs 16059 +gcplusg 16143 Scalarcsca 16146 ·𝑠 cvsca 16147 LHypclh 35773 LTrncltrn 35890 TEndoctendo 36542 EDRingcedring 36543 DVecAcdveca 36792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-plusg 16156 df-sca 16159 df-vsca 16160 df-dveca 36793 |
This theorem is referenced by: dvavadd 36805 |
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