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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 | β’ πΉ = (LFnlβπ) |
ldualvadd.r | β’ π = (Scalarβπ) |
ldualvadd.a | β’ + = (+gβπ ) |
ldualvadd.d | β’ π· = (LDualβπ) |
ldualvadd.p | β’ β = (+gβπ·) |
ldualvadd.w | β’ (π β π β π) |
ldualfvadd.q | ⒠⨣ = ( βf + βΎ (πΉ Γ πΉ)) |
Ref | Expression |
---|---|
ldualfvadd | β’ (π β β = ⨣ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ (Baseβπ) = (Baseβπ) | |
2 | ldualvadd.a | . . . 4 β’ + = (+gβπ ) | |
3 | ldualfvadd.q | . . . 4 ⒠⨣ = ( βf + βΎ (πΉ Γ πΉ)) | |
4 | ldualvadd.f | . . . 4 β’ πΉ = (LFnlβπ) | |
5 | ldualvadd.d | . . . 4 β’ π· = (LDualβπ) | |
6 | ldualvadd.r | . . . 4 β’ π = (Scalarβπ) | |
7 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
8 | eqid 2733 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
9 | eqid 2733 | . . . 4 β’ (opprβπ ) = (opprβπ ) | |
10 | eqid 2733 | . . . 4 β’ (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π}))) = (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π}))) | |
11 | ldualvadd.w | . . . 4 β’ (π β π β π) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | ldualset 37637 | . . 3 β’ (π β π· = ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©})) |
13 | 12 | fveq2d 6850 | . 2 β’ (π β (+gβπ·) = (+gβ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©}))) |
14 | ldualvadd.p | . 2 β’ β = (+gβπ·) | |
15 | 4 | fvexi 6860 | . . . . 5 β’ πΉ β V |
16 | id 22 | . . . . . 6 β’ (πΉ β V β πΉ β V) | |
17 | 16, 16 | ofmresex 7922 | . . . . 5 β’ (πΉ β V β ( βf + βΎ (πΉ Γ πΉ)) β V) |
18 | 15, 17 | ax-mp 5 | . . . 4 β’ ( βf + βΎ (πΉ Γ πΉ)) β V |
19 | 3, 18 | eqeltri 2830 | . . 3 ⒠⨣ β V |
20 | eqid 2733 | . . . 4 β’ ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©}) = ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©}) | |
21 | 20 | lmodplusg 17216 | . . 3 β’ ( ⨣ β V β ⨣ = (+gβ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©}))) |
22 | 19, 21 | ax-mp 5 | . 2 ⒠⨣ = (+gβ({β¨(Baseβndx), πΉβ©, β¨(+gβndx), ⨣ β©, β¨(Scalarβndx), (opprβπ )β©} βͺ {β¨( Β·π βndx), (π β (Baseβπ ), π β πΉ β¦ (π βf (.rβπ )((Baseβπ) Γ {π})))β©})) |
23 | 13, 14, 22 | 3eqtr4g 2798 | 1 β’ (π β β = ⨣ ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 βͺ cun 3912 {csn 4590 {ctp 4594 β¨cop 4596 Γ cxp 5635 βΎ cres 5639 βcfv 6500 (class class class)co 7361 β cmpo 7363 βf cof 7619 ndxcnx 17073 Basecbs 17091 +gcplusg 17141 .rcmulr 17142 Scalarcsca 17144 Β·π cvsca 17145 opprcoppr 20056 LFnlclfn 37569 LDualcld 37635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-sca 17157 df-vsca 17158 df-ldual 37636 |
This theorem is referenced by: ldualvadd 37641 |
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