![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualsmul | Structured version Visualization version GIF version |
Description: Scalar multiplication for the dual of a vector space. (Contributed by NM, 19-Oct-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
ldualsmul.f | ⊢ 𝐹 = (Scalar‘𝑊) |
ldualsmul.k | ⊢ 𝐾 = (Base‘𝐹) |
ldualsmul.t | ⊢ · = (.r‘𝐹) |
ldualsmul.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualsmul.r | ⊢ 𝑅 = (Scalar‘𝐷) |
ldualsmul.m | ⊢ ∙ = (.r‘𝑅) |
ldualsmul.w | ⊢ (𝜑 → 𝑊 ∈ 𝑉) |
ldualsmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualsmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
Ref | Expression |
---|---|
ldualsmul | ⊢ (𝜑 → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualsmul.m | . . . 4 ⊢ ∙ = (.r‘𝑅) | |
2 | ldualsmul.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2736 | . . . . . 6 ⊢ (oppr‘𝐹) = (oppr‘𝐹) | |
4 | ldualsmul.d | . . . . . 6 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | ldualsmul.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝐷) | |
6 | ldualsmul.w | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
7 | 2, 3, 4, 5, 6 | ldualsca 37561 | . . . . 5 ⊢ (𝜑 → 𝑅 = (oppr‘𝐹)) |
8 | 7 | fveq2d 6843 | . . . 4 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(oppr‘𝐹))) |
9 | 1, 8 | eqtrid 2788 | . . 3 ⊢ (𝜑 → ∙ = (.r‘(oppr‘𝐹))) |
10 | 9 | oveqd 7370 | . 2 ⊢ (𝜑 → (𝑋 ∙ 𝑌) = (𝑋(.r‘(oppr‘𝐹))𝑌)) |
11 | ldualsmul.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
12 | ldualsmul.t | . . 3 ⊢ · = (.r‘𝐹) | |
13 | eqid 2736 | . . 3 ⊢ (.r‘(oppr‘𝐹)) = (.r‘(oppr‘𝐹)) | |
14 | 11, 12, 3, 13 | opprmul 20037 | . 2 ⊢ (𝑋(.r‘(oppr‘𝐹))𝑌) = (𝑌 · 𝑋) |
15 | 10, 14 | eqtrdi 2792 | 1 ⊢ (𝜑 → (𝑋 ∙ 𝑌) = (𝑌 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 .rcmulr 17126 Scalarcsca 17128 opprcoppr 20033 LDualcld 37552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-plusg 17138 df-mulr 17139 df-sca 17141 df-vsca 17142 df-oppr 20034 df-ldual 37553 |
This theorem is referenced by: ldualvsass2 37571 |
Copyright terms: Public domain | W3C validator |