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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsdi2 | Structured version Visualization version GIF version |
Description: Reverse distributive law for scalar product operation, using operations from the dual space. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
ldualvsdi2.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsdi2.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsdi2.a | ⊢ + = (+g‘𝑅) |
ldualvsdi2.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualvsdi2.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsdi2.p | ⊢ ✚ = (+g‘𝐷) |
ldualvsdi2.s | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsdi2.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsdi2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualvsdi2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
ldualvsdi2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsdi2 | ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝐺) = ((𝑋 · 𝐺) ✚ (𝑌 · 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ldualvsdi2.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑊) | |
2 | eqid 2726 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | ldualvsdi2.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑊) | |
4 | ldualvsdi2.k | . . 3 ⊢ 𝐾 = (Base‘𝑅) | |
5 | eqid 2726 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | ldualvsdi2.d | . . 3 ⊢ 𝐷 = (LDual‘𝑊) | |
7 | ldualvsdi2.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐷) | |
8 | ldualvsdi2.w | . . 3 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
9 | ldualvsdi2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
10 | ldualvsdi2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
11 | ldualvsdi2.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
12 | 3, 4, 11 | lmodacl 20848 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾) |
13 | 8, 9, 10, 12 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐾) |
14 | ldualvsdi2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 13, 14 | ldualvs 38835 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝐺) = (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {(𝑋 + 𝑌)}))) |
16 | 2, 3, 4, 11, 5, 1, 8, 9, 10, 14 | lflvsdi2a 38778 | . 2 ⊢ (𝜑 → (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {(𝑋 + 𝑌)})) = ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f + (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑌})))) |
17 | ldualvsdi2.p | . . . 4 ⊢ ✚ = (+g‘𝐷) | |
18 | 1, 3, 4, 6, 7, 8, 9, 14 | ldualvscl 38837 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹) |
19 | 1, 3, 4, 6, 7, 8, 10, 14 | ldualvscl 38837 | . . . 4 ⊢ (𝜑 → (𝑌 · 𝐺) ∈ 𝐹) |
20 | 1, 3, 11, 6, 17, 8, 18, 19 | ldualvadd 38827 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝐺) ✚ (𝑌 · 𝐺)) = ((𝑋 · 𝐺) ∘f + (𝑌 · 𝐺))) |
21 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 14 | ldualvs 38835 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝐺) = (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋}))) |
22 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 14 | ldualvs 38835 | . . . 4 ⊢ (𝜑 → (𝑌 · 𝐺) = (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑌}))) |
23 | 21, 22 | oveq12d 7442 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝐺) ∘f + (𝑌 · 𝐺)) = ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f + (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑌})))) |
24 | 20, 23 | eqtr2d 2767 | . 2 ⊢ (𝜑 → ((𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑋})) ∘f + (𝐺 ∘f (.r‘𝑅)((Base‘𝑊) × {𝑌}))) = ((𝑋 · 𝐺) ✚ (𝑌 · 𝐺))) |
25 | 15, 16, 24 | 3eqtrd 2770 | 1 ⊢ (𝜑 → ((𝑋 + 𝑌) · 𝐺) = ((𝑋 · 𝐺) ✚ (𝑌 · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {csn 4633 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ∘f cof 7688 Basecbs 17213 +gcplusg 17266 .rcmulr 17267 Scalarcsca 17269 ·𝑠 cvsca 17270 LModclmod 20836 LFnlclfn 38755 LDualcld 38821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-sca 17282 df-vsca 17283 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-mgp 20118 df-ring 20218 df-lmod 20838 df-lfl 38756 df-ldual 38822 |
This theorem is referenced by: lduallmodlem 38850 |
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