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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsass | Structured version Visualization version GIF version |
Description: Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.) |
Ref | Expression |
---|---|
ldualvsass.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsass.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsass.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualvsass.t | ⊢ × = (.r‘𝑅) |
ldualvsass.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsass.s | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsass.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsass.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualvsass.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
ldualvsass.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsass | ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | ldualvsass.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | ldualvsass.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | ldualvsass.t | . . . 4 ⊢ × = (.r‘𝑅) | |
5 | ldualvsass.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
6 | ldualvsass.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | ldualvsass.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
8 | ldualvsass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
9 | ldualvsass.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lflvsass 36377 | . . 3 ⊢ (𝜑 → (𝐺 ∘f × ((Base‘𝑊) × {(𝑌 × 𝑋)})) = ((𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∘f × ((Base‘𝑊) × {𝑋}))) |
11 | ldualvsass.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
12 | ldualvsass.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
13 | 2 | lmodring 19635 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
14 | 6, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | 3, 4 | ringcl 19307 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐾) → (𝑌 × 𝑋) ∈ 𝐾) |
16 | 14, 7, 8, 15 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ 𝐾) |
17 | 5, 1, 2, 3, 4, 11, 12, 6, 16, 9 | ldualvs 36433 | . . 3 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝐺 ∘f × ((Base‘𝑊) × {(𝑌 × 𝑋)}))) |
18 | 1, 2, 3, 4, 5, 6, 9, 7 | lflvscl 36373 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∈ 𝐹) |
19 | 5, 1, 2, 3, 4, 11, 12, 6, 8, 18 | ldualvs 36433 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌}))) = ((𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∘f × ((Base‘𝑊) × {𝑋}))) |
20 | 10, 17, 19 | 3eqtr4d 2843 | . 2 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌})))) |
21 | 5, 1, 2, 3, 4, 11, 12, 6, 7, 9 | ldualvs 36433 | . . 3 ⊢ (𝜑 → (𝑌 · 𝐺) = (𝐺 ∘f × ((Base‘𝑊) × {𝑌}))) |
22 | 21 | oveq2d 7151 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · 𝐺)) = (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌})))) |
23 | 20, 22 | eqtr4d 2836 | 1 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 Basecbs 16475 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 Ringcrg 19290 LModclmod 19627 LFnlclfn 36353 LDualcld 36419 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-sca 16573 df-vsca 16574 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19233 df-ring 19292 df-lmod 19629 df-lfl 36354 df-ldual 36420 |
This theorem is referenced by: ldualvsass2 36438 |
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