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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflvsass | Structured version Visualization version GIF version |
Description: Associative law for (right vector space) scalar product of functionals. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
lflass.v | ⊢ 𝑉 = (Base‘𝑊) |
lflass.r | ⊢ 𝑅 = (Scalar‘𝑊) |
lflass.k | ⊢ 𝐾 = (Base‘𝑅) |
lflass.t | ⊢ · = (.r‘𝑅) |
lflass.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lflass.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lflass.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
lflass.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
lflass.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lflvsass | ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflass.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6936 | . . . 4 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lflass.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
5 | lflass.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
6 | lflass.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑊) | |
7 | lflass.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
8 | lflass.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑊) | |
9 | 6, 7, 1, 8 | lflf 39021 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | lflass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | fconst6g 6812 | . . . 4 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾) |
14 | lflass.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
15 | fconst6g 6812 | . . . 4 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
17 | 6 | lmodring 20890 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 4, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lflass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
20 | 7, 19 | ringass 20282 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
21 | 18, 20 | sylan 579 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | 3, 10, 13, 16, 21 | caofass 7754 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌})) = (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})))) |
23 | 3, 11, 14 | ofc12 7745 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})) = (𝑉 × {(𝑋 · 𝑌)})) |
24 | 23 | oveq2d 7466 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌}))) = (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)}))) |
25 | 22, 24 | eqtr2d 2781 | 1 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)})) = ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 {csn 4648 × cxp 5698 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 ∘f cof 7714 Basecbs 17260 .rcmulr 17314 Scalarcsca 17316 Ringcrg 20262 LModclmod 20882 LFnlclfn 39015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-plusg 17326 df-sgrp 18759 df-mnd 18775 df-mgp 20164 df-ring 20264 df-lmod 20884 df-lfl 39016 |
This theorem is referenced by: ldualvsass 39099 |
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