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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 6910 | . . . 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 38662 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | lflass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | fconst6g 6786 | . . . 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 6786 | . . . 4 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
17 | 6 | lmodring 20763 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 4, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lflass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
20 | 7, 19 | ringass 20205 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
21 | 18, 20 | sylan 578 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | 3, 10, 13, 16, 21 | caofass 7723 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌})) = (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})))) |
23 | 3, 11, 14 | ofc12 7714 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})) = (𝑉 × {(𝑋 · 𝑌)})) |
24 | 23 | oveq2d 7435 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌}))) = (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)}))) |
25 | 22, 24 | eqtr2d 2766 | 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 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 × cxp 5676 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∘f cof 7683 Basecbs 17183 .rcmulr 17237 Scalarcsca 17239 Ringcrg 20185 LModclmod 20755 LFnlclfn 38656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-sgrp 18682 df-mnd 18698 df-mgp 20087 df-ring 20187 df-lmod 20757 df-lfl 38657 |
This theorem is referenced by: ldualvsass 38740 |
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