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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 6462 | . . . 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 35226 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | lflass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | fconst6g 6346 | . . . 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 6346 | . . . 4 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
17 | 6 | lmodring 19274 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 4, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lflass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
20 | 7, 19 | ringass 18962 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
21 | 18, 20 | sylan 575 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | 3, 10, 13, 16, 21 | caofass 7210 | . 2 ⊢ (𝜑 → ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 · (𝑉 × {𝑌})) = (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌})))) |
23 | 3, 11, 14 | ofc12 7201 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌})) = (𝑉 × {(𝑋 · 𝑌)})) |
24 | 23 | oveq2d 6940 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌}))) = (𝐺 ∘𝑓 · (𝑉 × {(𝑋 · 𝑌)}))) |
25 | 22, 24 | eqtr2d 2815 | 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 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 {csn 4398 × cxp 5355 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ∘𝑓 cof 7174 Basecbs 16266 .rcmulr 16350 Scalarcsca 16352 Ringcrg 18945 LModclmod 19266 LFnlclfn 35220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-plusg 16362 df-sgrp 17681 df-mnd 17692 df-mgp 18888 df-ring 18947 df-lmod 19268 df-lfl 35221 |
This theorem is referenced by: ldualvsass 35304 |
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