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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 6687 | . . . 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 36203 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
10 | 4, 5, 9 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
11 | lflass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
12 | fconst6g 6571 | . . . 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 6571 | . . . 4 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
17 | 6 | lmodring 19645 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
18 | 4, 17 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | lflass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
20 | 7, 19 | ringass 19317 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
21 | 18, 20 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | 3, 10, 13, 16, 21 | caofass 7446 | . 2 ⊢ (𝜑 → ((𝐺 ∘f · (𝑉 × {𝑋})) ∘f · (𝑉 × {𝑌})) = (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})))) |
23 | 3, 11, 14 | ofc12 7437 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌})) = (𝑉 × {(𝑋 · 𝑌)})) |
24 | 23 | oveq2d 7175 | . 2 ⊢ (𝜑 → (𝐺 ∘f · ((𝑉 × {𝑋}) ∘f · (𝑉 × {𝑌}))) = (𝐺 ∘f · (𝑉 × {(𝑋 · 𝑌)}))) |
25 | 22, 24 | eqtr2d 2860 | 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 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 × cxp 5556 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 Basecbs 16486 .rcmulr 16569 Scalarcsca 16571 Ringcrg 19300 LModclmod 19637 LFnlclfn 36197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-sgrp 17904 df-mnd 17915 df-mgp 19243 df-ring 19302 df-lmod 19639 df-lfl 36198 |
This theorem is referenced by: ldualvsass 36281 |
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