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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 | fvex 6201 | . . . . 5 ⊢ (Base‘𝑊) ∈ V | |
3 | 1, 2 | eqeltri 2697 | . . . 4 ⊢ 𝑉 ∈ V |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑉 ∈ V) |
5 | lflass.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | lflass.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | lflass.r | . . . . 5 ⊢ 𝑅 = (Scalar‘𝑊) | |
8 | lflass.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑅) | |
9 | lflass.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑊) | |
10 | 7, 8, 1, 9 | lflf 34350 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
11 | 5, 6, 10 | syl2anc 693 | . . 3 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
12 | lflass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
13 | fconst6g 6094 | . . . 4 ⊢ (𝑋 ∈ 𝐾 → (𝑉 × {𝑋}):𝑉⟶𝐾) | |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑋}):𝑉⟶𝐾) |
15 | lflass.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
16 | fconst6g 6094 | . . . 4 ⊢ (𝑌 ∈ 𝐾 → (𝑉 × {𝑌}):𝑉⟶𝐾) | |
17 | 15, 16 | syl 17 | . . 3 ⊢ (𝜑 → (𝑉 × {𝑌}):𝑉⟶𝐾) |
18 | 7 | lmodring 18871 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
19 | 5, 18 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
20 | lflass.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
21 | 8, 20 | ringass 18564 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | 19, 21 | sylan 488 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
23 | 4, 11, 14, 17, 22 | caofass 6931 | . 2 ⊢ (𝜑 → ((𝐺 ∘𝑓 · (𝑉 × {𝑋})) ∘𝑓 · (𝑉 × {𝑌})) = (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌})))) |
24 | 4, 12, 15 | ofc12 6922 | . . 3 ⊢ (𝜑 → ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌})) = (𝑉 × {(𝑋 · 𝑌)})) |
25 | 24 | oveq2d 6666 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 · ((𝑉 × {𝑋}) ∘𝑓 · (𝑉 × {𝑌}))) = (𝐺 ∘𝑓 · (𝑉 × {(𝑋 · 𝑌)}))) |
26 | 23, 25 | eqtr2d 2657 | 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 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 Basecbs 15857 .rcmulr 15942 Scalarcsca 15944 Ringcrg 18547 LModclmod 18863 LFnlclfn 34344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-sgrp 17284 df-mnd 17295 df-mgp 18490 df-ring 18549 df-lmod 18865 df-lfl 34345 |
This theorem is referenced by: ldualvsass 34428 |
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