Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldualvsass | Structured version Visualization version GIF version |
Description: Associative law for scalar product operation. (Contributed by NM, 20-Oct-2014.) |
Ref | Expression |
---|---|
ldualvsass.f | ⊢ 𝐹 = (LFnl‘𝑊) |
ldualvsass.r | ⊢ 𝑅 = (Scalar‘𝑊) |
ldualvsass.k | ⊢ 𝐾 = (Base‘𝑅) |
ldualvsass.t | ⊢ × = (.r‘𝑅) |
ldualvsass.d | ⊢ 𝐷 = (LDual‘𝑊) |
ldualvsass.s | ⊢ · = ( ·𝑠 ‘𝐷) |
ldualvsass.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
ldualvsass.x | ⊢ (𝜑 → 𝑋 ∈ 𝐾) |
ldualvsass.y | ⊢ (𝜑 → 𝑌 ∈ 𝐾) |
ldualvsass.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
ldualvsass | ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | ldualvsass.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑊) | |
3 | ldualvsass.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
4 | ldualvsass.t | . . . 4 ⊢ × = (.r‘𝑅) | |
5 | ldualvsass.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
6 | ldualvsass.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
7 | ldualvsass.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
8 | ldualvsass.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐾) | |
9 | ldualvsass.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | lflvsass 36211 | . . 3 ⊢ (𝜑 → (𝐺 ∘f × ((Base‘𝑊) × {(𝑌 × 𝑋)})) = ((𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∘f × ((Base‘𝑊) × {𝑋}))) |
11 | ldualvsass.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑊) | |
12 | ldualvsass.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝐷) | |
13 | 2 | lmodring 19636 | . . . . . 6 ⊢ (𝑊 ∈ LMod → 𝑅 ∈ Ring) |
14 | 6, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
15 | 3, 4 | ringcl 19305 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐾 ∧ 𝑋 ∈ 𝐾) → (𝑌 × 𝑋) ∈ 𝐾) |
16 | 14, 7, 8, 15 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝑌 × 𝑋) ∈ 𝐾) |
17 | 5, 1, 2, 3, 4, 11, 12, 6, 16, 9 | ldualvs 36267 | . . 3 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝐺 ∘f × ((Base‘𝑊) × {(𝑌 × 𝑋)}))) |
18 | 1, 2, 3, 4, 5, 6, 9, 7 | lflvscl 36207 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∈ 𝐹) |
19 | 5, 1, 2, 3, 4, 11, 12, 6, 8, 18 | ldualvs 36267 | . . 3 ⊢ (𝜑 → (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌}))) = ((𝐺 ∘f × ((Base‘𝑊) × {𝑌})) ∘f × ((Base‘𝑊) × {𝑋}))) |
20 | 10, 17, 19 | 3eqtr4d 2866 | . 2 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌})))) |
21 | 5, 1, 2, 3, 4, 11, 12, 6, 7, 9 | ldualvs 36267 | . . 3 ⊢ (𝜑 → (𝑌 · 𝐺) = (𝐺 ∘f × ((Base‘𝑊) × {𝑌}))) |
22 | 21 | oveq2d 7166 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · 𝐺)) = (𝑋 · (𝐺 ∘f × ((Base‘𝑊) × {𝑌})))) |
23 | 20, 22 | eqtr4d 2859 | 1 ⊢ (𝜑 → ((𝑌 × 𝑋) · 𝐺) = (𝑋 · (𝑌 · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {csn 4561 × cxp 5548 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 Ringcrg 19291 LModclmod 19628 LFnlclfn 36187 LDualcld 36253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-sca 16575 df-vsca 16576 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ring 19293 df-lmod 19630 df-lfl 36188 df-ldual 36254 |
This theorem is referenced by: ldualvsass2 36272 |
Copyright terms: Public domain | W3C validator |