Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrsc | Structured version Visualization version GIF version |
Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.) |
Ref | Expression |
---|---|
lkrsc.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrsc.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrsc.k | ⊢ 𝐾 = (Base‘𝐷) |
lkrsc.t | ⊢ · = (.r‘𝐷) |
lkrsc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrsc.l | ⊢ 𝐿 = (LKer‘𝑊) |
lkrsc.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrsc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lkrsc.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
lkrsc.o | ⊢ 0 = (0g‘𝐷) |
lkrsc.e | ⊢ (𝜑 → 𝑅 ≠ 0 ) |
Ref | Expression |
---|---|
lkrsc | ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrsc.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
2 | 1 | fvexi 6689 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
3 | 2 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V) |
4 | lkrsc.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
5 | lkrsc.w | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
6 | lkrsc.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
7 | lkrsc.d | . . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊) | |
8 | lkrsc.k | . . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐷) | |
9 | lkrsc.f | . . . . . . . . . 10 ⊢ 𝐹 = (LFnl‘𝑊) | |
10 | 7, 8, 1, 9 | lflf 36697 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
11 | 5, 6, 10 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
12 | 11 | ffnd 6506 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝑉) |
13 | eqidd 2739 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) = (𝐺‘𝑣)) | |
14 | 3, 4, 12, 13 | ofc2 7452 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = ((𝐺‘𝑣) · 𝑅)) |
15 | 14 | eqeq1d 2740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ ((𝐺‘𝑣) · 𝑅) = 0 )) |
16 | lkrsc.o | . . . . . 6 ⊢ 0 = (0g‘𝐷) | |
17 | lkrsc.t | . . . . . 6 ⊢ · = (.r‘𝐷) | |
18 | 7 | lvecdrng 19997 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ DivRing) |
19 | 5, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ DivRing) |
20 | 19 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐷 ∈ DivRing) |
21 | 5 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
22 | 6 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐺 ∈ 𝐹) |
23 | simpr 488 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
24 | 7, 8, 1, 9 | lflcl 36698 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
25 | 21, 22, 23, 24 | syl3anc 1372 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
26 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ∈ 𝐾) |
27 | lkrsc.e | . . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 0 ) | |
28 | 27 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ≠ 0 ) |
29 | 8, 16, 17, 20, 25, 26, 28 | drngmuleq0 19645 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺‘𝑣) · 𝑅) = 0 ↔ (𝐺‘𝑣) = 0 )) |
30 | 15, 29 | bitrd 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ (𝐺‘𝑣) = 0 )) |
31 | 30 | pm5.32da 582 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
32 | lveclmod 19998 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
33 | 5, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
34 | 1, 7, 8, 17, 9, 33, 6, 4 | lflvscl 36711 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) |
35 | lkrsc.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
36 | 1, 7, 16, 9, 35 | ellkr 36723 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
37 | 5, 34, 36 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
38 | 1, 7, 16, 9, 35 | ellkr 36723 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
39 | 5, 6, 38 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
40 | 31, 37, 39 | 3bitr4d 314 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ 𝑣 ∈ (𝐿‘𝐺))) |
41 | 40 | eqrdv 2736 | 1 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 Vcvv 3398 {csn 4517 × cxp 5524 ⟶wf 6336 ‘cfv 6340 (class class class)co 7171 ∘f cof 7424 Basecbs 16587 .rcmulr 16670 Scalarcsca 16672 0gc0g 16817 DivRingcdr 19622 LModclmod 19754 LVecclvec 19994 LFnlclfn 36691 LKerclk 36719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-of 7426 df-om 7601 df-tpos 7922 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-map 8440 df-en 8557 df-dom 8558 df-sdom 8559 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-2 11780 df-3 11781 df-ndx 16590 df-slot 16591 df-base 16593 df-sets 16594 df-ress 16595 df-plusg 16682 df-mulr 16683 df-0g 16819 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-grp 18223 df-minusg 18224 df-mgp 19360 df-ur 19372 df-ring 19419 df-oppr 19496 df-dvdsr 19514 df-unit 19515 df-invr 19545 df-drng 19624 df-lmod 19756 df-lvec 19995 df-lfl 36692 df-lkr 36720 |
This theorem is referenced by: lkrscss 36732 ldualkrsc 36801 |
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