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 6677 | . . . . . . . 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 36079 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
11 | 5, 6, 10 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
12 | 11 | ffnd 6508 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝑉) |
13 | eqidd 2819 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) = (𝐺‘𝑣)) | |
14 | 3, 4, 12, 13 | ofc2 7422 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = ((𝐺‘𝑣) · 𝑅)) |
15 | 14 | eqeq1d 2820 | . . . . 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 19806 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ DivRing) |
19 | 5, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ DivRing) |
20 | 19 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐷 ∈ DivRing) |
21 | 5 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
22 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐺 ∈ 𝐹) |
23 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
24 | 7, 8, 1, 9 | lflcl 36080 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
25 | 21, 22, 23, 24 | syl3anc 1363 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
26 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ∈ 𝐾) |
27 | lkrsc.e | . . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 0 ) | |
28 | 27 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ≠ 0 ) |
29 | 8, 16, 17, 20, 25, 26, 28 | drngmuleq0 19454 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺‘𝑣) · 𝑅) = 0 ↔ (𝐺‘𝑣) = 0 )) |
30 | 15, 29 | bitrd 280 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ (𝐺‘𝑣) = 0 )) |
31 | 30 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
32 | lveclmod 19807 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
33 | 5, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
34 | 1, 7, 8, 17, 9, 33, 6, 4 | lflvscl 36093 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) |
35 | lkrsc.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
36 | 1, 7, 16, 9, 35 | ellkr 36105 | . . . 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 584 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
38 | 1, 7, 16, 9, 35 | ellkr 36105 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
39 | 5, 6, 38 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
40 | 31, 37, 39 | 3bitr4d 312 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ 𝑣 ∈ (𝐿‘𝐺))) |
41 | 40 | eqrdv 2816 | 1 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 0gc0g 16701 DivRingcdr 19431 LModclmod 19563 LVecclvec 19803 LFnlclfn 36073 LKerclk 36101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lvec 19804 df-lfl 36074 df-lkr 36102 |
This theorem is referenced by: lkrscss 36114 ldualkrsc 36183 |
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