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 6679 | . . . . . . . 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 36193 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
11 | 5, 6, 10 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
12 | 11 | ffnd 6510 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝑉) |
13 | eqidd 2822 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) = (𝐺‘𝑣)) | |
14 | 3, 4, 12, 13 | ofc2 7427 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = ((𝐺‘𝑣) · 𝑅)) |
15 | 14 | eqeq1d 2823 | . . . . 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 19871 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ DivRing) |
19 | 5, 18 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ DivRing) |
20 | 19 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐷 ∈ DivRing) |
21 | 5 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
22 | 6 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐺 ∈ 𝐹) |
23 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
24 | 7, 8, 1, 9 | lflcl 36194 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
25 | 21, 22, 23, 24 | syl3anc 1367 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
26 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ∈ 𝐾) |
27 | lkrsc.e | . . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 0 ) | |
28 | 27 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ≠ 0 ) |
29 | 8, 16, 17, 20, 25, 26, 28 | drngmuleq0 19519 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺‘𝑣) · 𝑅) = 0 ↔ (𝐺‘𝑣) = 0 )) |
30 | 15, 29 | bitrd 281 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ (𝐺‘𝑣) = 0 )) |
31 | 30 | pm5.32da 581 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
32 | lveclmod 19872 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
33 | 5, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
34 | 1, 7, 8, 17, 9, 33, 6, 4 | lflvscl 36207 | . . . 4 ⊢ (𝜑 → (𝐺 ∘f · (𝑉 × {𝑅})) ∈ 𝐹) |
35 | lkrsc.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
36 | 1, 7, 16, 9, 35 | ellkr 36219 | . . . 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 586 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘f · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
38 | 1, 7, 16, 9, 35 | ellkr 36219 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
39 | 5, 6, 38 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
40 | 31, 37, 39 | 3bitr4d 313 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) ↔ 𝑣 ∈ (𝐿‘𝐺))) |
41 | 40 | eqrdv 2819 | 1 ⊢ (𝜑 → (𝐿‘(𝐺 ∘f · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 {csn 4561 × cxp 5548 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∘f cof 7401 Basecbs 16477 .rcmulr 16560 Scalarcsca 16562 0gc0g 16707 DivRingcdr 19496 LModclmod 19628 LVecclvec 19868 LFnlclfn 36187 LKerclk 36215 |
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-rmo 3146 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-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-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 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-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-drng 19498 df-lmod 19630 df-lvec 19869 df-lfl 36188 df-lkr 36216 |
This theorem is referenced by: lkrscss 36228 ldualkrsc 36297 |
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