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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochfln0 | Structured version Visualization version GIF version |
Description: The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.) |
Ref | Expression |
---|---|
dochfln0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochfln0.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochfln0.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochfln0.v | ⊢ 𝑉 = (Base‘𝑈) |
dochfln0.r | ⊢ 𝑅 = (Scalar‘𝑈) |
dochfln0.n | ⊢ 𝑁 = (0g‘𝑅) |
dochfln0.z | ⊢ 0 = (0g‘𝑈) |
dochfln0.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochfln0.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochfln0.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochfln0.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
dochfln0.x | ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) |
Ref | Expression |
---|---|
dochfln0 | ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochfln0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dochfln0.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | dochfln0.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | dochfln0.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | dochfln0.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | dochfln0.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
7 | dochfln0.f | . . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈) | |
8 | dochfln0.l | . . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈) | |
9 | 1, 3, 6 | dvhlmod 40811 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | dochfln0.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 4, 7, 8, 9, 10 | lkrssv 38796 | . . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
12 | 1, 3, 4, 2 | dochssv 41056 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
13 | 6, 11, 12 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
14 | 13 | ssdifd 4140 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ⊆ (𝑉 ∖ { 0 })) |
15 | dochfln0.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
16 | 14, 15 | sseldd 3980 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
17 | 1, 2, 3, 4, 5, 6, 16 | dochnel 41094 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋})) |
18 | 15 | eldifad 3959 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 13, 18 | sseldd 3980 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
20 | 19 | biantrurd 531 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝑋) = 𝑁 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁))) |
21 | dochfln0.r | . . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈) | |
22 | dochfln0.n | . . . . . 6 ⊢ 𝑁 = (0g‘𝑅) | |
23 | 4, 21, 22, 7, 8 | ellkr 38789 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁))) |
24 | 9, 10, 23 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁))) |
25 | 1, 2, 3, 4, 5, 7, 8, 6, 10, 15 | dochsnkr 41173 | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
26 | 25 | eleq2d 2812 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ 𝑋 ∈ ( ⊥ ‘{𝑋}))) |
27 | 20, 24, 26 | 3bitr2rd 307 | . . 3 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) = 𝑁)) |
28 | 27 | necon3bbid 2968 | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) ≠ 𝑁)) |
29 | 17, 28 | mpbid 231 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∖ cdif 3944 ⊆ wss 3947 {csn 4633 ‘cfv 6556 Basecbs 17215 Scalarcsca 17271 0gc0g 17456 LModclmod 20838 LFnlclfn 38757 LKerclk 38785 HLchlt 39050 LHypclh 39685 DVecHcdvh 40779 ocHcoch 41048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-riotaBAD 38653 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-undef 8290 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-0g 17458 df-proset 18322 df-poset 18340 df-plt 18357 df-lub 18373 df-glb 18374 df-join 18375 df-meet 18376 df-p0 18452 df-p1 18453 df-lat 18459 df-clat 18526 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-grp 18933 df-minusg 18934 df-sbg 18935 df-subg 19119 df-cntz 19313 df-lsm 19636 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-dvr 20385 df-drng 20711 df-lmod 20840 df-lss 20911 df-lsp 20951 df-lvec 21083 df-lsatoms 38676 df-lshyp 38677 df-lfl 38758 df-lkr 38786 df-oposet 38876 df-ol 38878 df-oml 38879 df-covers 38966 df-ats 38967 df-atl 38998 df-cvlat 39022 df-hlat 39051 df-llines 39199 df-lplanes 39200 df-lvols 39201 df-lines 39202 df-psubsp 39204 df-pmap 39205 df-padd 39497 df-lhyp 39689 df-laut 39690 df-ldil 39805 df-ltrn 39806 df-trl 39860 df-tgrp 40444 df-tendo 40456 df-edring 40458 df-dveca 40704 df-disoa 40730 df-dvech 40780 df-dib 40840 df-dic 40874 df-dih 40930 df-doch 41049 df-djh 41096 |
This theorem is referenced by: dochkr1 41179 dochkr1OLDN 41180 lcfl6lem 41199 |
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