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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 38245 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod) |
10 | dochfln0.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
11 | 4, 7, 8, 9, 10 | lkrssv 36231 | . . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉) |
12 | 1, 3, 4, 2 | dochssv 38490 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
13 | 6, 11, 12 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉) |
14 | 13 | ssdifd 4116 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ⊆ (𝑉 ∖ { 0 })) |
15 | dochfln0.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | |
16 | 14, 15 | sseldd 3967 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
17 | 1, 2, 3, 4, 5, 6, 16 | dochnel 38528 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋})) |
18 | 15 | eldifad 3947 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺))) |
19 | 13, 18 | sseldd 3967 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
20 | 19 | biantrurd 535 | . . . 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 36224 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁))) |
24 | 9, 10, 23 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁))) |
25 | 1, 2, 3, 4, 5, 7, 8, 6, 10, 15 | dochsnkr 38607 | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) |
26 | 25 | eleq2d 2898 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ 𝑋 ∈ ( ⊥ ‘{𝑋}))) |
27 | 20, 24, 26 | 3bitr2rd 310 | . . 3 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) = 𝑁)) |
28 | 27 | necon3bbid 3053 | . 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) ≠ 𝑁)) |
29 | 17, 28 | mpbid 234 | 1 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∖ cdif 3932 ⊆ wss 3935 {csn 4566 ‘cfv 6354 Basecbs 16482 Scalarcsca 16567 0gc0g 16712 LModclmod 19633 LFnlclfn 36192 LKerclk 36220 HLchlt 36485 LHypclh 37119 DVecHcdvh 38213 ocHcoch 38482 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-riotaBAD 36088 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-tpos 7891 df-undef 7938 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-0g 16714 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-cntz 18446 df-lsm 18760 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-drng 19503 df-lmod 19635 df-lss 19703 df-lsp 19743 df-lvec 19874 df-lsatoms 36111 df-lshyp 36112 df-lfl 36193 df-lkr 36221 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-llines 36633 df-lplanes 36634 df-lvols 36635 df-lines 36636 df-psubsp 36638 df-pmap 36639 df-padd 36931 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 df-trl 37294 df-tgrp 37878 df-tendo 37890 df-edring 37892 df-dveca 38138 df-disoa 38164 df-dvech 38214 df-dib 38274 df-dic 38308 df-dih 38364 df-doch 38483 df-djh 38530 |
This theorem is referenced by: dochkr1 38613 dochkr1OLDN 38614 lcfl6lem 38633 |
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