Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrshp | Structured version Visualization version GIF version |
Description: The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.) |
Ref | Expression |
---|---|
dochkrshp.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochkrshp.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
dochkrshp.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dochkrshp.v | ⊢ 𝑉 = (Base‘𝑈) |
dochkrshp.y | ⊢ 𝑌 = (LSHyp‘𝑈) |
dochkrshp.f | ⊢ 𝐹 = (LFnl‘𝑈) |
dochkrshp.l | ⊢ 𝐿 = (LKer‘𝑈) |
dochkrshp.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochkrshp.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
dochkrshp | ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) | |
2 | dochkrshp.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dochkrshp.o | . . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | dochkrshp.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | dochkrshp.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈) | |
6 | dochkrshp.y | . . . . . . . 8 ⊢ 𝑌 = (LSHyp‘𝑈) | |
7 | dochkrshp.k | . . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
8 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
9 | 2fveq3 6670 | . . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉))) | |
10 | 2, 4, 3, 5, 7 | dochoc1 38491 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) |
11 | 9, 10 | sylan9eqr 2878 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉) |
12 | simpr 487 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉) | |
13 | 11, 12 | eqtr4d 2859 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) |
14 | 13 | ex 415 | . . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
15 | 14 | necon3d 3037 | . . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) → (𝐿‘𝐺) ≠ 𝑉)) |
16 | df-ne 3017 | . . . . . . . . . . 11 ⊢ ((𝐿‘𝐺) ≠ 𝑉 ↔ ¬ (𝐿‘𝐺) = 𝑉) | |
17 | dochkrshp.f | . . . . . . . . . . . . . 14 ⊢ 𝐹 = (LFnl‘𝑈) | |
18 | dochkrshp.l | . . . . . . . . . . . . . 14 ⊢ 𝐿 = (LKer‘𝑈) | |
19 | 2, 4, 7 | dvhlvec 38239 | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LVec) |
20 | dochkrshp.g | . . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
21 | 5, 6, 17, 18, 19, 20 | lkrshpor 36237 | . . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉)) |
22 | 21 | orcomd 867 | . . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ∨ (𝐿‘𝐺) ∈ 𝑌)) |
23 | 22 | ord 860 | . . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) = 𝑉 → (𝐿‘𝐺) ∈ 𝑌)) |
24 | 16, 23 | syl5bi 244 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 → (𝐿‘𝐺) ∈ 𝑌)) |
25 | 15, 24 | syld 47 | . . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) → (𝐿‘𝐺) ∈ 𝑌)) |
26 | 25 | imp 409 | . . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (𝐿‘𝐺) ∈ 𝑌) |
27 | 2, 3, 4, 5, 6, 8, 26 | dochshpncl 38514 | . . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉)) |
28 | 1, 27 | mpbid 234 | . . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉) |
29 | 28 | ex 415 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉)) |
30 | 29 | necon1d 3038 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
31 | 11 | ex 415 | . . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉)) |
32 | 31 | necon3ad 3029 | . . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → ¬ (𝐿‘𝐺) = 𝑉)) |
33 | 32, 23 | syld 47 | . . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → (𝐿‘𝐺) ∈ 𝑌)) |
34 | 30, 33 | jcad 515 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
35 | 2, 3, 4, 17, 6, 18, 7, 20 | dochlkr 38515 | . . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) |
36 | 34, 35 | sylibrd 261 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) |
37 | 2, 4, 7 | dvhlmod 38240 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
38 | 37 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod) |
39 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) | |
40 | 5, 6, 38, 39 | lshpne 36112 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) |
41 | 40 | ex 415 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)) |
42 | 36, 41 | impbid 214 | 1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6350 Basecbs 16477 LModclmod 19628 LSHypclsh 36105 LFnlclfn 36187 LKerclk 36215 HLchlt 36480 LHypclh 37114 DVecHcdvh 38208 ocHcoch 38477 |
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 ax-riotaBAD 36083 |
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 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-int 4870 df-iun 4914 df-iin 4915 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-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 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-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-lsatoms 36106 df-lshyp 36107 df-lfl 36188 df-lkr 36216 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tendo 37885 df-edring 37887 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dic 38303 df-dih 38359 df-doch 38478 |
This theorem is referenced by: dochkrshp2 38517 dochkrsat 38585 |
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