dochkrshp - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochkrshp		Structured version Visualization version GIF version

Theorem dochkrshp 41098

Description: The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dochkrshp	⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌))

**Proof of Theorem dochkrshp**
Step	Hyp	Ref	Expression
1		simpr 483	. . . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9		2fveq3 6898	. . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
10	2, 4, 3, 5, 7	dochoc1 41073	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
11	9, 10	sylan9eqr 2788	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉)
12		simpr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
13	11, 12	eqtr4d 2769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
14	13	ex 411	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
15	14	necon3d 2951	. . . . . . . . . 10 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) → (𝐿‘𝐺) ≠ 𝑉))
16		df-ne 2931	. . . . . . . . . . 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 40821	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LVec)
20		dochkrshp.g	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
21	5, 6, 17, 18, 19, 20	lkrshpor 38818	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = 𝑉))
22	21	orcomd 869	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ∨ (𝐿‘𝐺) ∈ 𝑌))
23	22	ord 862	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) = 𝑉 → (𝐿‘𝐺) ∈ 𝑌))
24	16, 23	biimtrid 241	. . . . . . . . . 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 405	. . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (𝐿‘𝐺) ∈ 𝑌)
27	2, 3, 4, 5, 6, 8, 26	dochshpncl 41096	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉))
28	1, 27	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉)
29	28	ex 411	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (𝐿‘𝐺) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉))
30	29	necon1d 2952	. . . 4 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
31	11	ex 411	. . . . . 6 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = 𝑉))
32	31	necon3ad 2943	. . . . 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 511	. . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
35	2, 3, 4, 17, 6, 18, 7, 20	dochlkr 41097	. . 3 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
36	34, 35	sylibrd 258	. 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌))
37	2, 4, 7	dvhlmod 40822	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
38	37	adantr 479	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod)
39		simpr 483	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
40	5, 6, 38, 39	lshpne 38693	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
41	40	ex 411	. 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉))
42	36, 41	impbid 211	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6546 Basecbs 17208 LModclmod 20832 LSHypclsh 38686 LFnlclfn 38768 LKerclk 38796 HLchlt 39061 LHypclh 39696 DVecHcdvh 40790 ocHcoch 41059

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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-riotaBAD 38664

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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-map 8849 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-sca 17277 df-vsca 17278 df-0g 17451 df-proset 18315 df-poset 18333 df-plt 18350 df-lub 18366 df-glb 18367 df-join 18368 df-meet 18369 df-p0 18445 df-p1 18446 df-lat 18452 df-clat 18519 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-grp 18926 df-minusg 18927 df-sbg 18928 df-subg 19113 df-cntz 19307 df-lsm 19630 df-cmn 19776 df-abl 19777 df-mgp 20114 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20312 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20705 df-lmod 20834 df-lss 20905 df-lsp 20945 df-lvec 21077 df-lsatoms 38687 df-lshyp 38688 df-lfl 38769 df-lkr 38797 df-oposet 38887 df-ol 38889 df-oml 38890 df-covers 38977 df-ats 38978 df-atl 39009 df-cvlat 39033 df-hlat 39062 df-llines 39210 df-lplanes 39211 df-lvols 39212 df-lines 39213 df-psubsp 39215 df-pmap 39216 df-padd 39508 df-lhyp 39700 df-laut 39701 df-ldil 39816 df-ltrn 39817 df-trl 39871 df-tendo 40467 df-edring 40469 df-disoa 40741 df-dvech 40791 df-dib 40851 df-dic 40885 df-dih 40941 df-doch 41060

This theorem is referenced by: dochkrshp2 41099 dochkrsat 41167

W3C validator