dochlkr - Mathbox for Norm Megill

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Theorem dochlkr 37188

Description: Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)

**Hypotheses**
Ref	Expression
dochlkr.h	⊢ 𝐻 = (LHyp‘𝐾)
dochlkr.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochlkr.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochlkr.f	⊢ 𝐹 = (LFnl‘𝑈)
dochlkr.y	⊢ 𝑌 = (LSHyp‘𝑈)
dochlkr.l	⊢ 𝐿 = (LKer‘𝑈)
dochlkr.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochlkr.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)

**Assertion**
Ref	Expression
dochlkr	⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))

**Proof of Theorem dochlkr**
Step	Hyp	Ref	Expression
1		dochlkr.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		eqid 2770	. . . . . . . . 9 ⊢ (Base‘𝑈) = (Base‘𝑈)
3		dochlkr.f	. . . . . . . . 9 ⊢ 𝐹 = (LFnl‘𝑈)
4		dochlkr.l	. . . . . . . . 9 ⊢ 𝐿 = (LKer‘𝑈)
5		dochlkr.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
6		dochlkr.u	. . . . . . . . . 10 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7	5, 6, 1	dvhlmod 36913	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		dochlkr.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
9	2, 3, 4, 7, 8	lkrssv 34898	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
10		dochlkr.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	5, 6, 2, 10	dochocss 37169	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
12	1, 9, 11	syl2anc 565	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
13	12	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
14		dochlkr.y	. . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈)
15	5, 6, 1	dvhlvec 36912	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
16	15	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LVec)
17	7	adantr 466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod)
18		simpr 471	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
19	2, 14, 17, 18	lshpne 34784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))
20	19	ex 397	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈)))
21	2, 14, 3, 4, 15, 8	lkrshpor 34909	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = (Base‘𝑈)))
22	21	ord 844	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → (𝐿‘𝐺) = (Base‘𝑈)))
23		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘(Base‘𝑈)))
24	23	fveq2d 6336	. . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
25	24	adantl 467	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
26	1	adantr 466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27	5, 6, 10, 2, 26	dochoc1 37164	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
28	25, 27	eqtrd 2804	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈))
29	28	ex 397	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
30	22, 29	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
31	30	necon1ad 2959	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈) → (𝐿‘𝐺) ∈ 𝑌))
32	20, 31	syld 47	. . . . . . . 8 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (𝐿‘𝐺) ∈ 𝑌))
33	32	imp 393	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ∈ 𝑌)
34	14, 16, 33, 18	lshpcmp 34790	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ((𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ↔ (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))))
35	13, 34	mpbid 222	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
36	35	eqcomd 2776	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
37	36, 33	jca 495	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))
38	37	ex 397	. 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
39		eleq1 2837	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (𝐿‘𝐺) ∈ 𝑌))
40	39	biimpar 463	. 2 ⊢ ((( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
41	38, 40	impbid1 215	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 ⊆ wss 3721 ‘cfv 6031 Basecbs 16063 LModclmod 19072 LVecclvec 19314 LSHypclsh 34777 LFnlclfn 34859 LKerclk 34887 HLchlt 35152 LHypclh 35785 DVecHcdvh 36881 ocHcoch 37150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34754

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-fal 1636 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-lsatoms 34778 df-lshyp 34779 df-lfl 34860 df-lkr 34888 df-oposet 34978 df-ol 34980 df-oml 34981 df-covers 35068 df-ats 35069 df-atl 35100 df-cvlat 35124 df-hlat 35153 df-llines 35299 df-lplanes 35300 df-lvols 35301 df-lines 35302 df-psubsp 35304 df-pmap 35305 df-padd 35597 df-lhyp 35789 df-laut 35790 df-ldil 35905 df-ltrn 35906 df-trl 35961 df-tendo 36557 df-edring 36559 df-disoa 36832 df-dvech 36882 df-dib 36942 df-dic 36976 df-dih 37032 df-doch 37151

This theorem is referenced by: dochkrshp 37189 dochkrshp2 37190 mapdordlem1a 37437 mapdordlem2 37440

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