dochlkr - Mathbox for Norm Megill

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Theorem dochlkr 42009

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 2762	. . . . . . . . 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 41734	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		dochlkr.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
9	2, 3, 4, 7, 8	lkrssv 39720	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
10		dochlkr.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	5, 6, 2, 10	dochocss 41990	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
12	1, 9, 11	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
13	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
14		dochlkr.y	. . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈)
15	5, 6, 1	dvhlvec 41733	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
16	15	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LVec)
17	7	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod)
18		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
19	2, 14, 17, 18	lshpne 39606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))
20	19	ex 416	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈)))
21	2, 14, 3, 4, 15, 8	lkrshpor 39731	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = (Base‘𝑈)))
22	21	ord 875	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → (𝐿‘𝐺) = (Base‘𝑈)))
23		2fveq3 6872	. . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
24	23	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
25	1	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26	5, 6, 10, 2, 25	dochoc1 41985	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
27	24, 26	eqtrd 2797	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈))
28	27	ex 416	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
29	22, 28	syld 47	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
30	29	necon1ad 2974	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈) → (𝐿‘𝐺) ∈ 𝑌))
31	20, 30	syld 47	. . . . . . . 8 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (𝐿‘𝐺) ∈ 𝑌))
32	31	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ∈ 𝑌)
33	14, 16, 32, 18	lshpcmp 39612	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ((𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ↔ (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))))
34	13, 33	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
35	34	eqcomd 2768	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
36	35, 32	jca 519	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))
37	36	ex 416	. 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
38		eleq1 2850	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (𝐿‘𝐺) ∈ 𝑌))
39	38	biimpar 481	. 2 ⊢ ((( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
40	37, 39	impbid1 227	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 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ⊆ wss 3904 ‘cfv 6521 Basecbs 17245 LModclmod 20927 LVecclvec 21169 LSHypclsh 39599 LFnlclfn 39681 LKerclk 39709 HLchlt 39974 LHypclh 40608 DVecHcdvh 41702 ocHcoch 41971

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-riotaBAD 39577

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-undef 8253 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-sca 17302 df-vsca 17303 df-0g 17470 df-proset 18326 df-poset 18345 df-plt 18360 df-lub 18376 df-glb 18377 df-join 18378 df-meet 18379 df-p0 18455 df-p1 18456 df-lat 18464 df-clat 18531 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19357 df-lsm 19676 df-cmn 19822 df-abl 19823 df-mgp 20187 df-rng 20199 df-ur 20232 df-ring 20285 df-oppr 20386 df-dvdsr 20406 df-unit 20407 df-invr 20437 df-dvr 20450 df-drng 20781 df-lmod 20929 df-lss 20999 df-lsp 21039 df-lvec 21170 df-lsatoms 39600 df-lshyp 39601 df-lfl 39682 df-lkr 39710 df-oposet 39800 df-ol 39802 df-oml 39803 df-covers 39890 df-ats 39891 df-atl 39922 df-cvlat 39946 df-hlat 39975 df-llines 40122 df-lplanes 40123 df-lvols 40124 df-lines 40125 df-psubsp 40127 df-pmap 40128 df-padd 40420 df-lhyp 40612 df-laut 40613 df-ldil 40728 df-ltrn 40729 df-trl 40783 df-tendo 41379 df-edring 41381 df-disoa 41653 df-dvech 41703 df-dib 41763 df-dic 41797 df-dih 41853 df-doch 41972

This theorem is referenced by: dochkrshp 42010 dochkrshp2 42011 mapdordlem1a 42258 mapdordlem2 42261

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