dochlkr - Mathbox for Norm Megill

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Theorem dochlkr 42442

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 2761	. . . . . . . . 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 42167	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		dochlkr.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
9	2, 3, 4, 7, 8	lkrssv 40153	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ (Base‘𝑈))
10		dochlkr.o	. . . . . . . . 9 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11	5, 6, 2, 10	dochocss 42423	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ (Base‘𝑈)) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
12	1, 9, 11	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
13	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
14		dochlkr.y	. . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈)
15	5, 6, 1	dvhlvec 42166	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
16	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LVec)
17	7	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → 𝑈 ∈ LMod)
18		simpr 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
19	2, 14, 17, 18	lshpne 40039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈))
20	19	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈)))
21	2, 14, 3, 4, 15, 8	lkrshpor 40164	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ∈ 𝑌 ∨ (𝐿‘𝐺) = (Base‘𝑈)))
22	21	ord 878	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → (𝐿‘𝐺) = (Base‘𝑈)))
23		2fveq3 6890	. . . . . . . . . . . . . 14 ⊢ ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
24	23	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
25	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
26	5, 6, 10, 2, 25	dochoc1 42418	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
27	24, 26	eqtrd 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈))
28	27	ex 418	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐿‘𝐺) = (Base‘𝑈) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
29	22, 28	syld 48	. . . . . . . . . 10 ⊢ (𝜑 → (¬ (𝐿‘𝐺) ∈ 𝑌 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (Base‘𝑈)))
30	29	necon1ad 2973	. . . . . . . . 9 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ (Base‘𝑈) → (𝐿‘𝐺) ∈ 𝑌))
31	20, 30	syld 48	. . . . . . . 8 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (𝐿‘𝐺) ∈ 𝑌))
32	31	imp 412	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) ∈ 𝑌)
33	14, 16, 32, 18	lshpcmp 40045	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ((𝐿‘𝐺) ⊆ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ↔ (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))))
34	13, 33	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (𝐿‘𝐺) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
35	34	eqcomd 2767	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
36	35, 32	jca 521	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))
37	36	ex 418	. 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
38		eleq1 2849	. . 3 ⊢ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (𝐿‘𝐺) ∈ 𝑌))
39	38	biimpar 483	. 2 ⊢ ((( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)
40	37, 39	impbid1 228	1 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ⊆ wss 3899 ‘cfv 6538 Basecbs 17387 LModclmod 21135 LVecclvec 21377 LSHypclsh 40032 LFnlclfn 40114 LKerclk 40142 HLchlt 40407 LHypclh 41041 DVecHcdvh 42135 ocHcoch 42404

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-riotaBAD 40010

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-tpos 8243 df-undef 8290 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-sca 17444 df-vsca 17445 df-0g 17612 df-proset 18468 df-poset 18487 df-plt 18502 df-lub 18518 df-glb 18519 df-join 18520 df-meet 18521 df-p0 18597 df-p1 18598 df-lat 18606 df-clat 18673 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-subg 19333 df-cntz 19531 df-lsm 19850 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-drng 20982 df-lmod 21137 df-lss 21207 df-lsp 21247 df-lvec 21378 df-lsatoms 40033 df-lshyp 40034 df-lfl 40115 df-lkr 40143 df-oposet 40233 df-ol 40235 df-oml 40236 df-covers 40323 df-ats 40324 df-atl 40355 df-cvlat 40379 df-hlat 40408 df-llines 40555 df-lplanes 40556 df-lvols 40557 df-lines 40558 df-psubsp 40560 df-pmap 40561 df-padd 40853 df-lhyp 41045 df-laut 41046 df-ldil 41161 df-ltrn 41162 df-trl 41216 df-tendo 41812 df-edring 41814 df-disoa 42086 df-dvech 42136 df-dib 42196 df-dic 42230 df-dih 42286 df-doch 42405

This theorem is used by: dochkrshp 42443 dochkrshp2 42444 mapdordlem1a 42691 mapdordlem2 42694

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