dochsnkr2cl - Mathbox for Norm Megill

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Theorem dochsnkr2cl 41473

Description: The 𝑋 determining functional 𝐺 belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.)

**Hypotheses**
Ref	Expression
dochsnkr2.h	⊢ 𝐻 = (LHyp‘𝐾)
dochsnkr2.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochsnkr2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochsnkr2.v	⊢ 𝑉 = (Base‘𝑈)
dochsnkr2.z	⊢ 0 = (0_g‘𝑈)
dochsnkr2.a	⊢ + = (+_g‘𝑈)
dochsnkr2.t	⊢ · = ( ·_𝑠 ‘𝑈)
dochsnkr2.l	⊢ 𝐿 = (LKer‘𝑈)
dochsnkr2.d	⊢ 𝐷 = (Scalar‘𝑈)
dochsnkr2.r	⊢ 𝑅 = (Base‘𝐷)
dochsnkr2.g	⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
dochsnkr2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochsnkr2.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
dochsnkr2cl	⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))

Distinct variable groups: 𝑣,𝑘,𝑤, + 𝐷,𝑘 ⊥ ,𝑘,𝑣,𝑤 𝑅,𝑘,𝑣 · ,𝑘,𝑣,𝑤 𝑣,𝑉 𝑘,𝑋,𝑣,𝑤 Allowed substitution hints: 𝜑(𝑤,𝑣,𝑘) 𝐷(𝑤,𝑣) 𝑅(𝑤) 𝑈(𝑤,𝑣,𝑘) 𝐺(𝑤,𝑣,𝑘) 𝐻(𝑤,𝑣,𝑘) 𝐾(𝑤,𝑣,𝑘) 𝐿(𝑤,𝑣,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑤,𝑣,𝑘) 0 (𝑤,𝑣,𝑘)

**Proof of Theorem dochsnkr2cl**
Step	Hyp	Ref	Expression
1		dochsnkr2.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		dochsnkr2.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		dochsnkr2.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 41109	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		dochsnkr2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
6	5	eldifad 3917	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		dochsnkr2.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2729	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9	7, 8	lspsnid 20915	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
10	4, 6, 9	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
11		dochsnkr2.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12		dochsnkr2.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
13		dochsnkr2.a	. . . . . . 7 ⊢ + = (+_g‘𝑈)
14		dochsnkr2.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
15		dochsnkr2.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
16		dochsnkr2.d	. . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑈)
17		dochsnkr2.r	. . . . . . 7 ⊢ 𝑅 = (Base‘𝐷)
18		dochsnkr2.g	. . . . . . 7 ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
19	1, 11, 2, 7, 12, 13, 14, 15, 16, 17, 18, 3, 5	dochsnkr2 41472	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
20	6	snssd 4763	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
21	1, 2, 11, 7, 8, 3, 20	dochocsp 41378	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋}))
22	19, 21	eqtr4d 2767	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})))
23	22	fveq2d 6830	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))))
24		eqid 2729	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
25	1, 2, 7, 8, 24	dihlsprn 41330	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	3, 6, 25	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	1, 24, 11	dochoc 41366	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
28	3, 26, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
29	23, 28	eqtr2d 2765	. . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) = ( ⊥ ‘(𝐿‘𝐺)))
30	10, 29	eleqtrd 2830	. 2 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)))
31		eldifsni 4744	. . 3 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
32	5, 31	syl 17	. 2 ⊢ (𝜑 → 𝑋 ≠ 0 )
33		eldifsn 4740	. 2 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑋 ≠ 0 ))
34	30, 32, 33	sylanbrc 583	1 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 ∖ cdif 3902 {csn 4579 ↦ cmpt 5176 ran crn 5624 ‘cfv 6486 ℩crio 7309 (class class class)co 7353 Basecbs 17139 +_gcplusg 17180 Scalarcsca 17183 ·_𝑠 cvsca 17184 0_gc0g 17362 LModclmod 20782 LSpanclspn 20893 LKerclk 39083 HLchlt 39348 LHypclh 39983 DVecHcdvh 41077 DIsoHcdih 41227 ocHcoch 41346

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-riotaBAD 38951

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-n0 12404 df-z 12491 df-uz 12755 df-fz 13430 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-sca 17196 df-vsca 17197 df-0g 17364 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-grp 18834 df-minusg 18835 df-sbg 18836 df-subg 19021 df-cntz 19215 df-lsm 19534 df-cmn 19680 df-abl 19681 df-mgp 20045 df-rng 20057 df-ur 20086 df-ring 20139 df-oppr 20241 df-dvdsr 20261 df-unit 20262 df-invr 20292 df-dvr 20305 df-drng 20635 df-lmod 20784 df-lss 20854 df-lsp 20894 df-lvec 21026 df-lsatoms 38974 df-lshyp 38975 df-lfl 39056 df-lkr 39084 df-oposet 39174 df-ol 39176 df-oml 39177 df-covers 39264 df-ats 39265 df-atl 39296 df-cvlat 39320 df-hlat 39349 df-llines 39497 df-lplanes 39498 df-lvols 39499 df-lines 39500 df-psubsp 39502 df-pmap 39503 df-padd 39795 df-lhyp 39987 df-laut 39988 df-ldil 40103 df-ltrn 40104 df-trl 40158 df-tgrp 40742 df-tendo 40754 df-edring 40756 df-dveca 41002 df-disoa 41028 df-dvech 41078 df-dib 41138 df-dic 41172 df-dih 41228 df-doch 41347 df-djh 41394

This theorem is referenced by: lcfl7lem 41498 lcfrlem9 41549

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