dochsnkr2cl - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochsnkr2cl		Structured version Visualization version GIF version

Theorem dochsnkr2cl 41966

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 41602	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		dochsnkr2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
6	5	eldifad 3895	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		dochsnkr2.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2739	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9	7, 8	lspsnid 20983	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
10	4, 6, 9	syl2anc 590	. . 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 41965	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
20	6	snssd 4718	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
21	1, 2, 11, 7, 8, 3, 20	dochocsp 41871	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋}))
22	19, 21	eqtr4d 2777	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})))
23	22	fveq2d 6831	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))))
24		eqid 2739	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
25	1, 2, 7, 8, 24	dihlsprn 41823	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	3, 6, 25	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	1, 24, 11	dochoc 41859	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
28	3, 26, 27	syl2anc 590	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
29	23, 28	eqtr2d 2775	. . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) = ( ⊥ ‘(𝐿‘𝐺)))
30	10, 29	eleqtrd 2841	. 2 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)))
31		eldifsni 4723	. . 3 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
32	5, 31	syl 17	. 2 ⊢ (𝜑 → 𝑋 ≠ 0 )
33		eldifsn 4719	. 2 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑋 ≠ 0 ))
34	30, 32, 33	sylanbrc 589	1 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∃wrex 3063 ∖ cdif 3880 {csn 4555 ↦ cmpt 5153 ran crn 5619 ‘cfv 6485 ℩crio 7312 (class class class)co 7356 Basecbs 17170 +_gcplusg 17211 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 LModclmod 20850 LSpanclspn 20961 LKerclk 39577 HLchlt 39842 LHypclh 40476 DVecHcdvh 41570 DIsoHcdih 41720 ocHcoch 41839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39445

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 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 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093 df-lsatoms 39468 df-lshyp 39469 df-lfl 39550 df-lkr 39578 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991 df-lvols 39992 df-lines 39993 df-psubsp 39995 df-pmap 39996 df-padd 40288 df-lhyp 40480 df-laut 40481 df-ldil 40596 df-ltrn 40597 df-trl 40651 df-tgrp 41235 df-tendo 41247 df-edring 41249 df-dveca 41495 df-disoa 41521 df-dvech 41571 df-dib 41631 df-dic 41665 df-dih 41721 df-doch 41840 df-djh 41887

This theorem is referenced by: lcfl7lem 41991 lcfrlem9 42042

W3C validator