dochsnkr2cl - Mathbox for Norm Megill

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Theorem dochsnkr2cl 40811

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 40447	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		dochsnkr2.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
6	5	eldifad 3960	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		dochsnkr2.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2731	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9	7, 8	lspsnid 20836	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
10	4, 6, 9	syl2anc 583	. . 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 40810	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
20	6	snssd 4812	. . . . . . 7 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
21	1, 2, 11, 7, 8, 3, 20	dochocsp 40716	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})) = ( ⊥ ‘{𝑋}))
22	19, 21	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑋})))
23	22	fveq2d 6895	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))))
24		eqid 2731	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
25	1, 2, 7, 8, 24	dihlsprn 40668	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26	3, 6, 25	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	1, 24, 11	dochoc 40704	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LSpan‘𝑈)‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
28	3, 26, 27	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑋}))) = ((LSpan‘𝑈)‘{𝑋}))
29	23, 28	eqtr2d 2772	. . 3 ⊢ (𝜑 → ((LSpan‘𝑈)‘{𝑋}) = ( ⊥ ‘(𝐿‘𝐺)))
30	10, 29	eleqtrd 2834	. 2 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)))
31		eldifsni 4793	. . 3 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
32	5, 31	syl 17	. 2 ⊢ (𝜑 → 𝑋 ≠ 0 )
33		eldifsn 4790	. 2 ⊢ (𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ↔ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)) ∧ 𝑋 ≠ 0 ))
34	30, 32, 33	sylanbrc 582	1 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 ∖ cdif 3945 {csn 4628 ↦ cmpt 5231 ran crn 5677 ‘cfv 6543 ℩crio 7367 (class class class)co 7412 Basecbs 17151 +_gcplusg 17204 Scalarcsca 17207 ·_𝑠 cvsca 17208 0_gc0g 17392 LModclmod 20702 LSpanclspn 20814 LKerclk 38421 HLchlt 38686 LHypclh 39321 DVecHcdvh 40415 DIsoHcdih 40565 ocHcoch 40684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-riotaBAD 38289

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-undef 8264 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-0g 17394 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-p1 18389 df-lat 18395 df-clat 18462 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19046 df-cntz 19229 df-lsm 19552 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-drng 20585 df-lmod 20704 df-lss 20775 df-lsp 20815 df-lvec 20947 df-lsatoms 38312 df-lshyp 38313 df-lfl 38394 df-lkr 38422 df-oposet 38512 df-ol 38514 df-oml 38515 df-covers 38602 df-ats 38603 df-atl 38634 df-cvlat 38658 df-hlat 38687 df-llines 38835 df-lplanes 38836 df-lvols 38837 df-lines 38838 df-psubsp 38840 df-pmap 38841 df-padd 39133 df-lhyp 39325 df-laut 39326 df-ldil 39441 df-ltrn 39442 df-trl 39496 df-tgrp 40080 df-tendo 40092 df-edring 40094 df-dveca 40340 df-disoa 40366 df-dvech 40416 df-dib 40476 df-dic 40510 df-dih 40566 df-doch 40685 df-djh 40732

This theorem is referenced by: lcfl7lem 40836 lcfrlem9 40887

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