lclkrlem2g - Mathbox for Norm Megill

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Theorem lclkrlem2g 42550

Description: Lemma for lclkr 42570. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
lclkrlem2f.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem2f.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem2f.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem2f.v	⊢ 𝑉 = (Base‘𝑈)
lclkrlem2f.s	⊢ 𝑆 = (Scalar‘𝑈)
lclkrlem2f.q	⊢ 𝑄 = (0_g‘𝑆)
lclkrlem2f.z	⊢ 0 = (0_g‘𝑈)
lclkrlem2f.a	⊢ ⊕ = (LSSum‘𝑈)
lclkrlem2f.n	⊢ 𝑁 = (LSpan‘𝑈)
lclkrlem2f.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem2f.j	⊢ 𝐽 = (LSHyp‘𝑈)
lclkrlem2f.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem2f.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem2f.p	⊢ + = (+_g‘𝐷)
lclkrlem2f.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem2f.b	⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
lclkrlem2f.e	⊢ (𝜑 → 𝐸 ∈ 𝐹)
lclkrlem2f.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lclkrlem2f.le	⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
lclkrlem2f.lg	⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
lclkrlem2f.kb	⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
lclkrlem2f.nx	⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
lclkrlem2f.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lclkrlem2f.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lclkrlem2f.ne	⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
lclkrlem2f.lp	⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)

**Assertion**
Ref	Expression
lclkrlem2g	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

**Proof of Theorem lclkrlem2g**
Step	Hyp	Ref	Expression
1		lclkrlem2f.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lclkrlem2f.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lclkrlem2f.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lclkrlem2f.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lclkrlem2f.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑈)
6		lclkrlem2f.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑆)
7		lclkrlem2f.z	. . . . 5 ⊢ 0 = (0_g‘𝑈)
8		lclkrlem2f.a	. . . . 5 ⊢ ⊕ = (LSSum‘𝑈)
9		lclkrlem2f.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
10		lclkrlem2f.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑈)
11		lclkrlem2f.j	. . . . 5 ⊢ 𝐽 = (LSHyp‘𝑈)
12		lclkrlem2f.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
13		lclkrlem2f.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
14		lclkrlem2f.p	. . . . 5 ⊢ + = (+_g‘𝐷)
15		lclkrlem2f.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		lclkrlem2f.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 }))
17		lclkrlem2f.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
18		lclkrlem2f.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
19		lclkrlem2f.le	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋}))
20		lclkrlem2f.lg	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌}))
21		lclkrlem2f.kb	. . . . 5 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄)
22		lclkrlem2f.nx	. . . . 5 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵})))
23		lclkrlem2f.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
24		lclkrlem2f.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
25		lclkrlem2f.ne	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺))
26		lclkrlem2f.lp	. . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽)
27	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26	lclkrlem2f 42549	. . . 4 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)))
28	1, 3, 15	dvhlvec 42146	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
29	19, 20	ineq12d 4167	. . . . . . 7 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
30	29	oveq1d 7433	. . . . . 6 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})))
31		eqid 2761	. . . . . . 7 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
32	25, 19, 20	3netr3d 3032	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌}))
33	1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 11	lclkrlem2c 42546	. . . . . 6 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)
34	30, 33	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ∈ 𝐽)
35	11, 28, 34, 26	lshpcmp 40025	. . . 4 ⊢ (𝜑 → ((((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)) ↔ (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = (𝐿‘(𝐸 + 𝐺))))
36	27, 35	mpbid 235	. . 3 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = (𝐿‘(𝐸 + 𝐺)))
37		eqid 2761	. . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
38	1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 37	lclkrlem2d 42547	. . . 4 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
39	30, 38	eqeltrd 2861	. . 3 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ∈ ran ((DIsoH‘𝐾)‘𝑊))
40	36, 39	eqeltrrd 2862	. 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
41	1, 3, 37, 4	dihrnss 42315	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
42	15, 40, 41	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉)
43	1, 37, 3, 4, 2, 15, 42	dochoccl 42406	. 2 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))))
44	40, 43	mpbid 235	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 {csn 4584 ran crn 5652 ‘cfv 6537 (class class class)co 7418 Basecbs 17380 +_gcplusg 17421 Scalarcsca 17424 0_gc0g 17603 LSSumclsm 19841 LSpanclspn 21239 LSAtomsclsa 40011 LSHypclsh 40012 LFnlclfn 40094 LKerclk 40122 LDualcld 40160 HLchlt 40387 LHypclh 41021 DVecHcdvh 42115 DIsoHcdih 42265 ocHcoch 42384

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 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-riotaBAD 39990

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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-tpos 8236 df-undef 8283 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-0g 17605 df-mre 17749 df-mrc 17750 df-acs 17752 df-proset 18461 df-poset 18480 df-plt 18495 df-lub 18511 df-glb 18512 df-join 18513 df-meet 18514 df-p0 18590 df-p1 18591 df-lat 18599 df-clat 18666 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cntz 19524 df-oppg 19553 df-lsm 19843 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-drng 20975 df-lmod 21130 df-lss 21200 df-lsp 21240 df-lvec 21371 df-lsatoms 40013 df-lshyp 40014 df-lcv 40056 df-lfl 40095 df-lkr 40123 df-ldual 40161 df-oposet 40213 df-ol 40215 df-oml 40216 df-covers 40303 df-ats 40304 df-atl 40335 df-cvlat 40359 df-hlat 40388 df-llines 40535 df-lplanes 40536 df-lvols 40537 df-lines 40538 df-psubsp 40540 df-pmap 40541 df-padd 40833 df-lhyp 41025 df-laut 41026 df-ldil 41141 df-ltrn 41142 df-trl 41196 df-tgrp 41780 df-tendo 41792 df-edring 41794 df-dveca 42040 df-disoa 42066 df-dvech 42116 df-dib 42176 df-dic 42210 df-dih 42266 df-doch 42385 df-djh 42432

This theorem is used by: lclkrlem2h 42551

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