lcfrlem6 - Mathbox for Norm Megill

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Theorem lcfrlem6 41993

Description: Lemma for lcfr 42031. Closure of vector sum with colinear vectors. TODO: Move down 𝑁 definition so top hypotheses can be shared. (Contributed by NM, 10-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem6.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem6.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem6.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem6.p	⊢ + = (+_g‘𝑈)
lcfrlem6.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem6.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem6.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem6.q	⊢ 𝑄 = (LSubSp‘𝐷)
lcfrlem6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem6.g	⊢ (𝜑 → 𝐺 ∈ 𝑄)
lcfrlem6.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem6.x	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem6.y	⊢ (𝜑 → 𝑌 ∈ 𝐸)
lcfrlem6.en	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lcfrlem6	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Distinct variable groups: + ,𝑔 𝑈,𝑔 𝑔,𝑋 𝑔,𝑌 𝜑,𝑔 Allowed substitution hints: 𝐷(𝑔) 𝑄(𝑔) 𝐸(𝑔) 𝐺(𝑔) 𝐻(𝑔) 𝐾(𝑔) 𝐿(𝑔) 𝑁(𝑔) ⊥ (𝑔) 𝑊(𝑔)

**Proof of Theorem lcfrlem6**
Step	Hyp	Ref	Expression
1		lcfrlem6.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
2		lcfrlem6.e	. . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
3	1, 2	eleqtrdi 2846	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
4		eliun 4937	. . . . 5 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
5	3, 4	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
6		lcfrlem6.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
7		lcfrlem6.u	. . . . . . . . . 10 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		lcfrlem6.k	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	6, 7, 8	dvhlmod 41556	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod)
11	10	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod)
12	8	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈)
14		eqid 2736	. . . . . . . . . 10 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
15		lcfrlem6.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
16		lcfrlem6.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
17		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝐷) = (Base‘𝐷)
18		lcfrlem6.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (LSubSp‘𝐷)
19	17, 18	lssel 20932	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
20	16, 19	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
21		lcfrlem6.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (LDual‘𝑈)
22	14, 21, 17, 9	ldualvbase 39572	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐷) = (LFnl‘𝑈))
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = (LFnl‘𝑈))
24	20, 23	eleqtrd 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (LFnl‘𝑈))
25	13, 14, 15, 10, 24	lkrssv 39542	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ (Base‘𝑈))
26		eqid 2736	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
27		lcfrlem6.o	. . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
28	6, 7, 13, 26, 27	dochlss 41800	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
29	12, 25, 28	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
30	29	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
31		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
32		lcfrlem6.en	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
33	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
34	33	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
35		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
36	34, 35	eqsstrrd 3957	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
37	36	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ((𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔))))
38		lcfrlem6.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
39	6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 1	lcfrlem4 41991	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑋 ∈ (Base‘𝑈))
41	13, 26, 38, 10, 29, 40	ellspsn5b 20990	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))))
42		lcfrlem6.y	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43	6, 27, 7, 13, 15, 21, 18, 2, 8, 16, 42	lcfrlem4 41991	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
44	43	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ (Base‘𝑈))
45	13, 26, 38, 10, 29, 44	ellspsn5b 20990	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔))))
46	37, 41, 45	3imtr4d 294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔))))
47	46	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))
48		lcfrlem6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
49	48, 26	lssvacl 20938	. . . . . . 7 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
50	11, 30, 31, 47, 49	syl22anc 839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
51	50	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))))
52	51	reximdva 3150	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))))
53	5, 52	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
54		eliun 4937	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	53, 54	sylibr 234	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
56	55, 2	eleqtrrdi 2847	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 ⊆ wss 3889 {csn 4567 ∪ ciun 4933 ‘cfv 6498 (class class class)co 7367 Basecbs 17179 +_gcplusg 17220 LModclmod 20855 LSubSpclss 20926 LSpanclspn 20966 LFnlclfn 39503 LKerclk 39531 LDualcld 39569 HLchlt 39796 LHypclh 40430 DVecHcdvh 41524 ocHcoch 41793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lfl 39504 df-lkr 39532 df-ldual 39570 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tendo 41201 df-edring 41203 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794

This theorem is referenced by: lcfrlem41 42029

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