lcfrlem6 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem6		Structured version Visualization version GIF version

Theorem lcfrlem6 42183

Description: Lemma for lcfr 42221. 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 2875	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
4		eliun 4956	. . . . 5 ⊢ (𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
5	3, 4	sylib 221	. . . 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 41746	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
10	9	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod)
11	10	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod)
12	8	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		eqid 2765	. . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈)
14		eqid 2765	. . . . . . . . . 10 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
15		lcfrlem6.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
16		lcfrlem6.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
17		eqid 2765	. . . . . . . . . . . . 13 ⊢ (Base‘𝐷) = (Base‘𝐷)
18		lcfrlem6.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (LSubSp‘𝐷)
19	17, 18	lssel 21027	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
20	16, 19	sylan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
21		lcfrlem6.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (LDual‘𝑈)
22	14, 21, 17, 9	ldualvbase 39762	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐷) = (LFnl‘𝑈))
23	22	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = (LFnl‘𝑈))
24	20, 23	eleqtrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (LFnl‘𝑈))
25	13, 14, 15, 10, 24	lkrssv 39732	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ (Base‘𝑈))
26		eqid 2765	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
27		lcfrlem6.o	. . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
28	6, 7, 13, 26, 27	dochlss 41990	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
29	12, 25, 28	syl2anc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
30	29	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
31		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
32		lcfrlem6.en	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
33	32	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
34	33	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
35		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
36	34, 35	eqsstrrd 3974	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
37	36	ex 417	. . . . . . . . 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 42181	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
40	39	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑋 ∈ (Base‘𝑈))
41	13, 26, 38, 10, 29, 40	ellspsn5b 21085	. . . . . . . . 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 42181	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
44	43	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ (Base‘𝑈))
45	13, 26, 38, 10, 29, 44	ellspsn5b 21085	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔))))
46	37, 41, 45	3imtr4d 297	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔))))
47	46	imp 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))
48		lcfrlem6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
49	48, 26	lssvacl 21033	. . . . . . 7 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
50	11, 30, 31, 47, 49	syl22anc 851	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
51	50	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))))
52	51	reximdva 3178	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝐺 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))))
53	5, 52	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
54		eliun 4956	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	53, 54	sylibr 237	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
56	55, 2	eleqtrrdi 2876	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∃wrex 3089 ⊆ wss 3907 {csn 4585 ∪ ciun 4952 ‘cfv 6525 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 LModclmod 20950 LSubSpclss 21021 LSpanclspn 21061 LFnlclfn 39693 LKerclk 39721 LDualcld 39759 HLchlt 39986 LHypclh 40620 DVecHcdvh 41714 ocHcoch 41983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-riotaBAD 39589

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-0g 17484 df-proset 18340 df-poset 18359 df-plt 18374 df-lub 18390 df-glb 18391 df-join 18392 df-meet 18393 df-p0 18469 df-p1 18470 df-lat 18478 df-clat 18545 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-cntz 19378 df-lsm 19697 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-drng 20806 df-lmod 20952 df-lss 21022 df-lsp 21062 df-lvec 21193 df-lfl 39694 df-lkr 39722 df-ldual 39760 df-oposet 39812 df-ol 39814 df-oml 39815 df-covers 39902 df-ats 39903 df-atl 39934 df-cvlat 39958 df-hlat 39987 df-llines 40134 df-lplanes 40135 df-lvols 40136 df-lines 40137 df-psubsp 40139 df-pmap 40140 df-padd 40432 df-lhyp 40624 df-laut 40625 df-ldil 40740 df-ltrn 40741 df-trl 40795 df-tendo 41391 df-edring 41393 df-disoa 41665 df-dvech 41715 df-dib 41775 df-dic 41809 df-dih 41865 df-doch 41984

This theorem is referenced by: lcfrlem41 42219

W3C validator