lcfrlem6 - Mathbox for Norm Megill

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Theorem lcfrlem6 42428

Description: Lemma for lcfr 42466. 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 2872	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
4		eliun 4958	. . . . 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 41991	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
10	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑈 ∈ LMod)
11	10	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑈 ∈ LMod)
12	8	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘𝑈) = (Base‘𝑈)
14		eqid 2762	. . . . . . . . . 10 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
15		lcfrlem6.l	. . . . . . . . . 10 ⊢ 𝐿 = (LKer‘𝑈)
16		lcfrlem6.g	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
17		eqid 2762	. . . . . . . . . . . . 13 ⊢ (Base‘𝐷) = (Base‘𝐷)
18		lcfrlem6.q	. . . . . . . . . . . . 13 ⊢ 𝑄 = (LSubSp‘𝐷)
19	17, 18	lssel 21127	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑄 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
20	16, 19	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (Base‘𝐷))
21		lcfrlem6.d	. . . . . . . . . . . . 13 ⊢ 𝐷 = (LDual‘𝑈)
22	14, 21, 17, 9	ldualvbase 40007	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘𝐷) = (LFnl‘𝑈))
23	22	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (Base‘𝐷) = (LFnl‘𝑈))
24	20, 23	eleqtrd 2864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑔 ∈ (LFnl‘𝑈))
25	13, 14, 15, 10, 24	lkrssv 39977	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝐿‘𝑔) ⊆ (Base‘𝑈))
26		eqid 2762	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
27		lcfrlem6.o	. . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
28	6, 7, 13, 26, 27	dochlss 42235	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝑔) ⊆ (Base‘𝑈)) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
29	12, 25, 28	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
30	29	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈))
31		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)))
32		lcfrlem6.en	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
33	32	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
34	33	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
35		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
36	34, 35	eqsstrrd 3969	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘(𝐿‘𝑔))) → (𝑁‘{𝑌}) ⊆ ( ⊥ ‘(𝐿‘𝑔)))
37	36	ex 418	. . . . . . . . 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 42426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))
40	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑋 ∈ (Base‘𝑈))
41	13, 26, 38, 10, 29, 40	ellspsn5b 21185	. . . . . . . . 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 42426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑈))
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → 𝑌 ∈ (Base‘𝑈))
45	13, 26, 38, 10, 29, 44	ellspsn5b 21185	. . . . . . . . 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 412	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))
48		lcfrlem6.p	. . . . . . . 8 ⊢ + = (+_g‘𝑈)
49	48, 26	lssvacl 21133	. . . . . . 7 ⊢ (((𝑈 ∈ LMod ∧ ( ⊥ ‘(𝐿‘𝑔)) ∈ (LSubSp‘𝑈)) ∧ (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) ∧ 𝑌 ∈ ( ⊥ ‘(𝐿‘𝑔)))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
50	11, 30, 31, 47, 49	syl22anc 852	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝐺) ∧ 𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔))) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
51	50	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐺) → (𝑋 ∈ ( ⊥ ‘(𝐿‘𝑔)) → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔))))
52	51	reximdva 3177	. . . 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 4958	. . 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 2873	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃wrex 3088 ⊆ wss 3902 {csn 4587 ∪ ciun 4954 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 LModclmod 21050 LSubSpclss 21121 LSpanclspn 21161 LFnlclfn 39938 LKerclk 39966 LDualcld 40004 HLchlt 40231 LHypclh 40865 DVecHcdvh 41959 ocHcoch 42228

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-riotaBAD 39834

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-0g 17532 df-proset 18388 df-poset 18407 df-plt 18422 df-lub 18438 df-glb 18439 df-join 18440 df-meet 18441 df-p0 18517 df-p1 18518 df-lat 18526 df-clat 18593 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-cntz 19450 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lvec 21293 df-lfl 39939 df-lkr 39967 df-ldual 40005 df-oposet 40057 df-ol 40059 df-oml 40060 df-covers 40147 df-ats 40148 df-atl 40179 df-cvlat 40203 df-hlat 40232 df-llines 40379 df-lplanes 40380 df-lvols 40381 df-lines 40382 df-psubsp 40384 df-pmap 40385 df-padd 40677 df-lhyp 40869 df-laut 40870 df-ldil 40985 df-ltrn 40986 df-trl 41040 df-tendo 41636 df-edring 41638 df-disoa 41910 df-dvech 41960 df-dib 42020 df-dic 42054 df-dih 42110 df-doch 42229

This theorem is used by: lcfrlem41 42464

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