lcfrlem20 - Mathbox for Norm Megill

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Theorem lcfrlem20 41600

Description: Lemma for lcfr 41623. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+_g‘𝑈)
lcfrlem17.z	⊢ 0 = (0_g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem20.e	⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))

**Assertion**
Ref	Expression
lcfrlem20	⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

**Proof of Theorem lcfrlem20**
Step	Hyp	Ref	Expression
1		lcfrlem17.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
2		lcfrlem17.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
3		eqid 2731	. . . 4 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
4		lcfrlem17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		lcfrlem17.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcfrlem17.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	dvhlmod 41148	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		lcfrlem17.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
9	8	eldifad 3914	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10		lcfrlem17.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3914	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12	1, 2, 3, 7, 9, 11	lsmpr 21021	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})))
13	12	ineq1d 4169	. 2 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) = (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})))
14		eqid 2731	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
15		eqid 2731	. . 3 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
16		lcfrlem17.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
17	4, 5, 6	dvhlvec 41147	. . 3 ⊢ (𝜑 → 𝑈 ∈ LVec)
18		lcfrlem17.o	. . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
19		lcfrlem17.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
20		lcfrlem17.p	. . . . 5 ⊢ + = (+_g‘𝑈)
21		lcfrlem17.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
22	4, 18, 5, 1, 20, 19, 2, 16, 6, 8, 10, 21	lcfrlem17 41597	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
23	4, 18, 5, 1, 19, 15, 6, 22	dochsnshp 41491	. . 3 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSHyp‘𝑈))
24	1, 2, 19, 16, 7, 8	lsatlspsn 39031	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝐴)
25	1, 2, 19, 16, 7, 10	lsatlspsn 39031	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ 𝐴)
26		lcfrlem20.e	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}))
27	1, 20	lmodvacl 20806	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
28	7, 9, 11, 27	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
29	28	snssd 4761	. . . . . 6 ⊢ (𝜑 → {(𝑋 + 𝑌)} ⊆ 𝑉)
30	4, 5, 1, 14, 18	dochlss 41392	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {(𝑋 + 𝑌)} ⊆ 𝑉) → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
31	6, 29, 30	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) ∈ (LSubSp‘𝑈))
32	1, 14, 2, 7, 31, 9	ellspsn5b 20926	. . . 4 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{(𝑋 + 𝑌)}) ↔ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)})))
33	26, 32	mtbid 324	. . 3 ⊢ (𝜑 → ¬ (𝑁‘{𝑋}) ⊆ ( ⊥ ‘{(𝑋 + 𝑌)}))
34	14, 3, 15, 16, 17, 23, 24, 25, 21, 33	lshpat 39094	. 2 ⊢ (𝜑 → (((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)
35	13, 34	eqeltrd 2831	1 ⊢ (𝜑 → ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)})) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4576 {cpr 4578 ‘cfv 6481 (class class class)co 7346 Basecbs 17117 +_gcplusg 17158 0_gc0g 17340 LSSumclsm 19544 LModclmod 20791 LSubSpclss 20862 LSpanclspn 20902 LSAtomsclsa 39012 LSHypclsh 39013 HLchlt 39388 LHypclh 40022 DVecHcdvh 41116 ocHcoch 41385

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-riotaBAD 38991

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-n0 12379 df-z 12466 df-uz 12730 df-fz 13405 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-sca 17174 df-vsca 17175 df-0g 17342 df-mre 17485 df-mrc 17486 df-acs 17488 df-proset 18197 df-poset 18216 df-plt 18231 df-lub 18247 df-glb 18248 df-join 18249 df-meet 18250 df-p0 18326 df-p1 18327 df-lat 18335 df-clat 18402 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-grp 18846 df-minusg 18847 df-sbg 18848 df-subg 19033 df-cntz 19227 df-oppg 19256 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-drng 20644 df-lmod 20793 df-lss 20863 df-lsp 20903 df-lvec 21035 df-lsatoms 39014 df-lshyp 39015 df-lcv 39057 df-oposet 39214 df-ol 39216 df-oml 39217 df-covers 39304 df-ats 39305 df-atl 39336 df-cvlat 39360 df-hlat 39389 df-llines 39536 df-lplanes 39537 df-lvols 39538 df-lines 39539 df-psubsp 39541 df-pmap 39542 df-padd 39834 df-lhyp 40026 df-laut 40027 df-ldil 40142 df-ltrn 40143 df-trl 40197 df-tgrp 40781 df-tendo 40793 df-edring 40795 df-dveca 41041 df-disoa 41067 df-dvech 41117 df-dib 41177 df-dic 41211 df-dih 41267 df-doch 41386 df-djh 41433

This theorem is referenced by: lcfrlem21 41601

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