dochexmidlem5 - Mathbox for Norm Megill

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Theorem dochexmidlem5 39405

Description: Lemma for dochexmid 39409. (Contributed by NM, 15-Jan-2015.)

**Hypotheses**
Ref	Expression
dochexmidlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
dochexmidlem1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochexmidlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochexmidlem1.v	⊢ 𝑉 = (Base‘𝑈)
dochexmidlem1.s	⊢ 𝑆 = (LSubSp‘𝑈)
dochexmidlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
dochexmidlem1.p	⊢ ⊕ = (LSSum‘𝑈)
dochexmidlem1.a	⊢ 𝐴 = (LSAtoms‘𝑈)
dochexmidlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochexmidlem1.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
dochexmidlem5.pp	⊢ (𝜑 → 𝑝 ∈ 𝐴)
dochexmidlem5.z	⊢ 0 = (0_g‘𝑈)
dochexmidlem5.m	⊢ 𝑀 = (𝑋 ⊕ 𝑝)
dochexmidlem5.xn	⊢ (𝜑 → 𝑋 ≠ { 0 })
dochexmidlem5.pl	⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))

**Assertion**
Ref	Expression
dochexmidlem5	⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })

Proof of Theorem dochexmidlem5 Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochexmidlem5.pl	. 2 ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
2		dochexmidlem1.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3		dochexmidlem5.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
4		dochexmidlem1.a	. . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈)
5		dochexmidlem1.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
6		dochexmidlem1.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		dochexmidlem1.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlmod 39051	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → 𝑈 ∈ LMod)
10		dochexmidlem1.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
11		dochexmidlem1.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑈)
12	11, 2	lssss 20113	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
13	10, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
14		dochexmidlem1.o	. . . . . . . . . 10 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15	5, 6, 11, 2, 14	dochlss 39295	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
16	7, 13, 15	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
17		dochexmidlem5.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
18		dochexmidlem5.pp	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑝 ∈ 𝐴)
19	2, 4, 8, 18	lsatlssel 36938	. . . . . . . . . 10 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
20		dochexmidlem1.p	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
21	2, 20	lsmcl 20260	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
22	8, 10, 19, 21	syl3anc 1369	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
23	17, 22	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
24	2	lssincl 20142	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑋) ∈ 𝑆 ∧ 𝑀 ∈ 𝑆) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆)
25	8, 16, 23, 24	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆)
27		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 })
28	2, 3, 4, 9, 26, 27	lssatomic 36952	. . . . 5 ⊢ ((𝜑 ∧ (( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 }) → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))
29	28	ex 412	. . . 4 ⊢ (𝜑 → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 } → ∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)))
30		dochexmidlem1.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
31	7	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	10	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ∈ 𝑆)
33	18	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ∈ 𝐴)
34		simp2 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ∈ 𝐴)
35		dochexmidlem5.xn	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 })
36	35	3ad2ant1 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ≠ { 0 })
37		simp3 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀))
38	5, 14, 6, 11, 2, 30, 20, 4, 31, 32, 33, 34, 3, 17, 36, 37	dochexmidlem4 39404	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
39	38	rexlimdv3a 3214	. . . 4 ⊢ (𝜑 → (∃𝑞 ∈ 𝐴 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
40	29, 39	syld 47	. . 3 ⊢ (𝜑 → ((( ⊥ ‘𝑋) ∩ 𝑀) ≠ { 0 } → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
41	40	necon1bd 2960	. 2 ⊢ (𝜑 → (¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)) → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 }))
42	1, 41	mpd 15	1 ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 })

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ∩ cin 3882 ⊆ wss 3883 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 0_gc0g 17067 LSSumclsm 19154 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 LSAtomsclsa 36915 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 ocHcoch 39288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lcv 36960 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tendo 38696 df-edring 38698 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289

This theorem is referenced by: dochexmidlem6 39406

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