dochexmidlem5 - Mathbox for Norm Megill

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Theorem dochexmidlem5 41562

Description: Lemma for dochexmid 41566. (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 41208	. . . . . . 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 20869	. . . . . . . . . 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 41452	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
16	7, 13, 15	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
17		dochexmidlem5.m	. . . . . . . . 9 ⊢ 𝑀 = (𝑋 ⊕ 𝑝)
18		dochexmidlem5.pp	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑝 ∈ 𝐴)
19	2, 4, 8, 18	lsatlssel 39095	. . . . . . . . . 10 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
20		dochexmidlem1.p	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
21	2, 20	lsmcl 21017	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑝 ∈ 𝑆) → (𝑋 ⊕ 𝑝) ∈ 𝑆)
22	8, 10, 19, 21	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ⊕ 𝑝) ∈ 𝑆)
23	17, 22	eqeltrid 2835	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
24	2	lssincl 20898	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘𝑋) ∈ 𝑆 ∧ 𝑀 ∈ 𝑆) → (( ⊥ ‘𝑋) ∩ 𝑀) ∈ 𝑆)
25	8, 16, 23, 24	syl3anc 1373	. . . . . . 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 39109	. . . . 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 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	10	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ∈ 𝑆)
33	18	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ∈ 𝐴)
34		simp2 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ∈ 𝐴)
35		dochexmidlem5.xn	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 })
36	35	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ≠ { 0 })
37		simp3 1138	. . . . . 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 41561	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
39	38	rexlimdv3a 3137	. . . 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 2946	. 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 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∃wrex 3056 ∩ cin 3896 ⊆ wss 3897 {csn 4573 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 0_gc0g 17343 LSSumclsm 19546 LModclmod 20793 LSubSpclss 20864 LSpanclspn 20904 LSAtomsclsa 39072 HLchlt 39448 LHypclh 40082 DVecHcdvh 41176 ocHcoch 41445

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 39051

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 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 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 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19229 df-oppg 19258 df-lsm 19548 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lvec 21037 df-lsatoms 39074 df-lcv 39117 df-oposet 39274 df-ol 39276 df-oml 39277 df-covers 39364 df-ats 39365 df-atl 39396 df-cvlat 39420 df-hlat 39449 df-llines 39596 df-lplanes 39597 df-lvols 39598 df-lines 39599 df-psubsp 39601 df-pmap 39602 df-padd 39894 df-lhyp 40086 df-laut 40087 df-ldil 40202 df-ltrn 40203 df-trl 40257 df-tendo 40853 df-edring 40855 df-disoa 41127 df-dvech 41177 df-dib 41237 df-dic 41271 df-dih 41327 df-doch 41446

This theorem is referenced by: dochexmidlem6 41563

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