dochexmidlem5 - Mathbox for Norm Megill

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Theorem dochexmidlem5 41466

Description: Lemma for dochexmid 41470. (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 41112	. . . . . . 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 20934	. . . . . . . . . 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 41356	. . . . . . . . 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 38998	. . . . . . . . . 10 ⊢ (𝜑 → 𝑝 ∈ 𝑆)
20		dochexmidlem1.p	. . . . . . . . . . 11 ⊢ ⊕ = (LSSum‘𝑈)
21	2, 20	lsmcl 21082	. . . . . . . . . 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 2845	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ 𝑆)
24	2	lssincl 20963	. . . . . . . 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 39012	. . . . 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 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	10	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ∈ 𝑆)
33	18	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ∈ 𝐴)
34		simp2 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑞 ∈ 𝐴)
35		dochexmidlem5.xn	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ { 0 })
36	35	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑋 ≠ { 0 })
37		simp3 1139	. . . . . 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 41465	. . . . 5 ⊢ ((𝜑 ∧ 𝑞 ∈ 𝐴 ∧ 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
39	38	rexlimdv3a 3159	. . . 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 2958	. 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 0_gc0g 17484 LSSumclsm 19652 LModclmod 20858 LSubSpclss 20929 LSpanclspn 20969 LSAtomsclsa 38975 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 ocHcoch 41349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mre 17629 df-mrc 17630 df-acs 17632 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-oppg 19364 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-lcv 39020 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 df-edring 40759 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350

This theorem is referenced by: dochexmidlem6 41467

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