dochexmidlem8 - Mathbox for Norm Megill

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Theorem dochexmidlem8 39127

Description: Lemma for dochexmid 39128. The contradiction of dochexmidlem6 39125 and dochexmidlem7 39126 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (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)
dochexmidlem8.z	⊢ 0 = (0_g‘𝑈)
dochexmidlem8.xn	⊢ (𝜑 → 𝑋 ≠ { 0 })
dochexmidlem8.c	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)

**Assertion**
Ref	Expression
dochexmidlem8	⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)

Proof of Theorem dochexmidlem8 Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nonconne 2947	. 2 ⊢ ¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)
2		dochexmidlem1.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
3		dochexmidlem1.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		dochexmidlem1.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5	2, 3, 4	dvhlmod 38770	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
6		dochexmidlem1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
7		dochexmidlem1.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈)
8		dochexmidlem1.s	. . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑈)
9	7, 8	lssss 19830	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ 𝑉)
10	6, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑉)
11		dochexmidlem1.o	. . . . . . . 8 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12	2, 3, 7, 8, 11	dochlss 39014	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
13	4, 10, 12	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ 𝑆)
14		dochexmidlem1.p	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈)
15	8, 14	lsmcl 19977	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
16	5, 6, 13, 15	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
17	7, 8	lssss 19830	. . . . 5 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉)
18	16, 17	syl 17	. . . 4 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉)
19		dochexmidlem1.a	. . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈)
20	5	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑈 ∈ LMod)
21	16	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ∈ 𝑆)
22	7, 8	lss1 19832	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑉 ∈ 𝑆)
23	5, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ 𝑆)
24	23	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → 𝑉 ∈ 𝑆)
25		df-pss 3863	. . . . . . . . 9 ⊢ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉 ↔ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉))
26	25	biimpri 231	. . . . . . . 8 ⊢ (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉)
27	26	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → (𝑋 ⊕ ( ⊥ ‘𝑋)) ⊊ 𝑉)
28	8, 19, 20, 21, 24, 27	lpssat 36673	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉)) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))))
29	28	ex 416	. . . . 5 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → ∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))))
30		dochexmidlem1.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
31	4	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
32	6	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ∈ 𝑆)
33		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑝 ∈ 𝐴)
34		dochexmidlem8.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑈)
35		eqid 2739	. . . . . . . . 9 ⊢ (𝑋 ⊕ 𝑝) = (𝑋 ⊕ 𝑝)
36		dochexmidlem8.xn	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ { 0 })
37	36	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → 𝑋 ≠ { 0 })
38		dochexmidlem8.c	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
39	38	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
40		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))
41	2, 11, 3, 7, 8, 30, 14, 19, 31, 32, 33, 34, 35, 37, 39, 40	dochexmidlem6 39125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) = 𝑋)
42	2, 11, 3, 7, 8, 30, 14, 19, 31, 32, 33, 34, 35, 37, 39, 40	dochexmidlem7 39126	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 ⊕ 𝑝) ≠ 𝑋)
43	41, 42	pm2.21ddne 3019	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))
44	43	3adant3l 1181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋)))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋))
45	44	rexlimdv3a 3197	. . . . 5 ⊢ (𝜑 → (∃𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑉 ∧ ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
46	29, 45	syld 47	. . . 4 ⊢ (𝜑 → (((𝑋 ⊕ ( ⊥ ‘𝑋)) ⊆ 𝑉 ∧ (𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉) → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
47	18, 46	mpand 695	. . 3 ⊢ (𝜑 → ((𝑋 ⊕ ( ⊥ ‘𝑋)) ≠ 𝑉 → (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋)))
48	47	necon1bd 2953	. 2 ⊢ (𝜑 → (¬ (𝑋 = 𝑋 ∧ 𝑋 ≠ 𝑋) → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉))
49	1, 48	mpi 20	1 ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 ⊆ wss 3844 ⊊ wpss 3845 {csn 4517 ‘cfv 6340 (class class class)co 7173 Basecbs 16589 0_gc0g 16819 LSSumclsm 18880 LModclmod 19756 LSubSpclss 19825 LSpanclspn 19865 LSAtomsclsa 36634 HLchlt 37010 LHypclh 37644 DVecHcdvh 38738 ocHcoch 39007

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-riotaBAD 36613

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-undef 7971 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-0g 16821 df-mre 16963 df-mrc 16964 df-acs 16966 df-proset 17657 df-poset 17675 df-plt 17687 df-lub 17703 df-glb 17704 df-join 17705 df-meet 17706 df-p0 17768 df-p1 17769 df-lat 17775 df-clat 17837 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-cntz 18568 df-oppg 18595 df-lsm 18882 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-ring 19421 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-drng 19626 df-lmod 19758 df-lss 19826 df-lsp 19866 df-lvec 19997 df-lsatoms 36636 df-lcv 36679 df-oposet 36836 df-ol 36838 df-oml 36839 df-covers 36926 df-ats 36927 df-atl 36958 df-cvlat 36982 df-hlat 37011 df-llines 37158 df-lplanes 37159 df-lvols 37160 df-lines 37161 df-psubsp 37163 df-pmap 37164 df-padd 37456 df-lhyp 37648 df-laut 37649 df-ldil 37764 df-ltrn 37765 df-trl 37819 df-tgrp 38403 df-tendo 38415 df-edring 38417 df-dveca 38663 df-disoa 38689 df-dvech 38739 df-dib 38799 df-dic 38833 df-dih 38889 df-doch 39008 df-djh 39055

This theorem is referenced by: dochexmid 39128

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