Theorem lhpj1 37773

Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)

Hypotheses

Ref	Expression
lhpj1.b	⊢ 𝐵 = (Base‘𝐾)
lhpj1.l	⊢ ≤ = (le‘𝐾)
lhpj1.j	⊢ ∨ = (join‘𝐾)
lhpj1.u	⊢ 1 = (1.‘𝐾)
lhpj1.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
lhpj1	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )

Proof of Theorem lhpj1

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 767	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ HL)
2		simpr 488	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
3		lhpj1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
4		lhpj1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5	3, 4	lhpbase 37749	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
6	5	ad2antlr 727	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ 𝐵)
7		lhpj1.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
8		eqid 2737	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
9	3, 7, 8	hlrelat2 37154	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)))
10	1, 2, 6, 9	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 ↔ ∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)))
11		simp1l 1199	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp2 1139	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13		simp3r 1204	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → ¬ 𝑝 ≤ 𝑊)
14		lhpj1.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
15		lhpj1.u	. . . . . . . 8 ⊢ 1 = (1.‘𝐾)
16	7, 14, 15, 8, 4	lhpjat1 37771	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
17	11, 12, 13, 16	syl12anc 837	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
18		simp3l 1203	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ≤ 𝑋)
19		simp1ll 1238	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ HL)
20	19	hllatd 37115	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ Lat)
21	3, 8	atbase 37040	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
22	21	3ad2ant2 1136	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ 𝐵)
23		simp1r 1200	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
24	6	3ad2ant1 1135	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
25	3, 7, 14	latjlej2 17960	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑝 ≤ 𝑋 → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋)))
26	20, 22, 23, 24, 25	syl13anc 1374	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑝 ≤ 𝑋 → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋)))
27	18, 26	mpd 15	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) ≤ (𝑊 ∨ 𝑋))
28	17, 27	eqbrtrrd 5077	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 1 ≤ (𝑊 ∨ 𝑋))
29		hlop 37113	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
30	19, 29	syl 17	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ OP)
31	3, 14	latjcl 17945	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑊 ∨ 𝑋) ∈ 𝐵)
32	20, 24, 23, 31	syl3anc 1373	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑋) ∈ 𝐵)
33	3, 7, 15	op1le 36943	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (𝑊 ∨ 𝑋) ∈ 𝐵) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
34	30, 32, 33	syl2anc 587	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
35	28, 34	mpbid 235	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )
36	35	rexlimdv3a 3205	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊) → (𝑊 ∨ 𝑋) = 1 ))
37	10, 36	sylbid 243	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 → (𝑊 ∨ 𝑋) = 1 ))
38	37	impr 458	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∃wrex 3062   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  joincjn 17818  1.cp1 17930  Latclat 17937  OPcops 36923  Atomscatm 37014  HLchlt 37101  LHypclh 37735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-proset 17802  df-poset 17820  df-plt 17836  df-lub 17852  df-glb 17853  df-join 17854  df-meet 17855  df-p0 17931  df-p1 17932  df-lat 17938  df-clat 18005  df-oposet 36927  df-ol 36929  df-oml 36930  df-covers 37017  df-ats 37018  df-atl 37049  df-cvlat 37073  df-hlat 37102  df-lhyp 37739

This theorem is referenced by:  lhpmcvr  37774  cdleme30a  38129