Theorem lhpj1 40018

Description: The join of a co-atom (hyperplane) and an element not under it is the lattice unity. (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 766	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ HL)
2		simpr 484	. . . 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 39994	. . . . 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 2729	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
9	3, 7, 8	hlrelat2 39399	. . . 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 1198	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp2 1137	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13		simp3r 1203	. . . . . . 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 40016	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
17	11, 12, 13, 16	syl12anc 836	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑝) = 1 )
18		simp3l 1202	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ≤ 𝑋)
19		simp1ll 1237	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ HL)
20	19	hllatd 39360	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ Lat)
21	3, 8	atbase 39285	. . . . . . . . 9 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
22	21	3ad2ant2 1134	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑝 ∈ 𝐵)
23		simp1r 1199	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
24	6	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
25	3, 7, 14	latjlej2 18347	. . . . . . . 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 5112	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 1 ≤ (𝑊 ∨ 𝑋))
29		hlop 39358	. . . . . . 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 18332	. . . . . . 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 39188	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ (𝑊 ∨ 𝑋) ∈ 𝐵) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
34	30, 32, 33	syl2anc 584	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → ( 1 ≤ (𝑊 ∨ 𝑋) ↔ (𝑊 ∨ 𝑋) = 1 ))
35	28, 34	mpbid 232	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )
36	35	rexlimdv3a 3134	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (∃𝑝 ∈ (Atoms‘𝐾)(𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊) → (𝑊 ∨ 𝑋) = 1 ))
37	10, 36	sylbid 240	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑊 → (𝑊 ∨ 𝑋) = 1 ))
38	37	impr 454	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝑊 ∨ 𝑋) = 1 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5088  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  lecple 17155  joincjn 18204  1.cp1 18315  Latclat 18324  OPcops 39168  Atomscatm 39259  HLchlt 39346  LHypclh 39980

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-proset 18187  df-poset 18206  df-plt 18221  df-lub 18237  df-glb 18238  df-join 18239  df-meet 18240  df-p0 18316  df-p1 18317  df-lat 18325  df-clat 18392  df-oposet 39172  df-ol 39174  df-oml 39175  df-covers 39262  df-ats 39263  df-atl 39294  df-cvlat 39318  df-hlat 39347  df-lhyp 39984

This theorem is referenced by:  lhpmcvr  40019  cdleme30a  40374