Theorem lhpj1 40009

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 39985	. . . . 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 39390	. . . 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 40007	. . . . . . 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 39350	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 𝐾 ∈ Lat)
21	3, 8	atbase 39275	. . . . . . . . 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 18395	. . . . . . . 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 5126	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ (𝑝 ≤ 𝑋 ∧ ¬ 𝑝 ≤ 𝑊)) → 1 ≤ (𝑊 ∨ 𝑋))
29		hlop 39348	. . . . . . 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 18380	. . . . . . 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 39178	. . . . . 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 3138	. . 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 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203  joincjn 18252  1.cp1 18363  Latclat 18372  OPcops 39158  Atomscatm 39249  HLchlt 39336  LHypclh 39971

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18235  df-poset 18254  df-plt 18269  df-lub 18285  df-glb 18286  df-join 18287  df-meet 18288  df-p0 18364  df-p1 18365  df-lat 18373  df-clat 18440  df-oposet 39162  df-ol 39164  df-oml 39165  df-covers 39252  df-ats 39253  df-atl 39284  df-cvlat 39308  df-hlat 39337  df-lhyp 39975

This theorem is referenced by:  lhpmcvr  40010  cdleme30a  40365