Theorem 2llnm4 37270

Description: Two lattice lines that majorize the same atom always meet. (Contributed by NM, 20-Jul-2012.)

Hypotheses

Ref	Expression
2llnm4.l	⊢ ≤ = (le‘𝐾)
2llnm4.m	⊢ ∧ = (meet‘𝐾)
2llnm4.z	⊢ 0 = (0.‘𝐾)
2llnm4.a	⊢ 𝐴 = (Atoms‘𝐾)
2llnm4.n	⊢ 𝑁 = (LLines‘𝐾)

Assertion

Ref	Expression
2llnm4	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 )

Proof of Theorem 2llnm4

Step	Hyp	Ref	Expression
1		hlatl 37060	. . 3 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	3ad2ant1 1135	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝐾 ∈ AtLat)
3		hllat 37063	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1135	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝐾 ∈ Lat)
5		simp22 1209	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑋 ∈ 𝑁)
6		eqid 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		2llnm4.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	llnbase 37209	. . . 4 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
9	5, 8	syl 17	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
10		simp23 1210	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑌 ∈ 𝑁)
11	6, 7	llnbase 37209	. . . 4 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
12	10, 11	syl 17	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
13		2llnm4.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
14	6, 13	latmcl 17900	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
15	4, 9, 12, 14	syl3anc 1373	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
16		simp21 1208	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ∈ 𝐴)
17		simp3 1140	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌))
18		2llnm4.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
19	6, 18	atbase 36989	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
20	16, 19	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ∈ (Base‘𝐾))
21		2llnm4.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
22	6, 21, 13	latlem12 17926	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌)))
23	4, 20, 9, 12, 22	syl13anc 1374	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → ((𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌) ↔ 𝑃 ≤ (𝑋 ∧ 𝑌)))
24	17, 23	mpbid 235	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → 𝑃 ≤ (𝑋 ∧ 𝑌))
25		2llnm4.z	. . 3 ⊢ 0 = (0.‘𝐾)
26	6, 21, 25, 18	atlen0 37010	. 2 ⊢ (((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾) ∧ 𝑃 ∈ 𝐴) ∧ 𝑃 ≤ (𝑋 ∧ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 )
27	2, 15, 16, 24, 26	syl31anc 1375	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ≤ 𝑋 ∧ 𝑃 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≠ 0 )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932   class class class wbr 5039  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  lecple 16756  meetcmee 17773  0.cp0 17883  Latclat 17891  Atomscatm 36963  AtLatcal 36964  HLchlt 37050  LLinesclln 37191

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 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

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 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-proset 17756  df-poset 17774  df-plt 17790  df-lub 17806  df-glb 17807  df-join 17808  df-meet 17809  df-p0 17885  df-lat 17892  df-covers 36966  df-ats 36967  df-atl 36998  df-cvlat 37022  df-hlat 37051  df-llines 37198

This theorem is referenced by:  2llnmeqat  37271  dalem2  37361