Theorem 2llnm3N 36699

Description: Two lattice lines in a lattice plane always meet. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2llnm3.l	⊢ ≤ = (le‘𝐾)
2llnm3.m	⊢ ∧ = (meet‘𝐾)
2llnm3.z	⊢ 0 = (0.‘𝐾)
2llnm3.n	⊢ 𝑁 = (LLines‘𝐾)
2llnm3.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
2llnm3N	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≠ 0 )

Proof of Theorem 2llnm3N

Step	Hyp	Ref	Expression
1		oveq1 7157	. . 3 ⊢ (𝑋 = 𝑌 → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑌))
2	1	neeq1d 3075	. 2 ⊢ (𝑋 = 𝑌 → ((𝑋 ∧ 𝑌) ≠ 0 ↔ (𝑌 ∧ 𝑌) ≠ 0 ))
3		simpl1 1187	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → 𝐾 ∈ HL)
4		hlatl 36490	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
5	3, 4	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → 𝐾 ∈ AtLat)
6		simpl2 1188	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃))
7		simpl3l 1224	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → 𝑋 ≤ 𝑊)
8		simpl3r 1225	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → 𝑌 ≤ 𝑊)
9		simpr 487	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
10		2llnm3.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
11		2llnm3.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
12		eqid 2821	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
13		2llnm3.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
14		2llnm3.p	. . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾)
15	10, 11, 12, 13, 14	2llnm2N 36698	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Atoms‘𝐾))
16	3, 6, 7, 8, 9, 15	syl113anc 1378	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∧ 𝑌) ∈ (Atoms‘𝐾))
17		2llnm3.z	. . . 4 ⊢ 0 = (0.‘𝐾)
18	17, 12	atn0 36438	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ (Atoms‘𝐾)) → (𝑋 ∧ 𝑌) ≠ 0 )
19	5, 16, 18	syl2anc 586	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≠ 𝑌) → (𝑋 ∧ 𝑌) ≠ 0 )
20		hllat 36493	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
21	20	3ad2ant1 1129	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ Lat)
22		simp22 1203	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝑁)
23		eqid 2821	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
24	23, 13	llnbase 36639	. . . . 5 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
25	22, 24	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ (Base‘𝐾))
26	23, 11	latmidm 17690	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑌 ∧ 𝑌) = 𝑌)
27	21, 25, 26	syl2anc 586	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝑌 ∧ 𝑌) = 𝑌)
28		simp1 1132	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝐾 ∈ HL)
29	17, 13	llnn0 36646	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑁) → 𝑌 ≠ 0 )
30	28, 22, 29	syl2anc 586	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ≠ 0 )
31	27, 30	eqnetrd 3083	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝑌 ∧ 𝑌) ≠ 0 )
32	2, 19, 31	pm2.61ne 3102	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝑋 ∧ 𝑌) ≠ 0 )

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5059  ‘cfv 6350  (class class class)co 7150  Basecbs 16477  lecple 16566  meetcmee 17549  0.cp0 17641  Latclat 17649  Atomscatm 36393  AtLatcal 36394  HLchlt 36480  LLinesclln 36621  LPlanesclpl 36622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17532  df-poset 17550  df-plt 17562  df-lub 17578  df-glb 17579  df-join 17580  df-meet 17581  df-p0 17643  df-p1 17644  df-lat 17650  df-clat 17712  df-oposet 36306  df-ol 36308  df-oml 36309  df-covers 36396  df-ats 36397  df-atl 36428  df-cvlat 36452  df-hlat 36481  df-llines 36628  df-lplanes 36629

This theorem is referenced by: (None)