Theorem 2llnm2N 40014

Description: The meet of two different lattice lines in a lattice plane is an atom. (Contributed by NM, 5-Jul-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
2llnm2.l	⊢ ≤ = (le‘𝐾)
2llnm2.m	⊢ ∧ = (meet‘𝐾)
2llnm2.a	⊢ 𝐴 = (Atoms‘𝐾)
2llnm2.n	⊢ 𝑁 = (LLines‘𝐾)
2llnm2.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
2llnm2N	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐴)

Proof of Theorem 2llnm2N

Step	Hyp	Ref	Expression
1		simp22 1209	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ 𝑁)
2		simp1 1137	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝐾 ∈ HL)
3		hllat 39809	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
4	3	3ad2ant1 1134	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝐾 ∈ Lat)
5		simp21 1208	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ 𝑁)
6		eqid 2736	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
7		2llnm2.n	. . . . . 6 ⊢ 𝑁 = (LLines‘𝐾)
8	6, 7	llnbase 39955	. . . . 5 ⊢ (𝑋 ∈ 𝑁 → 𝑋 ∈ (Base‘𝐾))
9	5, 8	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ∈ (Base‘𝐾))
10	6, 7	llnbase 39955	. . . . 5 ⊢ (𝑌 ∈ 𝑁 → 𝑌 ∈ (Base‘𝐾))
11	1, 10	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑌 ∈ (Base‘𝐾))
12		2llnm2.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
13	6, 12	latmcl 18406	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
14	4, 9, 11, 13	syl3anc 1374	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌) ∈ (Base‘𝐾))
15		2llnm2.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
16		eqid 2736	. . . . . . 7 ⊢ (join‘𝐾) = (join‘𝐾)
17		2llnm2.p	. . . . . . 7 ⊢ 𝑃 = (LPlanes‘𝐾)
18	15, 16, 7, 17	2llnjN 40013	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋(join‘𝐾)𝑌) = 𝑊)
19		simp23 1210	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑊 ∈ 𝑃)
20	18, 19	eqeltrd 2836	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋(join‘𝐾)𝑌) ∈ 𝑃)
21	6, 15, 16	latlej1 18414	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → 𝑋 ≤ (𝑋(join‘𝐾)𝑌))
22	4, 9, 11, 21	syl3anc 1374	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑋 ≤ (𝑋(join‘𝐾)𝑌))
23		eqid 2736	. . . . . 6 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
24	15, 23, 7, 17	llncvrlpln2 40003	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ (𝑋(join‘𝐾)𝑌) ∈ 𝑃) ∧ 𝑋 ≤ (𝑋(join‘𝐾)𝑌)) → 𝑋( ⋖ ‘𝐾)(𝑋(join‘𝐾)𝑌))
25	2, 5, 20, 22, 24	syl31anc 1376	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → 𝑋( ⋖ ‘𝐾)(𝑋(join‘𝐾)𝑌))
26	6, 16, 12, 23	cvrexch 39866	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((𝑋 ∧ 𝑌)( ⋖ ‘𝐾)𝑌 ↔ 𝑋( ⋖ ‘𝐾)(𝑋(join‘𝐾)𝑌)))
27	2, 9, 11, 26	syl3anc 1374	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((𝑋 ∧ 𝑌)( ⋖ ‘𝐾)𝑌 ↔ 𝑋( ⋖ ‘𝐾)(𝑋(join‘𝐾)𝑌)))
28	25, 27	mpbird 257	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌)( ⋖ ‘𝐾)𝑌)
29		2llnm2.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
30	6, 23, 29, 7	atcvrlln 39966	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∧ 𝑌) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑋 ∧ 𝑌)( ⋖ ‘𝐾)𝑌) → ((𝑋 ∧ 𝑌) ∈ 𝐴 ↔ 𝑌 ∈ 𝑁))
31	2, 14, 11, 28, 30	syl31anc 1376	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → ((𝑋 ∧ 𝑌) ∈ 𝐴 ↔ 𝑌 ∈ 𝑁))
32	1, 31	mpbird 257	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑊 ∈ 𝑃) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊 ∧ 𝑋 ≠ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  lecple 17227  joincjn 18277  meetcmee 18278  Latclat 18397   ⋖ ccvr 39708  Atomscatm 39709  HLchlt 39796  LLinesclln 39937  LPlanesclpl 39938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-proset 18260  df-poset 18279  df-plt 18294  df-lub 18310  df-glb 18311  df-join 18312  df-meet 18313  df-p0 18389  df-lat 18398  df-clat 18465  df-oposet 39622  df-ol 39624  df-oml 39625  df-covers 39712  df-ats 39713  df-atl 39744  df-cvlat 39768  df-hlat 39797  df-llines 39944  df-lplanes 39945

This theorem is referenced by:  2llnm3N  40015