Theorem 2atjm 36575

Description: The meet of a line (expressed with 2 atoms) and a lattice element. (Contributed by NM, 30-Jul-2012.)

Hypotheses

Ref	Expression
2atjm.b	⊢ 𝐵 = (Base‘𝐾)
2atjm.l	⊢ ≤ = (le‘𝐾)
2atjm.j	⊢ ∨ = (join‘𝐾)
2atjm.m	⊢ ∧ = (meet‘𝐾)
2atjm.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
2atjm	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)

Proof of Theorem 2atjm

Step	Hyp	Ref	Expression
1		hllat 36493	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1129	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat)
3		simp21 1202	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
4		2atjm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
5		2atjm.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 36419	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	3, 6	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐵)
8		simp22 1203	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
9	4, 5	atbase 36419	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐵)
11		2atjm.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
12		2atjm.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
13	4, 11, 12	latlej1 17664	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑄))
14	2, 7, 10, 13	syl3anc 1367	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑄))
15		simp3l 1197	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ 𝑋)
16		simp1 1132	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
17	4, 12, 5	hlatjcl 36497	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
18	16, 3, 8, 17	syl3anc 1367	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
19		simp23 1204	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
20		2atjm.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
21	4, 11, 20	latlem12 17682	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≤ 𝑋) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
22	2, 7, 18, 19, 21	syl13anc 1368	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≤ 𝑋) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
23	14, 15, 22	mpbi2and 710	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋))
24		hlatl 36490	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
25	24	3ad2ant1 1129	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝐾 ∈ AtLat)
26	4, 20	latmcom 17679	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
27	2, 18, 19, 26	syl3anc 1367	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑃 ∨ 𝑄)))
28	19, 3, 8	3jca 1124	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
29		nbrne2 5078	. . . . . . 7 ⊢ ((𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋) → 𝑃 ≠ 𝑄)
30	29	3ad2ant3 1131	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≠ 𝑄)
31		simp3r 1198	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ¬ 𝑄 ≤ 𝑋)
32	4, 12	latjcl 17655	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
33	2, 19, 10, 32	syl3anc 1367	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
34	4, 11, 12	latlej1 17664	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑄))
35	2, 19, 10, 34	syl3anc 1367	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑋 ≤ (𝑋 ∨ 𝑄))
36	4, 11, 2, 7, 19, 33, 15, 35	lattrd 17662	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 ≤ (𝑋 ∨ 𝑄))
37	4, 11, 12, 20, 5	cvrat3 36572	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
38	37	imp 409	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑃 ≤ (𝑋 ∨ 𝑄))) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
39	16, 28, 30, 31, 36, 38	syl23anc 1373	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
40	27, 39	eqeltrd 2913	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)
41	11, 5	atcmp 36441	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴) → (𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
42	25, 3, 40, 41	syl3anc 1367	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → (𝑃 ≤ ((𝑃 ∨ 𝑄) ∧ 𝑋) ↔ 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋)))
43	23, 42	mpbid 234	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → 𝑃 = ((𝑃 ∨ 𝑄) ∧ 𝑋))
44	43	eqcomd 2827	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5058  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  lecple 16566  joincjn 17548  meetcmee 17549  Latclat 17649  Atomscatm 36393  AtLatcal 36394  HLchlt 36480

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 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  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 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  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-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

This theorem is referenced by:  atbtwn  36576  dalem24  36827  dalem25  36828