Theorem 2llnma1b 39782

Description: Generalization of 2llnma1 39783. (Contributed by NM, 26-Apr-2013.)

Hypotheses

Ref	Expression
2llnma1b.b	⊢ 𝐵 = (Base‘𝐾)
2llnma1b.l	⊢ ≤ = (le‘𝐾)
2llnma1b.j	⊢ ∨ = (join‘𝐾)
2llnma1b.m	⊢ ∧ = (meet‘𝐾)
2llnma1b.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
2llnma1b	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)

Proof of Theorem 2llnma1b

Step	Hyp	Ref	Expression
1		hllat 39359	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ Lat)
3		simp22 1208	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ∈ 𝐴)
4		2llnma1b.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
5		2llnma1b.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39285	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	3, 6	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ∈ 𝐵)
8		simp21 1207	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑋 ∈ 𝐵)
9		2llnma1b.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
10		2llnma1b.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
11	4, 9, 10	latlej1 18341	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑋))
12	2, 7, 8, 11	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑋))
13		simp23 1209	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑄 ∈ 𝐴)
14	4, 5	atbase 39285	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
15	13, 14	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑄 ∈ 𝐵)
16	4, 9, 10	latlej1 18341	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑄))
17	2, 7, 15, 16	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑄))
18	4, 10	latjcl 18332	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∨ 𝑋) ∈ 𝐵)
19	2, 7, 8, 18	syl3anc 1373	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → (𝑃 ∨ 𝑋) ∈ 𝐵)
20		simp1 1136	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ HL)
21	4, 10, 5	hlatjcl 39363	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
22	20, 3, 13, 21	syl3anc 1373	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
23		2llnma1b.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
24	4, 9, 23	latlem12 18359	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ (𝑃 ∨ 𝑋) ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵)) → ((𝑃 ≤ (𝑃 ∨ 𝑋) ∧ 𝑃 ≤ (𝑃 ∨ 𝑄)) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
25	2, 7, 19, 22, 24	syl13anc 1374	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ≤ (𝑃 ∨ 𝑋) ∧ 𝑃 ≤ (𝑃 ∨ 𝑄)) ↔ 𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
26	12, 17, 25	mpbi2and 712	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
27		hlatl 39356	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28	27	3ad2ant1 1133	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ AtLat)
29		simp3 1138	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑋))
30		nbrne2 5108	. . . . . 6 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑋) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≠ 𝑄)
31	12, 29, 30	syl2anc 584	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≠ 𝑄)
32	4, 10	latjcl 18332	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐵)
33	2, 19, 15, 32	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐵)
34	4, 9, 10	latlej1 18341	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑋) ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))
35	2, 19, 15, 34	syl3anc 1373	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → (𝑃 ∨ 𝑋) ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))
36	4, 9, 2, 7, 19, 33, 12, 35	lattrd 18339	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))
37	4, 9, 10, 23, 5	cvrat3 39438	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋) ∧ 𝑃 ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
38	37	3impia 1117	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋) ∧ 𝑃 ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
39	20, 19, 3, 13, 31, 29, 36, 38	syl133anc 1395	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
40	9, 5	atcmp 39307	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) → (𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ↔ 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
41	28, 3, 39, 40	syl3anc 1373	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → (𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ↔ 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄))))
42	26, 41	mpbid 232	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
43	42	eqcomd 2735	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5088  ‘cfv 6476  (class class class)co 7340  Basecbs 17107  lecple 17155  joincjn 18204  meetcmee 18205  Latclat 18324  Atomscatm 39259  AtLatcal 39260  HLchlt 39346

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 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662

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 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-proset 18187  df-poset 18206  df-plt 18221  df-lub 18237  df-glb 18238  df-join 18239  df-meet 18240  df-p0 18316  df-lat 18325  df-clat 18392  df-oposet 39172  df-ol 39174  df-oml 39175  df-covers 39262  df-ats 39263  df-atl 39294  df-cvlat 39318  df-hlat 39347

This theorem is referenced by:  2llnma1  39783  cdlemg4  40613  cdlemkfid1N  40917