Theorem 2llnma1b 40293

Description: Generalization of 2llnma1 40294. (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 39870	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1140	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ Lat)
3		simp22 1215	. . . . . 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 39796	. . . . . 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 1214	. . . . 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 18409	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑋))
12	2, 7, 8, 11	syl3anc 1380	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑋))
13		simp23 1216	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑄 ∈ 𝐴)
14	4, 5	atbase 39796	. . . . . 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 18409	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → 𝑃 ≤ (𝑃 ∨ 𝑄))
17	2, 7, 15, 16	syl3anc 1380	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ (𝑃 ∨ 𝑄))
18	4, 10	latjcl 18400	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∨ 𝑋) ∈ 𝐵)
19	2, 7, 8, 18	syl3anc 1380	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → (𝑃 ∨ 𝑋) ∈ 𝐵)
20		simp1 1143	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ HL)
21	4, 10, 5	hlatjcl 39874	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
22	20, 3, 13, 21	syl3anc 1380	. . . . 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 18427	. . . . 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 1381	. . . 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 719	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
27		hlatl 39867	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
28	27	3ad2ant1 1140	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝐾 ∈ AtLat)
29		simp3 1145	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑋))
30		nbrne2 5095	. . . . . 6 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑋) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≠ 𝑄)
31	12, 29, 30	syl2anc 591	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≠ 𝑄)
32	4, 10	latjcl 18400	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐵)
33	2, 19, 15, 32	syl3anc 1380	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∨ 𝑄) ∈ 𝐵)
34	4, 9, 10	latlej1 18409	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑋) ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑋) ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))
35	2, 19, 15, 34	syl3anc 1380	. . . . . 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 18407	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 ≤ ((𝑃 ∨ 𝑋) ∨ 𝑄))
37	4, 9, 10, 23, 5	cvrat3 39949	. . . . . 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 1124	. . . . 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 1402	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
40	9, 5	atcmp 39818	. . . 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 1380	. . 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 234	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → 𝑃 = ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)))
43	42	eqcomd 2747	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ≠ wne 2936   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  lecple 17222  joincjn 18272  meetcmee 18273  Latclat 18392  Atomscatm 39770  AtLatcal 39771  HLchlt 39857

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-lat 18393  df-clat 18460  df-oposet 39683  df-ol 39685  df-oml 39686  df-covers 39773  df-ats 39774  df-atl 39805  df-cvlat 39829  df-hlat 39858

This theorem is referenced by:  2llnma1  40294  cdlemg4  41124  cdlemkfid1N  41428