Theorem cdleme21b 40287

Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)

Hypotheses

Ref	Expression
cdleme21a.l	⊢ ≤ = (le‘𝐾)
cdleme21a.j	⊢ ∨ = (join‘𝐾)
cdleme21a.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cdleme21b	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))

Proof of Theorem cdleme21b

Step	Hyp	Ref	Expression
1		simp23 1208	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
2		simp11 1203	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝐾 ∈ HL)
3		hlcvl 39319	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat)
4	2, 3	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝐾 ∈ CvLat)
5		simp3l 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑧 ∈ 𝐴)
6		simp13 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑄 ∈ 𝐴)
7		simp12 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑃 ∈ 𝐴)
8		simp21 1206	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑆 ∈ 𝐴)
9		cdleme21a.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
10		cdleme21a.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
11		cdleme21a.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
12	9, 10, 11	atnlej1 39340	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≠ 𝑃)
13	12	necomd 2986	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑆)
14	2, 8, 7, 6, 1, 13	syl131anc 1384	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑃 ≠ 𝑆)
15		simp3r 1202	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))
16	11, 10	cvlsupr5 39306	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑃 ≠ 𝑆 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑧 ≠ 𝑃)
17	4, 7, 8, 5, 14, 15, 16	syl132anc 1389	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑧 ≠ 𝑃)
18	9, 10, 11	cvlatexch1 39296	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑧 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑧 ≠ 𝑃) → (𝑧 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑧)))
19	4, 5, 6, 7, 17, 18	syl131anc 1384	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑧 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑧)))
20	11, 10	cvlsupr8 39309	. . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑃 ≠ 𝑆 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑃 ∨ 𝑆) = (𝑃 ∨ 𝑧))
21	4, 7, 8, 5, 14, 15, 20	syl132anc 1389	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑃 ∨ 𝑆) = (𝑃 ∨ 𝑧))
22	21	breq2d 5135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑄 ≤ (𝑃 ∨ 𝑆) ↔ 𝑄 ≤ (𝑃 ∨ 𝑧)))
23	19, 22	sylibrd 259	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑧 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆)))
24		simp22 1207	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑃 ≠ 𝑄)
25	24	necomd 2986	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑄 ≠ 𝑃)
26	9, 10, 11	cvlatexch1 39296	. . . 4 ⊢ ((𝐾 ∈ CvLat ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝑄)))
27	4, 6, 8, 7, 25, 26	syl131anc 1384	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑄 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝑄)))
28	23, 27	syld 47	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → (𝑧 ≤ (𝑃 ∨ 𝑄) → 𝑆 ≤ (𝑃 ∨ 𝑄)))
29	1, 28	mtod 198	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → ¬ 𝑧 ≤ (𝑃 ∨ 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  lecple 17280  joincjn 18327  Atomscatm 39223  CvLatclc 39225  HLchlt 39310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-proset 18310  df-poset 18329  df-plt 18344  df-lub 18360  df-glb 18361  df-join 18362  df-meet 18363  df-p0 18439  df-lat 18446  df-covers 39226  df-ats 39227  df-atl 39258  df-cvlat 39282  df-hlat 39311

This theorem is referenced by:  cdleme21d  40291  cdleme21e  40292