Theorem cdleme21b 37622

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 1205	. 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 1200	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝐾 ∈ HL)
3		hlcvl 36655	. . . . . 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 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑧 ∈ 𝐴)
6		simp13 1202	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑄 ∈ 𝐴)
7		simp12 1201	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑃 ∈ 𝐴)
8		simp21 1203	. . . . . 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 36675	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑆 ≠ 𝑃)
13	12	necomd 3042	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑆)
14	2, 8, 7, 6, 1, 13	syl131anc 1380	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))) → 𝑃 ≠ 𝑆)
15		simp3r 1199	. . . . . 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 36642	. . . . . 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 1385	. . . . 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 36632	. . . . 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 1380	. . . 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 36645	. . . . . 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 1385	. . . . 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 5042	. . . 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 262	. . 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 1204	. . . . 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 3042	. . . 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 36632	. . . 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 1380	. . 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 201	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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  lecple 16564  joincjn 17546  Atomscatm 36559  CvLatclc 36561  HLchlt 36646

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647

This theorem is referenced by:  cdleme21d  37626  cdleme21e  37627