Theorem 3atlem4 34772

Description: Lemma for 3at 34776. (Contributed by NM, 23-Jun-2012.)

Hypotheses

Ref	Expression
3at.l	⊢ ≤ = (le‘𝐾)
3at.j	⊢ ∨ = (join‘𝐾)
3at.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3atlem4	⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑅))

Proof of Theorem 3atlem4

Step	Hyp	Ref	Expression
1		simp11 1091	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝐾 ∈ HL)
2		simp12 1092	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
3		simp13l 1176	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑆 ∈ 𝐴)
4		simp13r 1177	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑇 ∈ 𝐴)
5		simp123 1195	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑅 ∈ 𝐴)
6	3, 4, 5	3jca 1242	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴))
7		simp2l 1087	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
8		hllat 34650	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
9	1, 8	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝐾 ∈ Lat)
10		eqid 2622	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		3at.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11	atbase 34576	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
13	5, 12	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑅 ∈ (Base‘𝐾))
14		simp121 1193	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑃 ∈ 𝐴)
15	10, 11	atbase 34576	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
16	14, 15	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑃 ∈ (Base‘𝐾))
17		simp122 1194	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑄 ∈ 𝐴)
18	10, 11	atbase 34576	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
19	17, 18	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑄 ∈ (Base‘𝐾))
20		3at.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
21		3at.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
22	10, 20, 21	latnlej1l 17069	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ≠ 𝑃)
23	9, 13, 16, 19, 7, 22	syl131anc 1339	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑅 ≠ 𝑃)
24	23	necomd 2849	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑃 ≠ 𝑅)
25		simp2r 1088	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑃 ≠ 𝑄)
26	25	necomd 2849	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → 𝑄 ≠ 𝑃)
27	20, 21, 11	hlatexch1 34681	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑅) → 𝑅 ≤ (𝑃 ∨ 𝑄)))
28	1, 17, 5, 14, 26, 27	syl131anc 1339	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → (𝑄 ≤ (𝑃 ∨ 𝑅) → 𝑅 ≤ (𝑃 ∨ 𝑄)))
29	7, 28	mtod 189	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑅))
30		simp3 1063	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅))
31	20, 21, 11	3atlem3 34771	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑅 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑅)) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑅))
32	1, 2, 6, 7, 24, 29, 30, 31	syl331anc 1351	1 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ 𝑃 ≠ 𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑅)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  Latclat 17045  Atomscatm 34550  HLchlt 34637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  3atlem5  34773