ps-2 - Mathbox for Norm Megill

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Theorem ps-2 40351

Description: Lattice analogue for the projective geometry axiom, "if a line intersects two sides of a triangle at different points then it also intersects the third side." Projective space condition PS2 in [MaedaMaeda] p. 68 and part of Theorem 16.4 in [MaedaMaeda] p. 69. (Contributed by NM, 1-Dec-2011.)

**Hypotheses**
Ref	Expression
ps1.l	⊢ ≤ = (le‘𝐾)
ps1.j	⊢ ∨ = (join‘𝐾)
ps1.a	⊢ 𝐴 = (Atoms‘𝐾)

**Assertion**
Ref	Expression
ps-2	⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))

Distinct variable groups: 𝑢,𝐴 𝑢, ∨ 𝑢,𝐾 𝑢, ≤ 𝑢,𝑃 𝑢,𝑄 𝑢,𝑅 𝑢,𝑆 𝑢,𝑇

**Proof of Theorem ps-2**
Step	Hyp	Ref	Expression
1		simpl21 1270	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 = 𝑃) → 𝑃 ∈ 𝐴)
2		simp1 1154	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐾 ∈ HL)
3		simp21 1225	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
4		simp23 1227	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑅 ∈ 𝐴)
5		ps1.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
6		ps1.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
7		ps1.a	. . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾)
8	5, 6, 7	hlatlej1 40248	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑅))
9	2, 3, 4, 8	syl3anc 1398	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑃 ≤ (𝑃 ∨ 𝑅))
10	9	adantr 486	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 = 𝑃) → 𝑃 ≤ (𝑃 ∨ 𝑅))
11		simp3r 1221	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑇 ∈ 𝐴)
12	5, 6, 7	hlatlej1 40248	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑇))
13	2, 3, 11, 12	syl3anc 1398	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑃 ≤ (𝑃 ∨ 𝑇))
14		oveq1 7420	. . . . . . . 8 ⊢ (𝑆 = 𝑃 → (𝑆 ∨ 𝑇) = (𝑃 ∨ 𝑇))
15	14	breq2d 5115	. . . . . . 7 ⊢ (𝑆 = 𝑃 → (𝑃 ≤ (𝑆 ∨ 𝑇) ↔ 𝑃 ≤ (𝑃 ∨ 𝑇)))
16	13, 15	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑆 = 𝑃 → 𝑃 ≤ (𝑆 ∨ 𝑇)))
17	16	imp 412	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 = 𝑃) → 𝑃 ≤ (𝑆 ∨ 𝑇))
18		breq1 5106	. . . . . . 7 ⊢ (𝑢 = 𝑃 → (𝑢 ≤ (𝑃 ∨ 𝑅) ↔ 𝑃 ≤ (𝑃 ∨ 𝑅)))
19		breq1 5106	. . . . . . 7 ⊢ (𝑢 = 𝑃 → (𝑢 ≤ (𝑆 ∨ 𝑇) ↔ 𝑃 ≤ (𝑆 ∨ 𝑇)))
20	18, 19	anbi12d 644	. . . . . 6 ⊢ (𝑢 = 𝑃 → ((𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)) ↔ (𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑃 ≤ (𝑆 ∨ 𝑇))))
21	20	rspcev 3576	. . . . 5 ⊢ ((𝑃 ∈ 𝐴 ∧ (𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑃 ≤ (𝑆 ∨ 𝑇))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
22	1, 10, 17, 21	syl12anc 850	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 = 𝑃) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
23	22	a1d 26	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 = 𝑃) → (((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
24		hlop 40235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
25	24	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐾 ∈ OP)
26		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐾) = (Base‘𝐾)
27		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (0.‘𝐾) = (0.‘𝐾)
28	26, 27	op0cl 40057	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
29	25, 28	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (0.‘𝐾) ∈ (Base‘𝐾))
30	26, 7	atbase 40162	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
31	3, 30	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾))
32		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾)
33	27, 32, 7	atcvr0 40161	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃)
34	2, 3, 33	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (0.‘𝐾)( ⋖ ‘𝐾)𝑃)
35		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (lt‘𝐾) = (lt‘𝐾)
36	26, 35, 32	cvrlt 40143	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ (0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾)) ∧ (0.‘𝐾)( ⋖ ‘𝐾)𝑃) → (0.‘𝐾)(lt‘𝐾)𝑃)
37	2, 29, 31, 34, 36	syl31anc 1400	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (0.‘𝐾)(lt‘𝐾)𝑃)
38		hlpos 40239	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
39	38	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐾 ∈ Poset)
40		hllat 40236	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
41	40	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐾 ∈ Lat)
42	26, 7	atbase 40162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
43	4, 42	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑅 ∈ (Base‘𝐾))
44	26, 6	latjcl 18527	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
45	41, 31, 43, 44	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
46	26, 5, 35	pltletr 18429	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Poset ∧ ((0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑃 ∧ 𝑃 ≤ (𝑃 ∨ 𝑅)) → (0.‘𝐾)(lt‘𝐾)(𝑃 ∨ 𝑅)))
47	39, 29, 31, 45, 46	syl13anc 1399	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (((0.‘𝐾)(lt‘𝐾)𝑃 ∧ 𝑃 ≤ (𝑃 ∨ 𝑅)) → (0.‘𝐾)(lt‘𝐾)(𝑃 ∨ 𝑅)))
48	37, 9, 47	mp2and 712	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (0.‘𝐾)(lt‘𝐾)(𝑃 ∨ 𝑅))
49	35	pltne 18420	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (0.‘𝐾) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑃 ∨ 𝑅) → (0.‘𝐾) ≠ (𝑃 ∨ 𝑅)))
50	2, 29, 45, 49	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((0.‘𝐾)(lt‘𝐾)(𝑃 ∨ 𝑅) → (0.‘𝐾) ≠ (𝑃 ∨ 𝑅)))
51	48, 50	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (0.‘𝐾) ≠ (𝑃 ∨ 𝑅))
52	51	necomd 3010	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ≠ (0.‘𝐾))
53	52	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑃 ∨ 𝑅) ≠ (0.‘𝐾))
54		hlatl 40233	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
55	54	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝐾 ∈ AtLat)
56		simp3l 1220	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑆 ∈ 𝐴)
57	5, 7	atncmp 40185	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ AtLat ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (¬ 𝑆 ≤ 𝑃 ↔ 𝑆 ≠ 𝑃))
58	55, 56, 3, 57	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (¬ 𝑆 ≤ 𝑃 ↔ 𝑆 ≠ 𝑃))
59		simp22 1226	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
60	26, 5, 6, 7	hlexch1 40255	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ (Base‘𝐾)) ∧ ¬ 𝑆 ≤ 𝑃) → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆)))
61	60	3expia 1139	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ (Base‘𝐾))) → (¬ 𝑆 ≤ 𝑃 → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆))))
62	2, 56, 59, 31, 61	syl13anc 1399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (¬ 𝑆 ≤ 𝑃 → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆))))
63	58, 62	sylbird 263	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑆 ≠ 𝑃 → (𝑆 ≤ (𝑃 ∨ 𝑄) → 𝑄 ≤ (𝑃 ∨ 𝑆))))
64	63	imp32 424	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → 𝑄 ≤ (𝑃 ∨ 𝑆))
65	26, 7	atbase 40162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
66	59, 65	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑄 ∈ (Base‘𝐾))
67	26, 7	atbase 40162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
68	56, 67	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑆 ∈ (Base‘𝐾))
69	26, 6	latjcl 18527	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
70	41, 31, 68, 69	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
71	26, 5, 6	latjlej1 18541	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝑄 ≤ (𝑃 ∨ 𝑆) → (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
72	41, 66, 70, 43, 71	syl13anc 1399	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑆) → (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
73	72	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑄 ≤ (𝑃 ∨ 𝑆) → (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
74	64, 73	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ 𝑆 ≤ (𝑃 ∨ 𝑄))) → (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅))
75	74	adantrrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅))
76	26, 7	atbase 40162	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
77	11, 76	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → 𝑇 ∈ (Base‘𝐾))
78	26, 6	latjcl 18527	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
79	41, 66, 43, 78	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
80	26, 6	latjcl 18527	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑅) ∈ (Base‘𝐾))
81	41, 70, 43, 80	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∨ 𝑅) ∈ (Base‘𝐾))
82	26, 5	lattr 18532	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾 ∈ Lat ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑆) ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑇 ≤ (𝑄 ∨ 𝑅) ∧ (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
83	41, 77, 79, 81, 82	syl13anc 1399	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑇 ≤ (𝑄 ∨ 𝑅) ∧ (𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
84	83	expdimp 458	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)) → ((𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
85	84	adantrl 729	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ((𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
86	85	adantrl 729	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑄 ∨ 𝑅) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅)))
87	75, 86	mpd 16	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅))
88	6, 7	hlatj32 40245	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑆))
89	2, 3, 56, 4, 88	syl13anc 1399	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑃 ∨ 𝑆) ∨ 𝑅) = ((𝑃 ∨ 𝑅) ∨ 𝑆))
90	89	breq2d 5115	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅) ↔ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
91	90	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑅) ↔ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
92	87, 91	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆))
93	53, 92	jca 521	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ≠ (0.‘𝐾) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
94	93	adantrrl 737	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ (𝑆 ≠ 𝑃 ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅))))) → ((𝑃 ∨ 𝑅) ≠ (0.‘𝐾) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)))
95	94	ex 418	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑆 ≠ 𝑃 ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ((𝑃 ∨ 𝑅) ≠ (0.‘𝐾) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆))))
96	26, 5, 6, 27, 7	cvrat4 40316	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑅) ∈ (Base‘𝐾) ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (((𝑃 ∨ 𝑅) ≠ (0.‘𝐾) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢))))
97	2, 45, 11, 56, 96	syl13anc 1399	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (((𝑃 ∨ 𝑅) ≠ (0.‘𝐾) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑅) ∨ 𝑆)) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢))))
98	95, 97	syld 48	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑆 ≠ 𝑃 ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢))))
99	98	impl 461	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)))
100	99	adantrlr 736	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)))
101	5, 7	atncmp 40185	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ AtLat ∧ 𝑇 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (¬ 𝑇 ≤ 𝑆 ↔ 𝑇 ≠ 𝑆))
102	55, 11, 56, 101	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (¬ 𝑇 ≤ 𝑆 ↔ 𝑇 ≠ 𝑆))
103		necom 3008	. . . . . . . . . . . . . 14 ⊢ (𝑇 ≠ 𝑆 ↔ 𝑆 ≠ 𝑇)
104	102, 103	bitrdi 290	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (¬ 𝑇 ≤ 𝑆 ↔ 𝑆 ≠ 𝑇))
105	104	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (¬ 𝑇 ≤ 𝑆 ↔ 𝑆 ≠ 𝑇))
106		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝐾 ∈ HL)
107		simpl3r 1248	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑇 ∈ 𝐴)
108		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
109	68	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → 𝑆 ∈ (Base‘𝐾))
110	26, 5, 6, 7	hlexch1 40255	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑆 ∈ (Base‘𝐾)) ∧ ¬ 𝑇 ≤ 𝑆) → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇)))
111	110	3expia 1139	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑢 ∈ 𝐴 ∧ 𝑆 ∈ (Base‘𝐾))) → (¬ 𝑇 ≤ 𝑆 → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇))))
112	106, 107, 108, 109, 111	syl13anc 1399	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (¬ 𝑇 ≤ 𝑆 → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇))))
113	105, 112	sylbird 263	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) → (𝑆 ≠ 𝑇 → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇))))
114	113	imp 412	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑢 ∈ 𝐴) ∧ 𝑆 ≠ 𝑇) → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇)))
115	114	an32s 665	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑇) ∧ 𝑢 ∈ 𝐴) → (𝑇 ≤ (𝑆 ∨ 𝑢) → 𝑢 ≤ (𝑆 ∨ 𝑇)))
116	115	anim2d 624	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑇) ∧ 𝑢 ∈ 𝐴) → ((𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)) → (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
117	116	reximdva 3175	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑇) → (∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
118	117	ad2ant2rl 762	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) ∧ (¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇)) → (∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
119	118	adantrr 730	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → (∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑇 ≤ (𝑆 ∨ 𝑢)) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
120	100, 119	mpd 16	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))
121	120	ex 418	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ 𝑆 ≠ 𝑃) → (((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
122	23, 121	pm2.61dane 3042	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → (((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇))))
123	122	imp 412	1 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) ∧ ((¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑆 ≠ 𝑇) ∧ (𝑆 ≤ (𝑃 ∨ 𝑄) ∧ 𝑇 ≤ (𝑄 ∨ 𝑅)))) → ∃𝑢 ∈ 𝐴 (𝑢 ≤ (𝑃 ∨ 𝑅) ∧ 𝑢 ≤ (𝑆 ∨ 𝑇)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 class class class wbr 5103 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 lecple 17349 Posetcpo 18395 ltcplt 18396 joincjn 18399 0.cp0 18509 Latclat 18519 OPcops 40045 ⋖ ccvr 40135 Atomscatm 40136 AtLatcal 40137 HLchlt 40223

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-lat 18520 df-clat 18587 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224

This theorem is used by: ps-2b 40355 paddasslem3 40695

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