Theorem lneq2at 39802

Description: A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)

Hypotheses

Ref	Expression
lneq2at.b	⊢ 𝐵 = (Base‘𝐾)
lneq2at.l	⊢ ≤ = (le‘𝐾)
lneq2at.j	⊢ ∨ = (join‘𝐾)
lneq2at.a	⊢ 𝐴 = (Atoms‘𝐾)
lneq2at.n	⊢ 𝑁 = (Lines‘𝐾)
lneq2at.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
lneq2at	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 = (𝑃 ∨ 𝑄))

Proof of Theorem lneq2at

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1204	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simp12 1205	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
3	1, 2	jca 511	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵))
4		simp13 1206	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝑀‘𝑋) ∈ 𝑁)
5		lneq2at.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
6		lneq2at.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
7		lneq2at.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
8		lneq2at.n	. . . . 5 ⊢ 𝑁 = (Lines‘𝐾)
9		lneq2at.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
10	5, 6, 7, 8, 9	isline3 39800	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))))
11	10	biimpd 229	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))))
12	3, 4, 11	sylc 65	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠)))
13		simp3r 1203	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝑋 = (𝑟 ∨ 𝑠))
14		simp111 1303	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝐾 ∈ HL)
15		simp121 1306	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝑃 ∈ 𝐴)
16		simp122 1307	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝑄 ∈ 𝐴)
17	15, 16	jca 511	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
18		simp2 1137	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
19	14, 17, 18	3jca 1128	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)))
20		simp123 1308	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝑃 ≠ 𝑄)
21	19, 20	jca 511	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄))
22	1	hllatd 39387	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat)
23		simp21 1207	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
24	5, 7	atbase 39312	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐵)
26		simp22 1208	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
27	5, 7	atbase 39312	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐵)
29	25, 28, 2	3jca 1128	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
30	22, 29	jca 511	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)))
31		simp3 1138	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋))
32		lneq2at.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
33	5, 32, 6	latjle12 18465	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃 ∨ 𝑄) ≤ 𝑋))
34	33	biimpd 229	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) → (𝑃 ∨ 𝑄) ≤ 𝑋))
35	30, 31, 34	sylc 65	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝑃 ∨ 𝑄) ≤ 𝑋)
36	35	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝑃 ∨ 𝑄) ≤ 𝑋)
37	36, 13	breqtrd 5150	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠))
38		simpl1 1192	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
39		simpl2l 1227	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
40		simpl2r 1228	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
41		simpr 484	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
42		simpl3 1194	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
43	32, 6, 7	ps-1 39501	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))
44	38, 39, 40, 41, 42, 43	syl131anc 1385	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))
45	44	biimpd 229	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠) → (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))
46	21, 37, 45	sylc 65	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠))
47	13, 46	eqtr4d 2774	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))) → 𝑋 = (𝑃 ∨ 𝑄))
48	47	3exp 1119	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → ((𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) → ((𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠)) → 𝑋 = (𝑃 ∨ 𝑄))))
49	48	rexlimdvv 3201	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠)) → 𝑋 = (𝑃 ∨ 𝑄)))
50	12, 49	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 = (𝑃 ∨ 𝑄))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  lecple 17283  joincjn 18328  Latclat 18446  Atomscatm 39286  HLchlt 39373  Linesclines 39518  pmapcpmap 39521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  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 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18311  df-poset 18330  df-plt 18345  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-p0 18440  df-lat 18447  df-clat 18514  df-oposet 39199  df-ol 39201  df-oml 39202  df-covers 39289  df-ats 39290  df-atl 39321  df-cvlat 39345  df-hlat 39374  df-lines 39525  df-pmap 39528

This theorem is referenced by:  lnjatN  39804  lncmp  39807  cdlema1N  39815