Theorem lneq2at 40815

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 1222	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simp12 1223	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
3	1, 2	jca 521	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵))
4		simp13 1224	. . 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 40813	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))))
11	10	biimpd 232	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))))
12	3, 4, 11	sylc 66	. 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 1221	. . . . 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 1321	. . . . . . . 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 1324	. . . . . . . . 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 1325	. . . . . . . . 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 521	. . . . . . . 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 1155	. . . . . . . 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 1146	. . . . . . 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 1326	. . . . . . 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 521	. . . . . 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 40401	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat)
23		simp21 1225	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
24	5, 7	atbase 40326	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
25	23, 24	syl 18	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐵)
26		simp22 1226	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
27	5, 7	atbase 40326	. . . . . . . . . . . 12 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
28	26, 27	syl 18	. . . . . . . . . . 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 1146	. . . . . . . . . 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 521	. . . . . . . . 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 1156	. . . . . . . . 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 18617	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃 ∨ 𝑄) ≤ 𝑋))
34	33	biimpd 232	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) → (𝑃 ∨ 𝑄) ≤ 𝑋))
35	30, 31, 34	sylc 66	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝑃 ∨ 𝑄) ≤ 𝑋)
36	35	3ad2ant1 1151	. . . . . . 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 5131	. . . . . 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 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
39		simpl2l 1245	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
40		simpl2r 1246	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
41		simpr 490	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
42		simpl3 1212	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
43	32, 6, 7	ps-1 40514	. . . . . . . 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 1410	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))
45	44	biimpd 232	. . . . . 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 66	. . . . 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 2799	. . . 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 1137	. . 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 3219	. 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 16	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 = (𝑃 ∨ 𝑄))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  lecple 17428  joincjn 18478  Latclat 18598  Atomscatm 40300  HLchlt 40387  Linesclines 40531  pmapcpmap 40534

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-proset 18461  df-poset 18480  df-plt 18495  df-lub 18511  df-glb 18512  df-join 18513  df-meet 18514  df-p0 18590  df-lat 18599  df-clat 18666  df-oposet 40213  df-ol 40215  df-oml 40216  df-covers 40303  df-ats 40304  df-atl 40335  df-cvlat 40359  df-hlat 40388  df-lines 40538  df-pmap 40541

This theorem is used by:  lnjatN  40817  lncmp  40820  cdlema1N  40828