Theorem lneq2at 38241

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 1203	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ HL)
2		simp12 1204	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
3	1, 2	jca 512	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵))
4		simp13 1205	. . 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 38239	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ 𝑋 = (𝑟 ∨ 𝑠))))
11	10	biimpd 228	. . 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 1202	. . . . 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 1302	. . . . . . . 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 1305	. . . . . . . . 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 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))) → 𝑄 ∈ 𝐴)
17	15, 16	jca 512	. . . . . . . 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 1307	. . . . . . 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 512	. . . . . 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 37826	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝐾 ∈ Lat)
23		simp21 1206	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
24	5, 7	atbase 37751	. . . . . . . . . . . 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 1207	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
27	5, 7	atbase 37751	. . . . . . . . . . . 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 512	. . . . . . . . 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 18339	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋) ↔ (𝑃 ∨ 𝑄) ≤ 𝑋))
34	33	biimpd 228	. . . . . . . . 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 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 1191	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL)
39		simpl2l 1226	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴)
40		simpl2r 1227	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴)
41		simpr 485	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
42		simpl3 1193	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
43	32, 6, 7	ps-1 37940	. . . . . . . 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 1383	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑄) ≤ (𝑟 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))
45	44	biimpd 228	. . . . . 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 2779	. . . 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 3204	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  Latclat 18320  Atomscatm 37725  HLchlt 37812  Linesclines 37957  pmapcpmap 37960

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-lines 37964  df-pmap 37967

This theorem is referenced by:  lnjatN  38243  lncmp  38246  cdlema1N  38254