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Theorem lneq2at 39161
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 𝐡 = (Baseβ€˜πΎ)
lneq2at.l ≀ = (leβ€˜πΎ)
lneq2at.j ∨ = (joinβ€˜πΎ)
lneq2at.a 𝐴 = (Atomsβ€˜πΎ)
lneq2at.n 𝑁 = (Linesβ€˜πΎ)
lneq2at.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
lneq2at (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 = (𝑃 ∨ 𝑄))

Proof of Theorem lneq2at
Dummy variables 𝑠 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
2 simp12 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
31, 2jca 511 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡))
4 simp13 1202 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (π‘€β€˜π‘‹) ∈ 𝑁)
5 lneq2at.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
6 lneq2at.j . . . . 5 ∨ = (joinβ€˜πΎ)
7 lneq2at.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
8 lneq2at.n . . . . 5 𝑁 = (Linesβ€˜πΎ)
9 lneq2at.m . . . . 5 𝑀 = (pmapβ€˜πΎ)
105, 6, 7, 8, 9isline3 39159 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))))
1110biimpd 228 . . 3 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 β†’ βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))))
123, 4, 11sylc 65 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠)))
13 simp3r 1199 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝑋 = (π‘Ÿ ∨ 𝑠))
14 simp111 1299 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝐾 ∈ HL)
15 simp121 1302 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝑃 ∈ 𝐴)
16 simp122 1303 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝑄 ∈ 𝐴)
1715, 16jca 511 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
18 simp2 1134 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
1914, 17, 183jca 1125 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)))
20 simp123 1304 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝑃 β‰  𝑄)
2119, 20jca 511 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄))
221hllatd 38746 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
23 simp21 1203 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐴)
245, 7atbase 38671 . . . . . . . . . . . 12 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
2523, 24syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐡)
26 simp22 1204 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
275, 7atbase 38671 . . . . . . . . . . . 12 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
2826, 27syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
2925, 28, 23jca 1125 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡))
3022, 29jca 511 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)))
31 simp3 1135 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋))
32 lneq2at.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
335, 32, 6latjle12 18412 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) ↔ (𝑃 ∨ 𝑄) ≀ 𝑋))
3433biimpd 228 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋) β†’ (𝑃 ∨ 𝑄) ≀ 𝑋))
3530, 31, 34sylc 65 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (𝑃 ∨ 𝑄) ≀ 𝑋)
36353ad2ant1 1130 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (𝑃 ∨ 𝑄) ≀ 𝑋)
3736, 13breqtrd 5167 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (𝑃 ∨ 𝑄) ≀ (π‘Ÿ ∨ 𝑠))
38 simpl1 1188 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ 𝐾 ∈ HL)
39 simpl2l 1223 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 ∈ 𝐴)
40 simpl2r 1224 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ 𝑄 ∈ 𝐴)
41 simpr 484 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ 𝑃 β‰  𝑄)
42 simpl3 1190 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
4332, 6, 7ps-1 38860 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ≀ (π‘Ÿ ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (π‘Ÿ ∨ 𝑠)))
4438, 39, 40, 41, 42, 43syl131anc 1380 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑄) ≀ (π‘Ÿ ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (π‘Ÿ ∨ 𝑠)))
4544biimpd 228 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ 𝑄) ≀ (π‘Ÿ ∨ 𝑠) β†’ (𝑃 ∨ 𝑄) = (π‘Ÿ ∨ 𝑠)))
4621, 37, 45sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ (𝑃 ∨ 𝑄) = (π‘Ÿ ∨ 𝑠))
4713, 46eqtr4d 2769 . . . 4 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠))) β†’ 𝑋 = (𝑃 ∨ 𝑄))
48473exp 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ ((π‘Ÿ ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) β†’ ((π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠)) β†’ 𝑋 = (𝑃 ∨ 𝑄))))
4948rexlimdvv 3204 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 βˆƒπ‘  ∈ 𝐴 (π‘Ÿ β‰  𝑠 ∧ 𝑋 = (π‘Ÿ ∨ 𝑠)) β†’ 𝑋 = (𝑃 ∨ 𝑄)))
5012, 49mpd 15 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (π‘€β€˜π‘‹) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 ≀ 𝑋 ∧ 𝑄 ≀ 𝑋)) β†’ 𝑋 = (𝑃 ∨ 𝑄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  lecple 17210  joincjn 18273  Latclat 18393  Atomscatm 38645  HLchlt 38732  Linesclines 38877  pmapcpmap 38880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-lines 38884  df-pmap 38887
This theorem is referenced by:  lnjatN  39163  lncmp  39166  cdlema1N  39174
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