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Theorem lneq2at 39283
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 510 . . 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 39281 . . . 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 510 . . . . . . . 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 510 . . . . . 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 38868 . . . . . . . . . 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 38793 . . . . . . . . . . . 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 38793 . . . . . . . . . . . 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 510 . . . . . . . . 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 18449 . . . . . . . . . 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 5178 . . . . . 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 483 . . . . . . . 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 38982 . . . . . . . 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 2771 . . . 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 3208 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  Latclat 18430  Atomscatm 38767  HLchlt 38854  Linesclines 38999  pmapcpmap 39002
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-lines 39006  df-pmap 39009
This theorem is referenced by:  lnjatN  39285  lncmp  39288  cdlema1N  39296
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