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Theorem 4noncolr3 38312
Description: A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)
Hypotheses
Ref Expression
3noncol.l ≀ = (leβ€˜πΎ)
3noncol.j ∨ = (joinβ€˜πΎ)
3noncol.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
4noncolr3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑄 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
Proof of Theorem 4noncolr3
StepHypRef Expression
1 simp11 1203 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝐾 ∈ HL)
21hllatd 38222 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝐾 ∈ Lat)
3 simp2l 1199 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑅 ∈ 𝐴)
4 eqid 2732 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
5 3noncol.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38147 . . . . 5 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
73, 6syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
8 simp12 1204 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑃 ∈ 𝐴)
94, 5atbase 38147 . . . . 5 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
108, 9syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
11 simp13 1205 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑄 ∈ 𝐴)
124, 5atbase 38147 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
1311, 12syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
14 simp32 1210 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))
15 3noncol.l . . . . 5 ≀ = (leβ€˜πΎ)
16 3noncol.j . . . . 5 ∨ = (joinβ€˜πΎ)
174, 15, 16latnlej1r 18407 . . . 4 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑅 β‰  𝑄)
182, 7, 10, 13, 14, 17syl131anc 1383 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑅 β‰  𝑄)
1918necomd 2996 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑄 β‰  𝑅)
20 simp2r 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑆 ∈ 𝐴)
214, 5atbase 38147 . . . 4 (𝑆 ∈ 𝐴 β†’ 𝑆 ∈ (Baseβ€˜πΎ))
2220, 21syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑆 ∈ (Baseβ€˜πΎ))
234, 16latjcl 18388 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
242, 13, 7, 23syl3anc 1371 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ))
25 simp33 1211 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))
2616, 5hlatjass 38228 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
271, 8, 11, 3, 26syl13anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
2827breq2d 5159 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑆 ≀ (𝑃 ∨ (𝑄 ∨ 𝑅))))
2925, 28mtbid 323 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑆 ≀ (𝑃 ∨ (𝑄 ∨ 𝑅)))
304, 15, 16latnlej2r 18410 . . 3 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ (𝑄 ∨ 𝑅))) β†’ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅))
312, 22, 10, 24, 29, 30syl131anc 1383 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅))
32 simp31 1209 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ 𝑃 β‰  𝑄)
3315, 16, 5hlatexch1 38254 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ 𝑅 ≀ (𝑄 ∨ 𝑃)))
341, 8, 3, 11, 32, 33syl131anc 1383 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ 𝑅 ≀ (𝑄 ∨ 𝑃)))
354, 16latjcom 18396 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
362, 10, 13, 35syl3anc 1371 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
3736breq2d 5159 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑅 ≀ (𝑃 ∨ 𝑄) ↔ 𝑅 ≀ (𝑄 ∨ 𝑃)))
3834, 37sylibrd 258 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑃 ≀ (𝑄 ∨ 𝑅) β†’ 𝑅 ≀ (𝑃 ∨ 𝑄)))
3914, 38mtod 197 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅))
404, 15, 16, 5hlexch1 38241 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑄 ∨ 𝑅) ∈ (Baseβ€˜πΎ)) ∧ Β¬ 𝑃 ≀ (𝑄 ∨ 𝑅)) β†’ (𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆) β†’ 𝑆 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑃)))
411, 8, 20, 24, 39, 40syl131anc 1383 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆) β†’ 𝑆 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑃)))
424, 16latjcom 18396 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
432, 13, 7, 42syl3anc 1371 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
4443oveq1d 7420 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑅 ∨ 𝑄) ∨ 𝑃))
454, 16latj31 18436 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ) ∧ 𝑃 ∈ (Baseβ€˜πΎ))) β†’ ((𝑅 ∨ 𝑄) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
462, 7, 13, 10, 45syl13anc 1372 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ ((𝑅 ∨ 𝑄) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
4744, 46eqtrd 2772 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
4847breq2d 5159 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑆 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑃) ↔ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
4941, 48sylibd 238 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆) β†’ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
5025, 49mtod 197 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ Β¬ 𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆))
5119, 31, 503jca 1128 1 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))) β†’ (𝑄 β‰  𝑅 ∧ Β¬ 𝑆 ≀ (𝑄 ∨ 𝑅) ∧ Β¬ 𝑃 ≀ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  HLchlt 38208
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  4noncolr2  38313  4noncolr1  38314  4atlem12  38471
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