Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemb Structured version   Visualization version   GIF version

Theorem cdlemb 38303
Description: Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
Hypotheses
Ref Expression
cdlemb.b 𝐡 = (Baseβ€˜πΎ)
cdlemb.l ≀ = (leβ€˜πΎ)
cdlemb.j ∨ = (joinβ€˜πΎ)
cdlemb.u 1 = (1.β€˜πΎ)
cdlemb.c 𝐢 = ( β‹– β€˜πΎ)
cdlemb.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cdlemb (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
Distinct variable groups:   𝐴,π‘Ÿ   𝐡,π‘Ÿ   𝐢,π‘Ÿ   ∨ ,π‘Ÿ   𝐾,π‘Ÿ   ≀ ,π‘Ÿ   𝑃,π‘Ÿ   𝑄,π‘Ÿ   1 ,π‘Ÿ   𝑋,π‘Ÿ

Proof of Theorem cdlemb
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 simp11 1204 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ HL)
2 simp12 1205 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐴)
3 simp13 1206 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐴)
4 simp2l 1200 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋 ∈ 𝐡)
5 simp2r 1201 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 β‰  𝑄)
6 simp31 1210 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑋𝐢 1 )
7 simp32 1211 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑃 ≀ 𝑋)
8 cdlemb.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
9 cdlemb.l . . . . 5 ≀ = (leβ€˜πΎ)
10 cdlemb.j . . . . 5 ∨ = (joinβ€˜πΎ)
11 eqid 2733 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
12 cdlemb.u . . . . 5 1 = (1.β€˜πΎ)
13 cdlemb.c . . . . 5 𝐢 = ( β‹– β€˜πΎ)
14 cdlemb.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
158, 9, 10, 11, 12, 13, 141cvrat 37985 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 β‰  𝑄 ∧ 𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∈ 𝐴)
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1394 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∈ 𝐴)
171hllatd 37872 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝐾 ∈ Lat)
188, 14atbase 37797 . . . . . . 7 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
192, 18syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ∈ 𝐡)
208, 14atbase 37797 . . . . . . 7 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
213, 20syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑄 ∈ 𝐡)
228, 10latjcl 18333 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2317, 19, 21, 22syl3anc 1372 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
248, 9, 11latmle2 18359 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ≀ 𝑋)
2517, 23, 4, 24syl3anc 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ≀ 𝑋)
26 eqid 2733 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
278, 9, 26, 12, 13, 141cvratlt 37983 . . . 4 (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑋𝐢 1 ∧ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋)(ltβ€˜πΎ)𝑋)
281, 16, 4, 6, 25, 27syl32anc 1379 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋)(ltβ€˜πΎ)𝑋)
298, 26, 142atlt 37948 . . 3 (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋)(ltβ€˜πΎ)𝑋) β†’ βˆƒπ‘’ ∈ 𝐴 (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))
301, 16, 4, 28, 29syl31anc 1374 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘’ ∈ 𝐴 (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))
31 simpl11 1249 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝐾 ∈ HL)
32 simpl12 1250 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑃 ∈ 𝐴)
33 simprl 770 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑒 ∈ 𝐴)
34 simpl32 1256 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ Β¬ 𝑃 ≀ 𝑋)
35 simprrr 781 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑒(ltβ€˜πΎ)𝑋)
36 simpl2l 1227 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑋 ∈ 𝐡)
379, 26pltle 18227 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑒 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑒(ltβ€˜πΎ)𝑋 β†’ 𝑒 ≀ 𝑋))
3831, 33, 36, 37syl3anc 1372 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ (𝑒(ltβ€˜πΎ)𝑋 β†’ 𝑒 ≀ 𝑋))
3935, 38mpd 15 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑒 ≀ 𝑋)
40 breq1 5109 . . . . . . 7 (𝑃 = 𝑒 β†’ (𝑃 ≀ 𝑋 ↔ 𝑒 ≀ 𝑋))
4139, 40syl5ibrcom 247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ (𝑃 = 𝑒 β†’ 𝑃 ≀ 𝑋))
4241necon3bd 2954 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ 𝑃 β‰  𝑒))
4334, 42mpd 15 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ 𝑃 β‰  𝑒)
449, 10, 14hlsupr 37895 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑒 ∈ 𝐴) ∧ 𝑃 β‰  𝑒) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)))
4531, 32, 33, 43, 44syl31anc 1374 . . 3 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)))
46 eqid 2733 . . . . . . . 8 ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) = ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋)
478, 9, 10, 12, 13, 14, 26, 11, 46cdlemblem 38302 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋)) ∧ (π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)))) β†’ (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
48473exp 1120 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋)) β†’ ((π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒))) β†’ (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))))
4948exp4a 433 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ ((𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋)) β†’ (π‘Ÿ ∈ 𝐴 β†’ ((π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)) β†’ (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄))))))
5049imp 408 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ (π‘Ÿ ∈ 𝐴 β†’ ((π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)) β†’ (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))))
5150reximdvai 3159 . . 3 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑒 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑒)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄))))
5245, 51mpd 15 . 2 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ (𝑒 ∈ 𝐴 ∧ (𝑒 β‰  ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ∧ 𝑒(ltβ€˜πΎ)𝑋))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
5330, 52rexlimddv 3155 1 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 β‰  𝑄) ∧ (𝑋𝐢 1 ∧ Β¬ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑋 ∧ Β¬ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  ltcplt 18202  joincjn 18205  meetcmee 18206  1.cp1 18318  Latclat 18325   β‹– ccvr 37770  Atomscatm 37771  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-p1 18320  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  cdlemb2  38550
  Copyright terms: Public domain W3C validator