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Theorem cdlemb 38653
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 1203 . . 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 1204 . . . 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 1205 . . . 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 1199 . . . 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 1200 . . . 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 1209 . . . 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 1210 . . . 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 2732 . . . . 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 38335 . . . 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 1393 . . 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 38222 . . . . 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 38147 . . . . . . 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 38147 . . . . . . 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 18388 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
2317, 19, 21, 22syl3anc 1371 . . . . 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 18414 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) ≀ 𝑋)
2517, 23, 4, 24syl3anc 1371 . . . 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 2732 . . . . 5 (ltβ€˜πΎ) = (ltβ€˜πΎ)
278, 9, 26, 12, 13, 141cvratlt 38333 . . . 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 1378 . . 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 38298 . . 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 1373 . 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 1248 . . . 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 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†’ 𝑃 ∈ 𝐴)
33 simprl 769 . . . 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 1255 . . . . 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 780 . . . . . . . 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 1226 . . . . . . . . 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 18282 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑒 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑒(ltβ€˜πΎ)𝑋 β†’ 𝑒 ≀ 𝑋))
3831, 33, 36, 37syl3anc 1371 . . . . . . . 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 5150 . . . . . . 7 (𝑃 = 𝑒 β†’ (𝑃 ≀ 𝑋 ↔ 𝑒 ≀ 𝑋))
4139, 40syl5ibrcom 246 . . . . . 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 38245 . . . 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 1373 . . 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 2732 . . . . . . . 8 ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋) = ((𝑃 ∨ 𝑄)(meetβ€˜πΎ)𝑋)
478, 9, 10, 12, 13, 14, 26, 11, 46cdlemblem 38652 . . . . . . 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 1119 . . . . . 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 432 . . . . 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 407 . . . 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 3165 . . 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 3161 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  ltcplt 18257  joincjn 18260  meetcmee 18261  1.cp1 18373  Latclat 18380   β‹– ccvr 38120  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-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  cdlemb2  38900
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