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Theorem cdlemblem 34556
 Description: Lemma for cdlemb 34557. (Contributed by NM, 8-May-2012.)
Hypotheses
Ref Expression
cdlemb.b 𝐵 = (Base‘𝐾)
cdlemb.l = (le‘𝐾)
cdlemb.j = (join‘𝐾)
cdlemb.u 1 = (1.‘𝐾)
cdlemb.c 𝐶 = ( ⋖ ‘𝐾)
cdlemb.a 𝐴 = (Atoms‘𝐾)
cdlemblem.s < = (lt‘𝐾)
cdlemblem.m = (meet‘𝐾)
cdlemblem.v 𝑉 = ((𝑃 𝑄) 𝑋)
Assertion
Ref Expression
cdlemblem ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))

Proof of Theorem cdlemblem
StepHypRef Expression
1 simp132 1195 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑃 𝑋)
2 simp111 1188 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ HL)
3 simp2l 1085 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝐴)
4 simp12l 1172 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑋𝐵)
52, 3, 43jca 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵))
6 simp2rr 1129 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢 < 𝑋)
7 cdlemb.l . . . . . . 7 = (le‘𝐾)
8 cdlemblem.s . . . . . . 7 < = (lt‘𝐾)
97, 8pltle 16882 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵) → (𝑢 < 𝑋𝑢 𝑋))
105, 6, 9sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢 𝑋)
11 hllat 34127 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ Lat)
122, 11syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ Lat)
13 simp3l 1087 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝐴)
14 cdlemb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
15 cdlemb.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
1614, 15atbase 34053 . . . . . . . 8 (𝑟𝐴𝑟𝐵)
1713, 16syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝐵)
1814, 15atbase 34053 . . . . . . . 8 (𝑢𝐴𝑢𝐵)
193, 18syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝐵)
20 cdlemb.j . . . . . . . 8 = (join‘𝐾)
2114, 7, 20latjle12 16983 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑟𝐵𝑢𝐵𝑋𝐵)) → ((𝑟 𝑋𝑢 𝑋) ↔ (𝑟 𝑢) 𝑋))
2212, 17, 19, 4, 21syl13anc 1325 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑋𝑢 𝑋) ↔ (𝑟 𝑢) 𝑋))
2322biimpd 219 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑋𝑢 𝑋) → (𝑟 𝑢) 𝑋))
2410, 23mpan2d 709 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑋 → (𝑟 𝑢) 𝑋))
25 simp112 1189 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝐴)
2613, 25, 33jca 1240 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟𝐴𝑃𝐴𝑢𝐴))
27 simp3r2 1168 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝑢)
282, 26, 273jca 1240 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟𝐴𝑃𝐴𝑢𝐴) ∧ 𝑟𝑢))
29 simp3r3 1169 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟 (𝑃 𝑢))
307, 20, 15hlatexch2 34159 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑃𝐴𝑢𝐴) ∧ 𝑟𝑢) → (𝑟 (𝑃 𝑢) → 𝑃 (𝑟 𝑢)))
3128, 29, 30sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃 (𝑟 𝑢))
3214, 15atbase 34053 . . . . . . 7 (𝑃𝐴𝑃𝐵)
3325, 32syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝐵)
3414, 20latjcl 16972 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑟𝐵𝑢𝐵) → (𝑟 𝑢) ∈ 𝐵)
3512, 17, 19, 34syl3anc 1323 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑢) ∈ 𝐵)
3614, 7lattr 16977 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑟 𝑢) ∈ 𝐵𝑋𝐵)) → ((𝑃 (𝑟 𝑢) ∧ (𝑟 𝑢) 𝑋) → 𝑃 𝑋))
3712, 33, 35, 4, 36syl13anc 1325 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑃 (𝑟 𝑢) ∧ (𝑟 𝑢) 𝑋) → 𝑃 𝑋))
3831, 37mpand 710 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑢) 𝑋𝑃 𝑋))
3924, 38syld 47 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑋𝑃 𝑋))
401, 39mtod 189 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑟 𝑋)
41 simp2rl 1128 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝑉)
42 simp113 1190 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑄𝐴)
43 simp3r1 1167 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝑃)
447, 20, 15hlatexchb1 34156 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑃𝐴) ∧ 𝑟𝑃) → (𝑟 (𝑃 𝑄) ↔ (𝑃 𝑟) = (𝑃 𝑄)))
452, 13, 42, 25, 43, 44syl131anc 1336 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) ↔ (𝑃 𝑟) = (𝑃 𝑄)))
4613, 3, 253jca 1240 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟𝐴𝑢𝐴𝑃𝐴))
472, 46, 433jca 1240 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟𝐴𝑢𝐴𝑃𝐴) ∧ 𝑟𝑃))
487, 20, 15hlatexch1 34158 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑢𝐴𝑃𝐴) ∧ 𝑟𝑃) → (𝑟 (𝑃 𝑢) → 𝑢 (𝑃 𝑟)))
4947, 29, 48sylc 65 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢 (𝑃 𝑟))
50 breq2 4617 . . . . . . . . 9 ((𝑃 𝑟) = (𝑃 𝑄) → (𝑢 (𝑃 𝑟) ↔ 𝑢 (𝑃 𝑄)))
5149, 50syl5ibcom 235 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑃 𝑟) = (𝑃 𝑄) → 𝑢 (𝑃 𝑄)))
5245, 51sylbid 230 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 (𝑃 𝑄)))
5352, 10jctird 566 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → (𝑢 (𝑃 𝑄) ∧ 𝑢 𝑋)))
5414, 15atbase 34053 . . . . . . . . . 10 (𝑄𝐴𝑄𝐵)
5542, 54syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑄𝐵)
5614, 20latjcl 16972 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
5712, 33, 55, 56syl3anc 1323 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑃 𝑄) ∈ 𝐵)
58 cdlemblem.m . . . . . . . . 9 = (meet‘𝐾)
5914, 7, 58latlem12 16999 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑢𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑢 (𝑃 𝑄) ∧ 𝑢 𝑋) ↔ 𝑢 ((𝑃 𝑄) 𝑋)))
6012, 19, 57, 4, 59syl13anc 1325 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑢 (𝑃 𝑄) ∧ 𝑢 𝑋) ↔ 𝑢 ((𝑃 𝑄) 𝑋)))
61 cdlemblem.v . . . . . . . 8 𝑉 = ((𝑃 𝑄) 𝑋)
6261breq2i 4621 . . . . . . 7 (𝑢 𝑉𝑢 ((𝑃 𝑄) 𝑋))
6360, 62syl6bbr 278 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑢 (𝑃 𝑄) ∧ 𝑢 𝑋) ↔ 𝑢 𝑉))
6453, 63sylibd 229 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 𝑉))
65 hlatl 34124 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
662, 65syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ AtLat)
67 simp12r 1173 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝑄)
68 simp131 1194 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑋𝐶 1 )
69 cdlemb.u . . . . . . . . 9 1 = (1.‘𝐾)
70 cdlemb.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
7114, 7, 20, 58, 69, 70, 151cvrat 34239 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
722, 25, 42, 4, 67, 68, 1, 71syl133anc 1346 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
7361, 72syl5eqel 2702 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑉𝐴)
747, 15atcmp 34075 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑢𝐴𝑉𝐴) → (𝑢 𝑉𝑢 = 𝑉))
7566, 3, 73, 74syl3anc 1323 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑢 𝑉𝑢 = 𝑉))
7664, 75sylibd 229 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 = 𝑉))
7776necon3ad 2803 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑢𝑉 → ¬ 𝑟 (𝑃 𝑄)))
7841, 77mpd 15 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑟 (𝑃 𝑄))
7940, 78jca 554 1 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790   class class class wbr 4613  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  ltcplt 16862  joincjn 16865  meetcmee 16866  1.cp1 16959  Latclat 16966   ⋖ ccvr 34026  Atomscatm 34027  AtLatcal 34028  HLchlt 34114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33940  df-ol 33942  df-oml 33943  df-covers 34030  df-ats 34031  df-atl 34062  df-cvlat 34086  df-hlat 34115 This theorem is referenced by:  cdlemb  34557
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