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

Theorem cdlemblem 37082
Description: Lemma for cdlemb 37083. (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 1306 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑃 𝑋)
2 simp111 1299 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ HL)
3 simp2l 1196 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝐴)
4 simp12l 1283 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑋𝐵)
52, 3, 43jca 1125 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ 𝑢𝐴𝑋𝐵))
6 simp2rr 1240 . . . . . 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 17566 . . . . . 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)
112hllatd 36653 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ Lat)
12 simp3l 1198 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝐴)
13 cdlemb.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
14 cdlemb.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
1513, 14atbase 36578 . . . . . . . 8 (𝑟𝐴𝑟𝐵)
1612, 15syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝐵)
1713, 14atbase 36578 . . . . . . . 8 (𝑢𝐴𝑢𝐵)
183, 17syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝐵)
19 cdlemb.j . . . . . . . 8 = (join‘𝐾)
2013, 7, 19latjle12 17667 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑟𝐵𝑢𝐵𝑋𝐵)) → ((𝑟 𝑋𝑢 𝑋) ↔ (𝑟 𝑢) 𝑋))
2111, 16, 18, 4, 20syl13anc 1369 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑋𝑢 𝑋) ↔ (𝑟 𝑢) 𝑋))
2221biimpd 232 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑋𝑢 𝑋) → (𝑟 𝑢) 𝑋))
2310, 22mpan2d 693 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑋 → (𝑟 𝑢) 𝑋))
24 simp112 1300 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝐴)
2512, 24, 33jca 1125 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟𝐴𝑃𝐴𝑢𝐴))
26 simp3r2 1279 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝑢)
272, 25, 263jca 1125 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟𝐴𝑃𝐴𝑢𝐴) ∧ 𝑟𝑢))
28 simp3r3 1280 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟 (𝑃 𝑢))
297, 19, 14hlatexch2 36685 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑃𝐴𝑢𝐴) ∧ 𝑟𝑢) → (𝑟 (𝑃 𝑢) → 𝑃 (𝑟 𝑢)))
3027, 28, 29sylc 65 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃 (𝑟 𝑢))
3113, 14atbase 36578 . . . . . . 7 (𝑃𝐴𝑃𝐵)
3224, 31syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝐵)
3313, 19latjcl 17656 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑟𝐵𝑢𝐵) → (𝑟 𝑢) ∈ 𝐵)
3411, 16, 18, 33syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑢) ∈ 𝐵)
3513, 7lattr 17661 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐵 ∧ (𝑟 𝑢) ∈ 𝐵𝑋𝐵)) → ((𝑃 (𝑟 𝑢) ∧ (𝑟 𝑢) 𝑋) → 𝑃 𝑋))
3611, 32, 34, 4, 35syl13anc 1369 . . . . 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))
3730, 36mpand 694 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑟 𝑢) 𝑋𝑃 𝑋))
3823, 37syld 47 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 𝑋𝑃 𝑋))
391, 38mtod 201 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑟 𝑋)
40 simp2rl 1239 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢𝑉)
41 simp113 1301 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑄𝐴)
42 simp3r1 1278 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑟𝑃)
437, 19, 14hlatexchb1 36682 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑄𝐴𝑃𝐴) ∧ 𝑟𝑃) → (𝑟 (𝑃 𝑄) ↔ (𝑃 𝑟) = (𝑃 𝑄)))
442, 12, 41, 24, 42, 43syl131anc 1380 . . . . . . . 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)))
4512, 3, 243jca 1125 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟𝐴𝑢𝐴𝑃𝐴))
462, 45, 423jca 1125 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝐾 ∈ HL ∧ (𝑟𝐴𝑢𝐴𝑃𝐴) ∧ 𝑟𝑃))
477, 19, 14hlatexch1 36684 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑟𝐴𝑢𝐴𝑃𝐴) ∧ 𝑟𝑃) → (𝑟 (𝑃 𝑢) → 𝑢 (𝑃 𝑟)))
4846, 28, 47sylc 65 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑢 (𝑃 𝑟))
49 breq2 5037 . . . . . . . . 9 ((𝑃 𝑟) = (𝑃 𝑄) → (𝑢 (𝑃 𝑟) ↔ 𝑢 (𝑃 𝑄)))
5048, 49syl5ibcom 248 . . . . . . . 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)))
5144, 50sylbid 243 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 (𝑃 𝑄)))
5251, 10jctird 530 . . . . . 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)))
5313, 14atbase 36578 . . . . . . . . . 10 (𝑄𝐴𝑄𝐵)
5441, 53syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑄𝐵)
5513, 19latjcl 17656 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
5611, 32, 54, 55syl3anc 1368 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑃 𝑄) ∈ 𝐵)
57 cdlemblem.m . . . . . . . . 9 = (meet‘𝐾)
5813, 7, 57latlem12 17683 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑢𝐵 ∧ (𝑃 𝑄) ∈ 𝐵𝑋𝐵)) → ((𝑢 (𝑃 𝑄) ∧ 𝑢 𝑋) ↔ 𝑢 ((𝑃 𝑄) 𝑋)))
5911, 18, 56, 4, 58syl13anc 1369 . . . . . . 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)))
60 cdlemblem.v . . . . . . . 8 𝑉 = ((𝑃 𝑄) 𝑋)
6160breq2i 5041 . . . . . . 7 (𝑢 𝑉𝑢 ((𝑃 𝑄) 𝑋))
6259, 61syl6bbr 292 . . . . . 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))
6352, 62sylibd 242 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 𝑉))
64 hlatl 36649 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
652, 64syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝐾 ∈ AtLat)
66 simp12r 1284 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑃𝑄)
67 simp131 1305 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑋𝐶 1 )
68 cdlemb.u . . . . . . . . 9 1 = (1.‘𝐾)
69 cdlemb.c . . . . . . . . 9 𝐶 = ( ⋖ ‘𝐾)
7013, 7, 19, 57, 68, 69, 141cvrat 36765 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑋𝐵) ∧ (𝑃𝑄𝑋𝐶 1 ∧ ¬ 𝑃 𝑋)) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
712, 24, 41, 4, 66, 67, 1, 70syl133anc 1390 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ((𝑃 𝑄) 𝑋) ∈ 𝐴)
7260, 71eqeltrid 2897 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → 𝑉𝐴)
737, 14atcmp 36600 . . . . . 6 ((𝐾 ∈ AtLat ∧ 𝑢𝐴𝑉𝐴) → (𝑢 𝑉𝑢 = 𝑉))
7465, 3, 72, 73syl3anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑢 𝑉𝑢 = 𝑉))
7563, 74sylibd 242 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑟 (𝑃 𝑄) → 𝑢 = 𝑉))
7675necon3ad 3003 . . 3 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (𝑢𝑉 → ¬ 𝑟 (𝑃 𝑄)))
7740, 76mpd 15 . 2 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → ¬ 𝑟 (𝑃 𝑄))
7839, 77jca 515 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  ltcplt 17546  joincjn 17549  meetcmee 17550  1.cp1 17643  Latclat 17650  ccvr 36551  Atomscatm 36552  AtLatcal 36553  HLchlt 36639
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36465  df-ol 36467  df-oml 36468  df-covers 36555  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640
This theorem is referenced by:  cdlemb  37083
  Copyright terms: Public domain W3C validator