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Theorem cvrat4 34548
Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 in [PtakPulmannova] p. 68. Also Lemma 9.2(delta) in [MaedaMaeda] p. 41. (atcvat4i 29226 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat4.b 𝐵 = (Base‘𝐾)
cvrat4.l = (le‘𝐾)
cvrat4.j = (join‘𝐾)
cvrat4.z 0 = (0.‘𝐾)
cvrat4.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cvrat4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))
Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   ,𝑟   𝐾,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟
Allowed substitution hint:   0 (𝑟)

Proof of Theorem cvrat4
StepHypRef Expression
1 hlatl 34466 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
21adantr 481 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
3 simpr1 1065 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑋𝐵)
4 cvrat4.b . . . . . . . . . . 11 𝐵 = (Base‘𝐾)
5 cvrat4.l . . . . . . . . . . 11 = (le‘𝐾)
6 cvrat4.z . . . . . . . . . . 11 0 = (0.‘𝐾)
7 cvrat4.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
84, 5, 6, 7atlex 34422 . . . . . . . . . 10 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑟𝐴 𝑟 𝑋)
983exp 1262 . . . . . . . . 9 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋)))
102, 3, 9sylc 65 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
1110adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
12 simpll 789 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
13 simplr3 1103 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄𝐴)
14 simpr 477 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑟𝐴)
15 cvrat4.j . . . . . . . . . . . . . . 15 = (join‘𝐾)
165, 15, 7hlatlej1 34480 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑟𝐴) → 𝑄 (𝑄 𝑟))
1712, 13, 14, 16syl3anc 1324 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄 (𝑄 𝑟))
18 breq1 4647 . . . . . . . . . . . . 13 (𝑃 = 𝑄 → (𝑃 (𝑄 𝑟) ↔ 𝑄 (𝑄 𝑟)))
1917, 18syl5ibr 236 . . . . . . . . . . . 12 (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑃 (𝑄 𝑟)))
2019expd 452 . . . . . . . . . . 11 (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑟𝐴𝑃 (𝑄 𝑟))))
2120impcom 446 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴𝑃 (𝑄 𝑟)))
2221anim2d 588 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 𝑋𝑟𝐴) → (𝑟 𝑋𝑃 (𝑄 𝑟))))
2322expcomd 454 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴 → (𝑟 𝑋 → (𝑟 𝑋𝑃 (𝑄 𝑟)))))
2423reximdvai 3012 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (∃𝑟𝐴 𝑟 𝑋 → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))
2511, 24syld 47 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑋0 → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))
2625ex 450 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → (𝑋0 → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
2726a1i 11 . . . 4 (𝑃 (𝑋 𝑄) → ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → (𝑋0 → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))))
2827com4l 92 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → (𝑋0 → (𝑃 (𝑋 𝑄) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))))
2928imp4a 613 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
30 hllat 34469 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3130adantr 481 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
32 simpr3 1067 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
334, 7atbase 34395 . . . . . . . . . . . . . 14 (𝑄𝐴𝑄𝐵)
3432, 33syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
354, 5, 15latleeqj2 17045 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3631, 34, 3, 35syl3anc 1324 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3736biimpa 501 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑋 𝑄) = 𝑋)
3837breq2d 4656 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) ↔ 𝑃 𝑋))
3938biimpa 501 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 𝑋)
4039expl 647 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑃 𝑋))
41 simpl 473 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
42 simpr2 1066 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
435, 15, 7hlatlej2 34481 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑃𝐴) → 𝑃 (𝑄 𝑃))
4441, 32, 42, 43syl3anc 1324 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃 (𝑄 𝑃))
4540, 44jctird 566 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃 𝑋𝑃 (𝑄 𝑃))))
4645, 42jctild 565 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃)))))
4746impl 649 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃))))
48 breq1 4647 . . . . . . 7 (𝑟 = 𝑃 → (𝑟 𝑋𝑃 𝑋))
49 oveq2 6643 . . . . . . . 8 (𝑟 = 𝑃 → (𝑄 𝑟) = (𝑄 𝑃))
5049breq2d 4656 . . . . . . 7 (𝑟 = 𝑃 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑃)))
5148, 50anbi12d 746 . . . . . 6 (𝑟 = 𝑃 → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ (𝑃 𝑋𝑃 (𝑄 𝑃))))
5251rspcev 3304 . . . . 5 ((𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃))) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
5347, 52syl 17 . . . 4 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
5453adantrl 751 . . 3 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ (𝑋0𝑃 (𝑋 𝑄))) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
5554exp31 629 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
56 simpr 477 . . 3 ((𝑋0𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
57 ioran 511 . . . . 5 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
58 df-ne 2792 . . . . . 6 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
5958anbi1i 730 . . . . 5 ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
6057, 59bitr4i 267 . . . 4 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (𝑃𝑄 ∧ ¬ 𝑄 𝑋))
61 eqid 2620 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
624, 5, 15, 61, 7cvrat3 34547 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
63623expd 1282 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))))
6463imp4c 616 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
654, 7atbase 34395 . . . . . . . . . . . . 13 (𝑃𝐴𝑃𝐵)
6642, 65syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
674, 15latjcl 17032 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
6831, 66, 34, 67syl3anc 1324 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
694, 5, 61latmle1 17057 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7031, 3, 68, 69syl3anc 1324 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7170adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
72 simpll 789 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝐾 ∈ HL)
7363imp44 621 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
74 simplr2 1102 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑃𝐴)
7534adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑄𝐵)
7673, 74, 753jca 1240 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵))
7772, 76jca 554 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)))
784, 5, 61, 6, 7atnle 34423 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑋𝐵) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
792, 32, 3, 78syl3anc 1324 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
804, 61latmcom 17056 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8131, 34, 3, 80syl3anc 1324 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8281eqeq1d 2622 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
8379, 82bitrd 268 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
844, 61latmcl 17033 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8531, 3, 68, 84syl3anc 1324 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8685, 3, 343jca 1240 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵))
8731, 86jca 554 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵)))
884, 5, 61latmlem2 17063 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑄(meet‘𝐾)𝑋)))
8987, 70, 88sylc 65 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑄(meet‘𝐾)𝑋))
9089, 81breqtrd 4670 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑋(meet‘𝐾)𝑄))
91 breq2 4648 . . . . . . . . . . . . . . . 16 ((𝑋(meet‘𝐾)𝑄) = 0 → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑋(meet‘𝐾)𝑄) ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ))
9290, 91syl5ibcom 235 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ))
93 hlop 34468 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ HL → 𝐾 ∈ OP)
9493adantr 481 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ OP)
954, 61latmcl 17033 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
9631, 34, 85, 95syl3anc 1324 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
974, 5, 6ople0 34293 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
9894, 96, 97syl2anc 692 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
9992, 98sylibd 229 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
10083, 99sylbid 230 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
101100imp 445 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
102101adantrl 751 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
103102adantrr 752 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
1044, 5, 61latmle2 17058 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
10531, 3, 68, 104syl3anc 1324 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
1064, 15latjcom 17040 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
10731, 66, 34, 106syl3anc 1324 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
108105, 107breqtrd 4670 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
109108adantr 481 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
11030adantr 481 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝐾 ∈ Lat)
111 simpr3 1067 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝑄𝐵)
112 simpr1 1065 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
1134, 7atbase 34395 . . . . . . . . . . . . . 14 ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴 → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
114112, 113syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
1154, 61latmcom 17056 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 𝑄))(meet‘𝐾)𝑄))
116110, 111, 114, 115syl3anc 1324 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = ((𝑋(meet‘𝐾)(𝑃 𝑄))(meet‘𝐾)𝑄))
117116eqeq1d 2622 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ↔ ((𝑋(meet‘𝐾)(𝑃 𝑄))(meet‘𝐾)𝑄) = 0 ))
1184, 5, 15, 61, 6, 7hlexch3 34496 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵) ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄))(meet‘𝐾)𝑄) = 0 ) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃) → 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
1191183expia 1265 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (((𝑋(meet‘𝐾)(𝑃 𝑄))(meet‘𝐾)𝑄) = 0 → ((𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃) → 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
120117, 119sylbid 230 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 → ((𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃) → 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
12177, 103, 109, 120syl3c 66 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))
12271, 121jca 554 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
123122ex 450 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
12464, 123jcad 555 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))))
125 breq1 4647 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑟 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋))
126 oveq2 6643 . . . . . . . . 9 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑄 𝑟) = (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))
127126breq2d 4656 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
128125, 127anbi12d 746 . . . . . . 7 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
129128rspcev 3304 . . . . . 6 (((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴 ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
130124, 129syl6 35 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))
131130expd 452 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
13260, 131syl5bi 232 . . 3 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ (𝑃 = 𝑄𝑄 𝑋) → (𝑃 (𝑋 𝑄) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
13356, 132syl7 74 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ (𝑃 = 𝑄𝑄 𝑋) → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
13429, 55, 133ecase3d 983 1 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wrex 2910   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  lecple 15929  joincjn 16925  meetcmee 16926  0.cp0 17018  Latclat 17026  OPcops 34278  Atomscatm 34369  AtLatcal 34370  HLchlt 34456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-glb 16956  df-join 16957  df-meet 16958  df-p0 17020  df-lat 17027  df-clat 17089  df-oposet 34282  df-ol 34284  df-oml 34285  df-covers 34372  df-ats 34373  df-atl 34404  df-cvlat 34428  df-hlat 34457
This theorem is referenced by:  cvrat42  34549  ps-2  34583
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