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Theorem cvrat4 35047
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 29384 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 34965 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
21adantr 480 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
3 simpr1 1087 . . . . . . . . 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 34921 . . . . . . . . . 10 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑟𝐴 𝑟 𝑋)
983exp 1283 . . . . . . . . 9 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋)))
102, 3, 9sylc 65 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
1110adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
12 simpll 805 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
13 simplr3 1125 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄𝐴)
14 simpr 476 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑟𝐴)
15 cvrat4.j . . . . . . . . . . . . . . 15 = (join‘𝐾)
165, 15, 7hlatlej1 34979 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑟𝐴) → 𝑄 (𝑄 𝑟))
1712, 13, 14, 16syl3anc 1366 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄 (𝑄 𝑟))
18 breq1 4688 . . . . . . . . . . . . 13 (𝑃 = 𝑄 → (𝑃 (𝑄 𝑟) ↔ 𝑄 (𝑄 𝑟)))
1917, 18syl5ibr 236 . . . . . . . . . . . 12 (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑃 (𝑄 𝑟)))
2019expd 451 . . . . . . . . . . 11 (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑟𝐴𝑃 (𝑄 𝑟))))
2120impcom 445 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴𝑃 (𝑄 𝑟)))
2221anim2d 588 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 𝑋𝑟𝐴) → (𝑟 𝑋𝑃 (𝑄 𝑟))))
2322expcomd 453 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴 → (𝑟 𝑋 → (𝑟 𝑋𝑃 (𝑄 𝑟)))))
2423reximdvai 3044 . . . . . . 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 449 . . . . 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 34968 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3130adantr 480 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
32 simpr3 1089 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
334, 7atbase 34894 . . . . . . . . . . . . . 14 (𝑄𝐴𝑄𝐵)
3432, 33syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
354, 5, 15latleeqj2 17111 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3631, 34, 3, 35syl3anc 1366 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3736biimpa 500 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑋 𝑄) = 𝑋)
3837breq2d 4697 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) ↔ 𝑃 𝑋))
3938biimpa 500 . . . . . . . . 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 472 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
42 simpr2 1088 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
435, 15, 7hlatlej2 34980 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑃𝐴) → 𝑃 (𝑄 𝑃))
4441, 32, 42, 43syl3anc 1366 . . . . . . . 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 4688 . . . . . . 7 (𝑟 = 𝑃 → (𝑟 𝑋𝑃 𝑋))
49 oveq2 6698 . . . . . . . 8 (𝑟 = 𝑃 → (𝑄 𝑟) = (𝑄 𝑃))
5049breq2d 4697 . . . . . . 7 (𝑟 = 𝑃 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑃)))
5148, 50anbi12d 747 . . . . . 6 (𝑟 = 𝑃 → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ (𝑃 𝑋𝑃 (𝑄 𝑃))))
5251rspcev 3340 . . . . 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 752 . . 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 476 . . 3 ((𝑋0𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
57 ioran 510 . . . . 5 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
58 df-ne 2824 . . . . . 6 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
5958anbi1i 731 . . . . 5 ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
6057, 59bitr4i 267 . . . 4 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (𝑃𝑄 ∧ ¬ 𝑄 𝑋))
61 eqid 2651 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
624, 5, 15, 61, 7cvrat3 35046 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
63623expd 1306 . . . . . . . 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 34894 . . . . . . . . . . . . 13 (𝑃𝐴𝑃𝐵)
6642, 65syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
674, 15latjcl 17098 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
6831, 66, 34, 67syl3anc 1366 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
694, 5, 61latmle1 17123 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7031, 3, 68, 69syl3anc 1366 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7170adantr 480 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
72 simpll 805 . . . . . . . . . . 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 1124 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑃𝐴)
7534adantr 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑄𝐵)
7673, 74, 753jca 1261 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵))
7772, 76jca 553 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)))
784, 5, 61, 6, 7atnle 34922 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑋𝐵) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
792, 32, 3, 78syl3anc 1366 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
804, 61latmcom 17122 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8131, 34, 3, 80syl3anc 1366 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8281eqeq1d 2653 . . . . . . . . . . . . . . 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 17099 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8531, 3, 68, 84syl3anc 1366 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8685, 3, 343jca 1261 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵))
8731, 86jca 553 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵)))
884, 5, 61latmlem2 17129 . . . . . . . . . . . . . . . . . 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 4711 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑋(meet‘𝐾)𝑄))
91 breq2 4689 . . . . . . . . . . . . . . . 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 34967 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ HL → 𝐾 ∈ OP)
9493adantr 480 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ OP)
954, 61latmcl 17099 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
9631, 34, 85, 95syl3anc 1366 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
974, 5, 6ople0 34792 . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
102101adantrl 752 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
103102adantrr 753 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
1044, 5, 61latmle2 17124 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
10531, 3, 68, 104syl3anc 1366 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
1064, 15latjcom 17106 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
10731, 66, 34, 106syl3anc 1366 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
108105, 107breqtrd 4711 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
109108adantr 480 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
11030adantr 480 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝐾 ∈ Lat)
111 simpr3 1089 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝑄𝐵)
112 simpr1 1087 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
1134, 7atbase 34894 . . . . . . . . . . . . . 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 17122 . . . . . . . . . . . . 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 1366 . . . . . . . . . . . 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 2653 . . . . . . . . . . 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 34995 . . . . . . . . . . . 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 1286 . . . . . . . . . . 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 553 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
123122ex 449 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
12464, 123jcad 554 . . . . . 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 4688 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑟 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋))
126 oveq2 6698 . . . . . . . . 9 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑄 𝑟) = (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))
127126breq2d 4697 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
128125, 127anbi12d 747 . . . . . . 7 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
129128rspcev 3340 . . . . . 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 451 . . . 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 1003 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 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  0.cp0 17084  Latclat 17092  OPcops 34777  Atomscatm 34868  AtLatcal 34869  HLchlt 34955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956
This theorem is referenced by:  cvrat42  35048  ps-2  35082
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