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Theorem cvrat4 35519
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 29812 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 35436 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
21adantr 474 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
3 simpr1 1254 . . . . . . . . 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 35392 . . . . . . . . . 10 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑟𝐴 𝑟 𝑋)
983exp 1154 . . . . . . . . 9 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋)))
102, 3, 9sylc 65 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
1110adantr 474 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
12 simpll 785 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
13 simplr3 1285 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄𝐴)
14 simpr 479 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑟𝐴)
15 cvrat4.j . . . . . . . . . . . . . . 15 = (join‘𝐾)
165, 15, 7hlatlej1 35451 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑟𝐴) → 𝑄 (𝑄 𝑟))
1712, 13, 14, 16syl3anc 1496 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄 (𝑄 𝑟))
18 breq1 4877 . . . . . . . . . . . . 13 (𝑃 = 𝑄 → (𝑃 (𝑄 𝑟) ↔ 𝑄 (𝑄 𝑟)))
1917, 18syl5ibr 238 . . . . . . . . . . . 12 (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑃 (𝑄 𝑟)))
2019expd 406 . . . . . . . . . . 11 (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑟𝐴𝑃 (𝑄 𝑟))))
2120impcom 398 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴𝑃 (𝑄 𝑟)))
2221anim2d 607 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 𝑋𝑟𝐴) → (𝑟 𝑋𝑃 (𝑄 𝑟))))
2322expcomd 408 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴 → (𝑟 𝑋 → (𝑟 𝑋𝑃 (𝑄 𝑟)))))
2423reximdvai 3224 . . . . . . 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 403 . . . . 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 415 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
30 hllat 35439 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3130adantr 474 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
32 simpr3 1258 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
334, 7atbase 35365 . . . . . . . . . . . . . 14 (𝑄𝐴𝑄𝐵)
3432, 33syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
354, 5, 15latleeqj2 17418 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3631, 34, 3, 35syl3anc 1496 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3736biimpa 470 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑋 𝑄) = 𝑋)
3837breq2d 4886 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) ↔ 𝑃 𝑋))
3938biimpa 470 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 𝑋)
4039expl 451 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑃 𝑋))
41 simpl 476 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
42 simpr2 1256 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
435, 15, 7hlatlej2 35452 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑃𝐴) → 𝑃 (𝑄 𝑃))
4441, 32, 42, 43syl3anc 1496 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃 (𝑄 𝑃))
4540, 44jctird 524 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃 𝑋𝑃 (𝑄 𝑃))))
4645, 42jctild 523 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃)))))
4746impl 449 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃))))
48 breq1 4877 . . . . . . 7 (𝑟 = 𝑃 → (𝑟 𝑋𝑃 𝑋))
49 oveq2 6914 . . . . . . . 8 (𝑟 = 𝑃 → (𝑄 𝑟) = (𝑄 𝑃))
5049breq2d 4886 . . . . . . 7 (𝑟 = 𝑃 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑃)))
5148, 50anbi12d 626 . . . . . 6 (𝑟 = 𝑃 → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ (𝑃 𝑋𝑃 (𝑄 𝑃))))
5251rspcev 3527 . . . . 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 709 . . 3 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ (𝑋0𝑃 (𝑋 𝑄))) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
5554exp31 412 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
56 simpr 479 . . 3 ((𝑋0𝑃 (𝑋 𝑄)) → 𝑃 (𝑋 𝑄))
57 ioran 1013 . . . . 5 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
58 df-ne 3001 . . . . . 6 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
5958anbi1i 619 . . . . 5 ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
6057, 59bitr4i 270 . . . 4 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (𝑃𝑄 ∧ ¬ 𝑄 𝑋))
61 eqid 2826 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
624, 5, 15, 61, 7cvrat3 35518 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
63623expd 1468 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))))
6463imp4c 416 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
654, 7atbase 35365 . . . . . . . . . . . . 13 (𝑃𝐴𝑃𝐵)
6642, 65syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
674, 15latjcl 17405 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
6831, 66, 34, 67syl3anc 1496 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
694, 5, 61latmle1 17430 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7031, 3, 68, 69syl3anc 1496 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7170adantr 474 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
72 simpll 785 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝐾 ∈ HL)
7363imp44 421 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
74 simplr2 1283 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑃𝐴)
7534adantr 474 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑄𝐵)
7673, 74, 753jca 1164 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵))
7772, 76jca 509 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)))
784, 5, 61, 6, 7atnle 35393 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑋𝐵) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
792, 32, 3, 78syl3anc 1496 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
804, 61latmcom 17429 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8131, 34, 3, 80syl3anc 1496 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8281eqeq1d 2828 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
8379, 82bitrd 271 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
844, 61latmcl 17406 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8531, 3, 68, 84syl3anc 1496 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8685, 3, 343jca 1164 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵))
8731, 86jca 509 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵)))
884, 5, 61latmlem2 17436 . . . . . . . . . . . . . . . . . 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 4900 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑋(meet‘𝐾)𝑄))
91 breq2 4878 . . . . . . . . . . . . . . . 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 237 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ))
93 hlop 35438 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ HL → 𝐾 ∈ OP)
9493adantr 474 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ OP)
954, 61latmcl 17406 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
9631, 34, 85, 95syl3anc 1496 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
974, 5, 6ople0 35263 . . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
9992, 98sylibd 231 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
10083, 99sylbid 232 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
101100imp 397 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
102101adantrl 709 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
103102adantrr 710 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
1044, 5, 61latmle2 17431 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
10531, 3, 68, 104syl3anc 1496 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
1064, 15latjcom 17413 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
10731, 66, 34, 106syl3anc 1496 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
108105, 107breqtrd 4900 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
109108adantr 474 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
11030adantr 474 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝐾 ∈ Lat)
111 simpr3 1258 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝑄𝐵)
112 simpr1 1254 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
1134, 7atbase 35365 . . . . . . . . . . . . . 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 17429 . . . . . . . . . . . . 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 1496 . . . . . . . . . . . 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 2828 . . . . . . . . . . 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 35467 . . . . . . . . . . . 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 1156 . . . . . . . . . . 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 232 . . . . . . . . . 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 509 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
123122ex 403 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
12464, 123jcad 510 . . . . . 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 4877 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑟 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋))
126 oveq2 6914 . . . . . . . . 9 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑄 𝑟) = (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))
127126breq2d 4886 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
128125, 127anbi12d 626 . . . . . . 7 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
129128rspcev 3527 . . . . . 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 406 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
13260, 131syl5bi 234 . . 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 1063 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 198  wa 386  wo 880  w3a 1113   = wceq 1658  wcel 2166  wne 3000  wrex 3119   class class class wbr 4874  cfv 6124  (class class class)co 6906  Basecbs 16223  lecple 16313  joincjn 17298  meetcmee 17299  0.cp0 17391  Latclat 17399  OPcops 35248  Atomscatm 35339  AtLatcal 35340  HLchlt 35426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-proset 17282  df-poset 17300  df-plt 17312  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-p0 17393  df-lat 17400  df-clat 17462  df-oposet 35252  df-ol 35254  df-oml 35255  df-covers 35342  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427
This theorem is referenced by:  cvrat42  35520  ps-2  35554
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