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Theorem cvrat4 35399
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 29712 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 35316 . . . . . . . . . 10 (𝐾 ∈ HL → 𝐾 ∈ AtLat)
21adantr 472 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ AtLat)
3 simpr1 1248 . . . . . . . . 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 35272 . . . . . . . . . 10 ((𝐾 ∈ AtLat ∧ 𝑋𝐵𝑋0 ) → ∃𝑟𝐴 𝑟 𝑋)
983exp 1148 . . . . . . . . 9 (𝐾 ∈ AtLat → (𝑋𝐵 → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋)))
102, 3, 9sylc 65 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
1110adantr 472 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑋0 → ∃𝑟𝐴 𝑟 𝑋))
12 simpll 783 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝐾 ∈ HL)
13 simplr3 1279 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄𝐴)
14 simpr 477 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑟𝐴)
15 cvrat4.j . . . . . . . . . . . . . . 15 = (join‘𝐾)
165, 15, 7hlatlej1 35331 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑟𝐴) → 𝑄 (𝑄 𝑟))
1712, 13, 14, 16syl3anc 1490 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑄 (𝑄 𝑟))
18 breq1 4812 . . . . . . . . . . . . 13 (𝑃 = 𝑄 → (𝑃 (𝑄 𝑟) ↔ 𝑄 (𝑄 𝑟)))
1917, 18syl5ibr 237 . . . . . . . . . . . 12 (𝑃 = 𝑄 → (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑟𝐴) → 𝑃 (𝑄 𝑟)))
2019expd 404 . . . . . . . . . . 11 (𝑃 = 𝑄 → ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑟𝐴𝑃 (𝑄 𝑟))))
2120impcom 396 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴𝑃 (𝑄 𝑟)))
2221anim2d 605 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → ((𝑟 𝑋𝑟𝐴) → (𝑟 𝑋𝑃 (𝑄 𝑟))))
2322expcomd 406 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑃 = 𝑄) → (𝑟𝐴 → (𝑟 𝑋 → (𝑟 𝑋𝑃 (𝑄 𝑟)))))
2423reximdvai 3161 . . . . . . 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 401 . . . . 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 413 . 2 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 = 𝑄 → ((𝑋0𝑃 (𝑋 𝑄)) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
30 hllat 35319 . . . . . . . . . . . . . 14 (𝐾 ∈ HL → 𝐾 ∈ Lat)
3130adantr 472 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ Lat)
32 simpr3 1252 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐴)
334, 7atbase 35245 . . . . . . . . . . . . . 14 (𝑄𝐴𝑄𝐵)
3432, 33syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑄𝐵)
354, 5, 15latleeqj2 17330 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3631, 34, 3, 35syl3anc 1490 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄 𝑋 ↔ (𝑋 𝑄) = 𝑋))
3736biimpa 468 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑋 𝑄) = 𝑋)
3837breq2d 4821 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) ↔ 𝑃 𝑋))
3938biimpa 468 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → 𝑃 𝑋)
4039expl 449 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → 𝑃 𝑋))
41 simpl 474 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ HL)
42 simpr2 1250 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐴)
435, 15, 7hlatlej2 35332 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑃𝐴) → 𝑃 (𝑄 𝑃))
4441, 32, 42, 43syl3anc 1490 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃 (𝑄 𝑃))
4540, 44jctird 522 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃 𝑋𝑃 (𝑄 𝑃))))
4645, 42jctild 521 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃)))))
4746impl 447 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑃𝐴 ∧ (𝑃 𝑋𝑃 (𝑄 𝑃))))
48 breq1 4812 . . . . . . 7 (𝑟 = 𝑃 → (𝑟 𝑋𝑃 𝑋))
49 oveq2 6850 . . . . . . . 8 (𝑟 = 𝑃 → (𝑄 𝑟) = (𝑄 𝑃))
5049breq2d 4821 . . . . . . 7 (𝑟 = 𝑃 → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 𝑃)))
5148, 50anbi12d 624 . . . . . 6 (𝑟 = 𝑃 → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ (𝑃 𝑋𝑃 (𝑄 𝑃))))
5251rspcev 3461 . . . . 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 707 . . 3 ((((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ 𝑄 𝑋) ∧ (𝑋0𝑃 (𝑋 𝑄))) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))
5554exp31 410 . 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 1006 . . . . 5 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
58 df-ne 2938 . . . . . 6 (𝑃𝑄 ↔ ¬ 𝑃 = 𝑄)
5958anbi1i 617 . . . . 5 ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ↔ (¬ 𝑃 = 𝑄 ∧ ¬ 𝑄 𝑋))
6057, 59bitr4i 269 . . . 4 (¬ (𝑃 = 𝑄𝑄 𝑋) ↔ (𝑃𝑄 ∧ ¬ 𝑄 𝑋))
61 eqid 2765 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
624, 5, 15, 61, 7cvrat3 35398 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
63623expd 1462 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃𝑄 → (¬ 𝑄 𝑋 → (𝑃 (𝑋 𝑄) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))))
6463imp4c 414 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴))
654, 7atbase 35245 . . . . . . . . . . . . 13 (𝑃𝐴𝑃𝐵)
6642, 65syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝑃𝐵)
674, 15latjcl 17317 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) ∈ 𝐵)
6831, 66, 34, 67syl3anc 1490 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) ∈ 𝐵)
694, 5, 61latmle1 17342 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7031, 3, 68, 69syl3anc 1490 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
7170adantr 472 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋)
72 simpll 783 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝐾 ∈ HL)
7363imp44 419 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
74 simplr2 1277 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑃𝐴)
7534adantr 472 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → 𝑄𝐵)
7673, 74, 753jca 1158 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵))
7772, 76jca 507 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)))
784, 5, 61, 6, 7atnle 35273 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ AtLat ∧ 𝑄𝐴𝑋𝐵) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
792, 32, 3, 78syl3anc 1490 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑄(meet‘𝐾)𝑋) = 0 ))
804, 61latmcom 17341 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8131, 34, 3, 80syl3anc 1490 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)𝑋) = (𝑋(meet‘𝐾)𝑄))
8281eqeq1d 2767 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)𝑋) = 0 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
8379, 82bitrd 270 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 ↔ (𝑋(meet‘𝐾)𝑄) = 0 ))
844, 61latmcl 17318 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8531, 3, 68, 84syl3anc 1490 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵)
8685, 3, 343jca 1158 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵))
8731, 86jca 507 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝐾 ∈ Lat ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵𝑋𝐵𝑄𝐵)))
884, 5, 61latmlem2 17348 . . . . . . . . . . . . . . . . . 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 4835 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) (𝑋(meet‘𝐾)𝑄))
91 breq2 4813 . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ))
93 hlop 35318 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ HL → 𝐾 ∈ OP)
9493adantr 472 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → 𝐾 ∈ OP)
954, 61latmcl 17318 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄𝐵 ∧ (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐵) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
9631, 34, 85, 95syl3anc 1490 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) ∈ 𝐵)
974, 5, 6ople0 35143 . . . . . . . . . . . . . . . 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 579 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) 0 ↔ (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
9992, 98sylibd 230 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑋(meet‘𝐾)𝑄) = 0 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
10083, 99sylbid 231 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (¬ 𝑄 𝑋 → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 ))
101100imp 395 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ¬ 𝑄 𝑋) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
102101adantrl 707 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ (𝑃𝑄 ∧ ¬ 𝑄 𝑋)) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
103102adantrr 708 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑄(meet‘𝐾)(𝑋(meet‘𝐾)(𝑃 𝑄))) = 0 )
1044, 5, 61latmle2 17343 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑃 𝑄) ∈ 𝐵) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
10531, 3, 68, 104syl3anc 1490 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑃 𝑄))
1064, 15latjcom 17325 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑄𝐵) → (𝑃 𝑄) = (𝑄 𝑃))
10731, 66, 34, 106syl3anc 1490 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
108105, 107breqtrd 4835 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
109108adantr 472 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → (𝑋(meet‘𝐾)(𝑃 𝑄)) (𝑄 𝑃))
11030adantr 472 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝐾 ∈ Lat)
111 simpr3 1252 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → 𝑄𝐵)
112 simpr1 1248 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ ((𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴𝑃𝐴𝑄𝐵)) → (𝑋(meet‘𝐾)(𝑃 𝑄)) ∈ 𝐴)
1134, 7atbase 35245 . . . . . . . . . . . . . 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 17341 . . . . . . . . . . . . 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 1490 . . . . . . . . . . . 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 2767 . . . . . . . . . . 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 35347 . . . . . . . . . . . 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 1150 . . . . . . . . . . 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 231 . . . . . . . . . 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 507 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) ∧ ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄))) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
123122ex 401 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → (((𝑃𝑄 ∧ ¬ 𝑄 𝑋) ∧ 𝑃 (𝑋 𝑄)) → ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
12464, 123jcad 508 . . . . . 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 4812 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑟 𝑋 ↔ (𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋))
126 oveq2 6850 . . . . . . . . 9 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑄 𝑟) = (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))
127126breq2d 4821 . . . . . . . 8 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → (𝑃 (𝑄 𝑟) ↔ 𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄)))))
128125, 127anbi12d 624 . . . . . . 7 (𝑟 = (𝑋(meet‘𝐾)(𝑃 𝑄)) → ((𝑟 𝑋𝑃 (𝑄 𝑟)) ↔ ((𝑋(meet‘𝐾)(𝑃 𝑄)) 𝑋𝑃 (𝑄 (𝑋(meet‘𝐾)(𝑃 𝑄))))))
129128rspcev 3461 . . . . . 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 404 . . . 4 ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴)) → ((𝑃𝑄 ∧ ¬ 𝑄 𝑋) → (𝑃 (𝑋 𝑄) → ∃𝑟𝐴 (𝑟 𝑋𝑃 (𝑄 𝑟)))))
13260, 131syl5bi 233 . . 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 1057 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 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wrex 3056   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16130  lecple 16221  joincjn 17210  meetcmee 17211  0.cp0 17303  Latclat 17311  OPcops 35128  Atomscatm 35219  AtLatcal 35220  HLchlt 35306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-proset 17194  df-poset 17212  df-plt 17224  df-lub 17240  df-glb 17241  df-join 17242  df-meet 17243  df-p0 17305  df-lat 17312  df-clat 17374  df-oposet 35132  df-ol 35134  df-oml 35135  df-covers 35222  df-ats 35223  df-atl 35254  df-cvlat 35278  df-hlat 35307
This theorem is referenced by:  cvrat42  35400  ps-2  35434
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