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Theorem cvrat4 37952
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 31381 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat4.b 𝐡 = (Baseβ€˜πΎ)
cvrat4.l ≀ = (leβ€˜πΎ)
cvrat4.j ∨ = (joinβ€˜πΎ)
cvrat4.z 0 = (0.β€˜πΎ)
cvrat4.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvrat4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
Distinct variable groups:   𝐴,π‘Ÿ   𝐡,π‘Ÿ   ∨ ,π‘Ÿ   𝐾,π‘Ÿ   ≀ ,π‘Ÿ   𝑃,π‘Ÿ   𝑄,π‘Ÿ   𝑋,π‘Ÿ
Allowed substitution hint:   0 (π‘Ÿ)

Proof of Theorem cvrat4
StepHypRef Expression
1 hlatl 37868 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
21adantr 482 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ AtLat)
3 simpr1 1195 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
4 cvrat4.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜πΎ)
5 cvrat4.l . . . . . . . . . . 11 ≀ = (leβ€˜πΎ)
6 cvrat4.z . . . . . . . . . . 11 0 = (0.β€˜πΎ)
7 cvrat4.a . . . . . . . . . . 11 𝐴 = (Atomsβ€˜πΎ)
84, 5, 6, 7atlex 37824 . . . . . . . . . 10 ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐡 ∧ 𝑋 β‰  0 ) β†’ βˆƒπ‘Ÿ ∈ 𝐴 π‘Ÿ ≀ 𝑋)
983exp 1120 . . . . . . . . 9 (𝐾 ∈ AtLat β†’ (𝑋 ∈ 𝐡 β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 π‘Ÿ ≀ 𝑋)))
102, 3, 9sylc 65 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 π‘Ÿ ≀ 𝑋))
1110adantr 482 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 π‘Ÿ ≀ 𝑋))
12 simpll 766 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ π‘Ÿ ∈ 𝐴) β†’ 𝐾 ∈ HL)
13 simplr3 1218 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ π‘Ÿ ∈ 𝐴) β†’ 𝑄 ∈ 𝐴)
14 simpr 486 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ π‘Ÿ ∈ 𝐴) β†’ π‘Ÿ ∈ 𝐴)
15 cvrat4.j . . . . . . . . . . . . . . 15 ∨ = (joinβ€˜πΎ)
165, 15, 7hlatlej1 37883 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) β†’ 𝑄 ≀ (𝑄 ∨ π‘Ÿ))
1712, 13, 14, 16syl3anc 1372 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ π‘Ÿ ∈ 𝐴) β†’ 𝑄 ≀ (𝑄 ∨ π‘Ÿ))
18 breq1 5109 . . . . . . . . . . . . 13 (𝑃 = 𝑄 β†’ (𝑃 ≀ (𝑄 ∨ π‘Ÿ) ↔ 𝑄 ≀ (𝑄 ∨ π‘Ÿ)))
1917, 18syl5ibr 246 . . . . . . . . . . . 12 (𝑃 = 𝑄 β†’ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ π‘Ÿ ∈ 𝐴) β†’ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
2019expd 417 . . . . . . . . . . 11 (𝑃 = 𝑄 β†’ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (π‘Ÿ ∈ 𝐴 β†’ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
2120impcom 409 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ (π‘Ÿ ∈ 𝐴 β†’ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
2221anim2d 613 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ ((π‘Ÿ ≀ 𝑋 ∧ π‘Ÿ ∈ 𝐴) β†’ (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
2322expcomd 418 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ (π‘Ÿ ∈ 𝐴 β†’ (π‘Ÿ ≀ 𝑋 β†’ (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
2423reximdvai 3159 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 π‘Ÿ ≀ 𝑋 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
2511, 24syld 47 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 = 𝑄) β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
2625ex 414 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 = 𝑄 β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
2726a1i 11 . . . 4 (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 = 𝑄 β†’ (𝑋 β‰  0 β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))))
2827com4l 92 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 = 𝑄 β†’ (𝑋 β‰  0 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))))
2928imp4a 424 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 = 𝑄 β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
30 hllat 37871 . . . . . . . . . . . . . 14 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
3130adantr 482 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
32 simpr3 1197 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
334, 7atbase 37797 . . . . . . . . . . . . . 14 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
3432, 33syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
354, 5, 15latleeqj2 18346 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑄 ≀ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
3631, 34, 3, 35syl3anc 1372 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄 ≀ 𝑋 ↔ (𝑋 ∨ 𝑄) = 𝑋))
3736biimpa 478 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) = 𝑋)
3837breq2d 5118 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) ↔ 𝑃 ≀ 𝑋))
3938biimpa 478 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑃 ≀ 𝑋)
4039expl 459 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑃 ≀ 𝑋))
41 simpl 484 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
42 simpr2 1196 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
435, 15, 7hlatlej2 37884 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 ≀ (𝑄 ∨ 𝑃))
4441, 32, 42, 43syl3anc 1372 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ≀ (𝑄 ∨ 𝑃))
4540, 44jctird 528 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ 𝑃))))
4645, 42jctild 527 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑃 ∈ 𝐴 ∧ (𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ 𝑃)))))
4746impl 457 . . . . 5 ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑃 ∈ 𝐴 ∧ (𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ 𝑃))))
48 breq1 5109 . . . . . . 7 (π‘Ÿ = 𝑃 β†’ (π‘Ÿ ≀ 𝑋 ↔ 𝑃 ≀ 𝑋))
49 oveq2 7366 . . . . . . . 8 (π‘Ÿ = 𝑃 β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ 𝑃))
5049breq2d 5118 . . . . . . 7 (π‘Ÿ = 𝑃 β†’ (𝑃 ≀ (𝑄 ∨ π‘Ÿ) ↔ 𝑃 ≀ (𝑄 ∨ 𝑃)))
5148, 50anbi12d 632 . . . . . 6 (π‘Ÿ = 𝑃 β†’ ((π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)) ↔ (𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ 𝑃))))
5251rspcev 3580 . . . . 5 ((𝑃 ∈ 𝐴 ∧ (𝑃 ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ 𝑃))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
5347, 52syl 17 . . . 4 ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
5453adantrl 715 . . 3 ((((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑄 ≀ 𝑋) ∧ (𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
5554exp31 421 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄 ≀ 𝑋 β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
56 simpr 486 . . 3 ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑃 ≀ (𝑋 ∨ 𝑄))
57 ioran 983 . . . . 5 (Β¬ (𝑃 = 𝑄 ∨ 𝑄 ≀ 𝑋) ↔ (Β¬ 𝑃 = 𝑄 ∧ Β¬ 𝑄 ≀ 𝑋))
58 df-ne 2941 . . . . . 6 (𝑃 β‰  𝑄 ↔ Β¬ 𝑃 = 𝑄)
5958anbi1i 625 . . . . 5 ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ↔ (Β¬ 𝑃 = 𝑄 ∧ Β¬ 𝑄 ≀ 𝑋))
6057, 59bitr4i 278 . . . 4 (Β¬ (𝑃 = 𝑄 ∨ 𝑄 ≀ 𝑋) ↔ (𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋))
61 eqid 2733 . . . . . . . . . 10 (meetβ€˜πΎ) = (meetβ€˜πΎ)
624, 5, 15, 61, 7cvrat3 37951 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴))
63623expd 1354 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 β‰  𝑄 β†’ (Β¬ 𝑄 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴))))
6463imp4c 425 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴))
654, 7atbase 37797 . . . . . . . . . . . . 13 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
6642, 65syl 17 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
674, 15latjcl 18333 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
6831, 66, 34, 67syl3anc 1372 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
694, 5, 61latmle1 18358 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋)
7031, 3, 68, 69syl3anc 1372 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋)
7170adantr 482 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋)
72 simpll 766 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝐾 ∈ HL)
7363imp44 430 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴)
74 simplr2 1217 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝑃 ∈ 𝐴)
7534adantr 482 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝑄 ∈ 𝐡)
7673, 74, 753jca 1129 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡))
7772, 76jca 513 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)))
784, 5, 61, 6, 7atnle 37825 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ (𝑄(meetβ€˜πΎ)𝑋) = 0 ))
792, 32, 3, 78syl3anc 1372 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ (𝑄(meetβ€˜πΎ)𝑋) = 0 ))
804, 61latmcom 18357 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑄(meetβ€˜πΎ)𝑋) = (𝑋(meetβ€˜πΎ)𝑄))
8131, 34, 3, 80syl3anc 1372 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄(meetβ€˜πΎ)𝑋) = (𝑋(meetβ€˜πΎ)𝑄))
8281eqeq1d 2735 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑄(meetβ€˜πΎ)𝑋) = 0 ↔ (𝑋(meetβ€˜πΎ)𝑄) = 0 ))
8379, 82bitrd 279 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ (𝑋(meetβ€˜πΎ)𝑄) = 0 ))
844, 61latmcl 18334 . . . . . . . . . . . . . . . . . . . . 21 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡)
8531, 3, 68, 84syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡)
8685, 3, 343jca 1129 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡))
8731, 86jca 513 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝐾 ∈ Lat ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡)))
884, 5, 61latmlem2 18364 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡)) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ (𝑄(meetβ€˜πΎ)𝑋)))
8987, 70, 88sylc 65 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ (𝑄(meetβ€˜πΎ)𝑋))
9089, 81breqtrd 5132 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ (𝑋(meetβ€˜πΎ)𝑄))
91 breq2 5110 . . . . . . . . . . . . . . . 16 ((𝑋(meetβ€˜πΎ)𝑄) = 0 β†’ ((𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ (𝑋(meetβ€˜πΎ)𝑄) ↔ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ 0 ))
9290, 91syl5ibcom 244 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋(meetβ€˜πΎ)𝑄) = 0 β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ 0 ))
93 hlop 37870 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
9493adantr 482 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ OP)
954, 61latmcl 18334 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ∈ 𝐡)
9631, 34, 85, 95syl3anc 1372 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ∈ 𝐡)
974, 5, 6ople0 37695 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OP ∧ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ∈ 𝐡) β†’ ((𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ 0 ↔ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 ))
9894, 96, 97syl2anc 585 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) ≀ 0 ↔ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 ))
9992, 98sylibd 238 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋(meetβ€˜πΎ)𝑄) = 0 β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 ))
10083, 99sylbid 239 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ 𝑄 ≀ 𝑋 β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 ))
101100imp 408 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 )
102101adantrl 715 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 )
103102adantrr 716 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 )
1044, 5, 61latmle2 18359 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑃 ∨ 𝑄))
10531, 3, 68, 104syl3anc 1372 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑃 ∨ 𝑄))
1064, 15latjcom 18341 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
10731, 66, 34, 106syl3anc 1372 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
108105, 107breqtrd 5132 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑄 ∨ 𝑃))
109108adantr 482 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑄 ∨ 𝑃))
11030adantr 482 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ 𝐾 ∈ Lat)
111 simpr3 1197 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ 𝑄 ∈ 𝐡)
112 simpr1 1195 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴)
1134, 7atbase 37797 . . . . . . . . . . . . . 14 ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡)
114112, 113syl 17 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡)
1154, 61latmcom 18357 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐡 ∧ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐡) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))(meetβ€˜πΎ)𝑄))
116110, 111, 114, 115syl3anc 1372 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ (𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))(meetβ€˜πΎ)𝑄))
117116eqeq1d 2735 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ ((𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 ↔ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))(meetβ€˜πΎ)𝑄) = 0 ))
1184, 5, 15, 61, 6, 7hlexch3 37900 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡) ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))(meetβ€˜πΎ)𝑄) = 0 ) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑄 ∨ 𝑃) β†’ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)))))
1191183expia 1122 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ (((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))(meetβ€˜πΎ)𝑄) = 0 β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑄 ∨ 𝑃) β†’ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))))
120117, 119sylbid 239 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐡)) β†’ ((𝑄(meetβ€˜πΎ)(𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))) = 0 β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ (𝑄 ∨ 𝑃) β†’ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))))
12177, 103, 109, 120syl3c 66 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))
12271, 121jca 513 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)))))
123122ex 414 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))))
12464, 123jcad 514 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)))))))
125 breq1 5109 . . . . . . . 8 (π‘Ÿ = (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ (π‘Ÿ ≀ 𝑋 ↔ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋))
126 oveq2 7366 . . . . . . . . 9 (π‘Ÿ = (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ (𝑄 ∨ π‘Ÿ) = (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))
127126breq2d 5118 . . . . . . . 8 (π‘Ÿ = (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ (𝑃 ≀ (𝑄 ∨ π‘Ÿ) ↔ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)))))
128125, 127anbi12d 632 . . . . . . 7 (π‘Ÿ = (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) β†’ ((π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)) ↔ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))))
129128rspcev 3580 . . . . . 6 (((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄)) ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ (𝑋(meetβ€˜πΎ)(𝑃 ∨ 𝑄))))) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
130124, 129syl6 35 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
131130expd 417 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
13260, 131biimtrid 241 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ (𝑃 = 𝑄 ∨ 𝑄 ≀ 𝑋) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
13356, 132syl7 74 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ (𝑃 = 𝑄 ∨ 𝑄 ≀ 𝑋) β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
13429, 55, 133ecase3d 1033 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 β‰  0 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  meetcmee 18206  0.cp0 18317  Latclat 18325  OPcops 37680  Atomscatm 37771  AtLatcal 37772  HLchlt 37858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by:  cvrat42  37953  ps-2  37987
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