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Theorem cvrat3 38919
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 32224 analog.) (Contributed by NM, 30-Nov-2011.)
Hypotheses
Ref Expression
cvrat3.b 𝐡 = (Baseβ€˜πΎ)
cvrat3.l ≀ = (leβ€˜πΎ)
cvrat3.j ∨ = (joinβ€˜πΎ)
cvrat3.m ∧ = (meetβ€˜πΎ)
cvrat3.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
cvrat3 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
Proof of Theorem cvrat3
StepHypRef Expression
1 cvrat3.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΎ)
2 cvrat3.l . . . . . . . . . . . 12 ≀ = (leβ€˜πΎ)
3 cvrat3.j . . . . . . . . . . . 12 ∨ = (joinβ€˜πΎ)
4 eqid 2727 . . . . . . . . . . . 12 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5cvr1 38887 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
763adant3r2 1180 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
87biimpa 475 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄))
98adantrr 715 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄))
10 hllat 38839 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1110adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
12 simpr2 1192 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
131, 5atbase 38765 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
15 simpr3 1193 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
161, 5atbase 38765 . . . . . . . . . . . . . . . . . 18 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
181, 3latjcom 18444 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1911, 14, 17, 18syl3anc 1368 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
2019oveq2d 7440 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21 simpr1 1191 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
221, 3latjass 18480 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
2311, 21, 17, 14, 22syl13anc 1369 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
2420, 23eqtr4d 2770 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
2524adantr 479 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
261, 3latjcl 18436 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
2711, 21, 17, 26syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
281, 2, 3latjlej2 18451 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐡 ∧ (𝑋 ∨ 𝑄) ∈ 𝐡 ∧ (𝑋 ∨ 𝑄) ∈ 𝐡)) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
2911, 14, 27, 27, 28syl13anc 1369 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
3029imp 405 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
3125, 30eqbrtrd 5172 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
321, 3latjidm 18459 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑄) ∈ 𝐡) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3311, 27, 32syl2anc 582 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3433adantr 479 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3531, 34breqtrd 5176 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ (𝑋 ∨ 𝑄))
36 simpl 481 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
372, 3, 5hlatlej2 38852 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
3836, 12, 15, 37syl3anc 1368 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
391, 3latjcl 18436 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
4011, 14, 17, 39syl3anc 1368 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
411, 2, 3latjlej2 18451 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ (𝑄 ≀ (𝑃 ∨ 𝑄) β†’ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄))))
4211, 17, 40, 21, 41syl13anc 1369 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑄 ≀ (𝑃 ∨ 𝑄) β†’ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄))))
4338, 42mpd 15 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄)))
4443adantr 479 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄)))
451, 3latjcl 18436 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐡)
4611, 21, 40, 45syl3anc 1368 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐡)
471, 2latasymb 18439 . . . . . . . . . . . . 13 ((𝐾 ∈ Lat ∧ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ (𝑋 ∨ 𝑄) ∈ 𝐡) β†’ (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
4811, 46, 27, 47syl3anc 1368 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
4948adantr 479 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (((𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ (𝑋 ∨ 𝑄) ∧ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄))) ↔ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄)))
5035, 44, 49mpbi2and 710 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
5150breq2d 5162 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
5251adantrl 714 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ (𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄)) ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
539, 52mpbird 256 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄))) β†’ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄)))
5453ex 411 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄))))
55 cvrat3.m . . . . . . . 8 ∧ = (meetβ€˜πΎ)
561, 3, 55, 4cvrexch 38897 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ ((𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄) ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄))))
5736, 21, 40, 56syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄) ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ (𝑃 ∨ 𝑄))))
5854, 57sylibrd 258 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)))
5958adantr 479 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ ((Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)))
601, 55latmcl 18437 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡)
6111, 21, 40, 60syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡)
621, 3, 4, 5cvrat2 38906 . . . . . . 7 ((𝐾 ∈ HL ∧ ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑃 β‰  𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄))) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
63623expia 1118 . . . . . 6 ((𝐾 ∈ HL ∧ ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
6436, 61, 12, 15, 63syl13anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ (𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
6564expdimp 451 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ ((𝑋 ∧ (𝑃 ∨ 𝑄))( β‹– β€˜πΎ)(𝑃 ∨ 𝑄) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
6659, 65syld 47 . . 3 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 β‰  𝑄) β†’ ((Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
6766exp4b 429 . 2 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 β‰  𝑄 β†’ (Β¬ 𝑄 ≀ 𝑋 β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))))
68673impd 1345 1 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑃 β‰  𝑄 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936   class class class wbr 5150  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  lecple 17245  joincjn 18308  meetcmee 18309  Latclat 18428   β‹– ccvr 38738  Atomscatm 38739  HLchlt 38826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18292  df-poset 18310  df-plt 18327  df-lub 18343  df-glb 18344  df-join 18345  df-meet 18346  df-p0 18422  df-lat 18429  df-clat 18496  df-oposet 38652  df-ol 38654  df-oml 38655  df-covers 38742  df-ats 38743  df-atl 38774  df-cvlat 38798  df-hlat 38827
This theorem is referenced by:  cvrat4  38920  2atjm  38922  1cvrat  38953  2llnma1b  39263
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