Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cvrat3 Structured version   Visualization version   GIF version
Theorem cvrat3 38301
Description: A condition implying that a certain lattice element is an atom. Part of Lemma 3.2.20 of [PtakPulmannova] p. 68. (atcvat3i 31636 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 2732 . . . . . . . . . . . 12 ( β‹– β€˜πΎ) = ( β‹– β€˜πΎ)
5 cvrat3.a . . . . . . . . . . . 12 𝐴 = (Atomsβ€˜πΎ)
61, 2, 3, 4, 5cvr1 38269 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
763adant3r2 1183 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (Β¬ 𝑄 ≀ 𝑋 ↔ 𝑋( β‹– β€˜πΎ)(𝑋 ∨ 𝑄)))
87biimpa 477 . . . . . . . . 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 38221 . . . . . . . . . . . . . . . . . 18 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
1110adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
12 simpr2 1195 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
131, 5atbase 38147 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
1412, 13syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐡)
15 simpr3 1196 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
161, 5atbase 38147 . . . . . . . . . . . . . . . . . 18 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ 𝐡)
1715, 16syl 17 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐡)
181, 3latjcom 18396 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
1911, 14, 17, 18syl3anc 1371 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
2019oveq2d 7421 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
21 simpr1 1194 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑋 ∈ 𝐡)
221, 3latjass 18432 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
2311, 21, 17, 14, 22syl13anc 1372 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) = (𝑋 ∨ (𝑄 ∨ 𝑃)))
2420, 23eqtr4d 2775 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
2524adantr 481 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) = ((𝑋 ∨ 𝑄) ∨ 𝑃))
261, 3latjcl 18388 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
2711, 21, 17, 26syl3anc 1371 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ 𝑄) ∈ 𝐡)
281, 2, 3latjlej2 18403 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ≀ (𝑋 ∨ 𝑄) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄))))
3029imp 407 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋 ∨ 𝑄) ∨ 𝑃) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
3125, 30eqbrtrd 5169 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)))
321, 3latjidm 18411 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ (𝑋 ∨ 𝑄) ∈ 𝐡) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3311, 27, 32syl2anc 584 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3433adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ ((𝑋 ∨ 𝑄) ∨ (𝑋 ∨ 𝑄)) = (𝑋 ∨ 𝑄))
3531, 34breqtrd 5173 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ≀ (𝑋 ∨ 𝑄))
36 simpl 483 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝐾 ∈ HL)
372, 3, 5hlatlej2 38234 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
3836, 12, 15, 37syl3anc 1371 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ 𝑄 ≀ (𝑃 ∨ 𝑄))
391, 3latjcl 18388 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑄 ∈ 𝐡) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
4011, 14, 17, 39syl3anc 1371 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ 𝐡)
411, 2, 3latjlej2 18403 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑃 ≀ (𝑋 ∨ 𝑄)) β†’ (𝑋 ∨ 𝑄) ≀ (𝑋 ∨ (𝑃 ∨ 𝑄)))
451, 3latjcl 18388 . . . . . . . . . . . . . 14 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐡)
4611, 21, 40, 45syl3anc 1371 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐡)
471, 2latasymb 18391 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 5159 . . . . . . . . 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 413 . . . . . 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 38279 . . . . . . 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 1371 . . . . . 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 481 . . . 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 18389 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∨ 𝑄) ∈ 𝐡) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡)
6111, 21, 40, 60syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ (𝑋 ∧ (𝑃 ∨ 𝑄)) ∈ 𝐡)
621, 3, 4, 5cvrat2 38288 . . . . . . 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 1121 . . . . . 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 1372 . . . . 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 453 . . . 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 431 . 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 1348 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  meetcmee 18261  Latclat 18380   β‹– ccvr 38120  Atomscatm 38121  HLchlt 38208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  cvrat4  38302  2atjm  38304  1cvrat  38335  2llnma1b  38645
  Copyright terms: Public domain W3C validator