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Theorem hlatexch4 38865
Description: Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
Hypotheses
Ref Expression
hlatexch4.j ∨ = (joinβ€˜πΎ)
hlatexch4.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
hlatexch4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))

Proof of Theorem hlatexch4
StepHypRef Expression
1 simp11 1200 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝐾 ∈ HL)
2 simp2l 1196 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑅 ∈ 𝐴)
3 simp2r 1197 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑆 ∈ 𝐴)
4 eqid 2726 . . . . . . . 8 (leβ€˜πΎ) = (leβ€˜πΎ)
5 hlatexch4.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
6 hlatexch4.a . . . . . . . 8 𝐴 = (Atomsβ€˜πΎ)
74, 5, 6hlatlej2 38759 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ 𝑆(leβ€˜πΎ)(𝑅 ∨ 𝑆))
81, 2, 3, 7syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑆(leβ€˜πΎ)(𝑅 ∨ 𝑆))
9 simp33 1208 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))
108, 9breqtrrd 5169 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑆(leβ€˜πΎ)(𝑃 ∨ 𝑄))
11 simp12 1201 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑃 ∈ 𝐴)
12 simp13 1202 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑄 ∈ 𝐴)
13 simp32 1207 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑄 β‰  𝑆)
1413necomd 2990 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑆 β‰  𝑄)
154, 5, 6hlatexch2 38780 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑆 β‰  𝑄) β†’ (𝑆(leβ€˜πΎ)(𝑃 ∨ 𝑄) β†’ 𝑃(leβ€˜πΎ)(𝑆 ∨ 𝑄)))
161, 3, 11, 12, 14, 15syl131anc 1380 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑆(leβ€˜πΎ)(𝑃 ∨ 𝑄) β†’ 𝑃(leβ€˜πΎ)(𝑆 ∨ 𝑄)))
1710, 16mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑃(leβ€˜πΎ)(𝑆 ∨ 𝑄))
185, 6hlatjcom 38751 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ (𝑆 ∨ 𝑄) = (𝑄 ∨ 𝑆))
191, 3, 12, 18syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑆 ∨ 𝑄) = (𝑄 ∨ 𝑆))
2017, 19breqtrd 5167 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑆))
214, 5, 6hlatlej2 38759 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ 𝑄(leβ€˜πΎ)(𝑃 ∨ 𝑄))
221, 11, 12, 21syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑄(leβ€˜πΎ)(𝑃 ∨ 𝑄))
2322, 9breqtrd 5167 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑄(leβ€˜πΎ)(𝑅 ∨ 𝑆))
244, 5, 6hlatexch2 38780 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ 𝑄 β‰  𝑆) β†’ (𝑄(leβ€˜πΎ)(𝑅 ∨ 𝑆) β†’ 𝑅(leβ€˜πΎ)(𝑄 ∨ 𝑆)))
251, 12, 2, 3, 13, 24syl131anc 1380 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑄(leβ€˜πΎ)(𝑅 ∨ 𝑆) β†’ 𝑅(leβ€˜πΎ)(𝑄 ∨ 𝑆)))
2623, 25mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑅(leβ€˜πΎ)(𝑄 ∨ 𝑆))
271hllatd 38747 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝐾 ∈ Lat)
28 eqid 2726 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2928, 6atbase 38672 . . . . 5 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
3011, 29syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
3128, 6atbase 38672 . . . . 5 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
322, 31syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
3328, 5, 6hlatjcl 38750 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) β†’ (𝑄 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
341, 12, 3, 33syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑄 ∨ 𝑆) ∈ (Baseβ€˜πΎ))
3528, 4, 5latjle12 18415 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ) ∧ (𝑄 ∨ 𝑆) ∈ (Baseβ€˜πΎ))) β†’ ((𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑆) ∧ 𝑅(leβ€˜πΎ)(𝑄 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑅)(leβ€˜πΎ)(𝑄 ∨ 𝑆)))
3627, 30, 32, 34, 35syl13anc 1369 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ ((𝑃(leβ€˜πΎ)(𝑄 ∨ 𝑆) ∧ 𝑅(leβ€˜πΎ)(𝑄 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑅)(leβ€˜πΎ)(𝑄 ∨ 𝑆)))
3720, 26, 36mpbi2and 709 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑃 ∨ 𝑅)(leβ€˜πΎ)(𝑄 ∨ 𝑆))
38 simp31 1206 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ 𝑃 β‰  𝑅)
394, 5, 6ps-1 38861 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 β‰  𝑅) ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑅)(leβ€˜πΎ)(𝑄 ∨ 𝑆) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆)))
401, 11, 2, 38, 12, 3, 39syl132anc 1385 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ ((𝑃 ∨ 𝑅)(leβ€˜πΎ)(𝑄 ∨ 𝑆) ↔ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆)))
4137, 40mpbid 231 1 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 β‰  𝑅 ∧ 𝑄 β‰  𝑆 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))) β†’ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Latclat 18396  Atomscatm 38646  HLchlt 38733
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  cdlemg18a  40062
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