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Theorem hlatexch4 36010
Description: Exchange 2 atoms. (Contributed by NM, 13-May-2013.)
Hypotheses
Ref Expression
hlatexch4.j = (join‘𝐾)
hlatexch4.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlatexch4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅) = (𝑄 𝑆))

Proof of Theorem hlatexch4
StepHypRef Expression
1 simp11 1183 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ HL)
2 simp2l 1179 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅𝐴)
3 simp2r 1180 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝐴)
4 eqid 2772 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . . . 8 = (join‘𝐾)
6 hlatexch4.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 35905 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆(le‘𝐾)(𝑅 𝑆))
81, 2, 3, 7syl3anc 1351 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑅 𝑆))
9 simp33 1191 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑄) = (𝑅 𝑆))
108, 9breqtrrd 4951 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑃 𝑄))
11 simp12 1184 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝐴)
12 simp13 1185 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝐴)
13 simp32 1190 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝑆)
1413necomd 3016 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝑄)
154, 5, 6hlatexch2 35925 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ 𝑆𝑄) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
161, 3, 11, 12, 14, 15syl131anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
1710, 16mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑆 𝑄))
185, 6hlatjcom 35897 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑄𝐴) → (𝑆 𝑄) = (𝑄 𝑆))
191, 3, 12, 18syl3anc 1351 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆 𝑄) = (𝑄 𝑆))
2017, 19breqtrd 4949 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑄 𝑆))
214, 5, 6hlatlej2 35905 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
221, 11, 12, 21syl3anc 1351 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑃 𝑄))
2322, 9breqtrd 4949 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑅 𝑆))
244, 5, 6hlatexch2 35925 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑄𝑆) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
251, 12, 2, 3, 13, 24syl131anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
2623, 25mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅(le‘𝐾)(𝑄 𝑆))
271hllatd 35893 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ Lat)
28 eqid 2772 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2928, 6atbase 35818 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3011, 29syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
3128, 6atbase 35818 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
322, 31syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅 ∈ (Base‘𝐾))
3328, 5, 6hlatjcl 35896 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ (Base‘𝐾))
341, 12, 3, 33syl3anc 1351 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄 𝑆) ∈ (Base‘𝐾))
3528, 4, 5latjle12 17520 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑄 𝑆) ∈ (Base‘𝐾))) → ((𝑃(le‘𝐾)(𝑄 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 𝑆)) ↔ (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆)))
3627, 30, 32, 34, 35syl13anc 1352 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃(le‘𝐾)(𝑄 𝑆) ∧ 𝑅(le‘𝐾)(𝑄 𝑆)) ↔ (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆)))
3720, 26, 36mpbi2and 699 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆))
38 simp31 1189 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝑅)
394, 5, 6ps-1 36006 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑃𝑅) ∧ (𝑄𝐴𝑆𝐴)) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
401, 11, 2, 38, 12, 3, 39syl132anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
4137, 40mpbid 224 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅) = (𝑄 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2048  wne 2961   class class class wbr 4923  cfv 6182  (class class class)co 6970  Basecbs 16329  lecple 16418  joincjn 17402  Latclat 17503  Atomscatm 35792  HLchlt 35879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-proset 17386  df-poset 17404  df-plt 17416  df-lub 17432  df-glb 17433  df-join 17434  df-meet 17435  df-p0 17497  df-lat 17504  df-covers 35795  df-ats 35796  df-atl 35827  df-cvlat 35851  df-hlat 35880
This theorem is referenced by:  cdlemg18a  37207
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