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Theorem hlatexch4 35085
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 1111 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ HL)
2 simp2l 1107 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅𝐴)
3 simp2r 1108 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝐴)
4 eqid 2651 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . . . 8 = (join‘𝐾)
6 hlatexch4.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 34980 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆(le‘𝐾)(𝑅 𝑆))
81, 2, 3, 7syl3anc 1366 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑅 𝑆))
9 simp33 1119 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑄) = (𝑅 𝑆))
108, 9breqtrrd 4713 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑃 𝑄))
11 simp12 1112 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝐴)
12 simp13 1113 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝐴)
13 simp32 1118 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝑆)
1413necomd 2878 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝑄)
154, 5, 6hlatexch2 35000 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ 𝑆𝑄) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
161, 3, 11, 12, 14, 15syl131anc 1379 . . . . 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 34972 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑄𝐴) → (𝑆 𝑄) = (𝑄 𝑆))
191, 3, 12, 18syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆 𝑄) = (𝑄 𝑆))
2017, 19breqtrd 4711 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑄 𝑆))
214, 5, 6hlatlej2 34980 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
221, 11, 12, 21syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑃 𝑄))
2322, 9breqtrd 4711 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑅 𝑆))
244, 5, 6hlatexch2 35000 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑄𝑆) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
251, 12, 2, 3, 13, 24syl131anc 1379 . . . 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))
27 hllat 34968 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
281, 27syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ Lat)
29 eqid 2651 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3029, 6atbase 34894 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3111, 30syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
3229, 6atbase 34894 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
332, 32syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅 ∈ (Base‘𝐾))
3429, 5, 6hlatjcl 34971 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ (Base‘𝐾))
351, 12, 3, 34syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄 𝑆) ∈ (Base‘𝐾))
3629, 4, 5latjle12 17109 . . . 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)))
3728, 31, 33, 35, 36syl13anc 1368 . . 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)))
3820, 26, 37mpbi2and 976 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆))
39 simp31 1117 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝑅)
404, 5, 6ps-1 35081 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑃𝑅) ∧ (𝑄𝐴𝑆𝐴)) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
411, 11, 2, 39, 12, 3, 40syl132anc 1384 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
4238, 41mpbid 222 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  HLchlt 34955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956
This theorem is referenced by:  cdlemg18a  36283
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