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Theorem hlatexch4 33785
 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 1084 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ HL)
2 simp2l 1080 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅𝐴)
3 simp2r 1081 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝐴)
4 eqid 2610 . . . . . . . 8 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . . . 8 = (join‘𝐾)
6 hlatexch4.a . . . . . . . 8 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 33680 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑆(le‘𝐾)(𝑅 𝑆))
81, 2, 3, 7syl3anc 1318 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑅 𝑆))
9 simp33 1092 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑄) = (𝑅 𝑆))
108, 9breqtrrd 4611 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆(le‘𝐾)(𝑃 𝑄))
11 simp12 1085 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝐴)
12 simp13 1086 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝐴)
13 simp32 1091 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄𝑆)
1413necomd 2837 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑆𝑄)
154, 5, 6hlatexch2 33700 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ 𝑆𝑄) → (𝑆(le‘𝐾)(𝑃 𝑄) → 𝑃(le‘𝐾)(𝑆 𝑄)))
161, 3, 11, 12, 14, 15syl131anc 1331 . . . . 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 33672 . . . . 5 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑄𝐴) → (𝑆 𝑄) = (𝑄 𝑆))
191, 3, 12, 18syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑆 𝑄) = (𝑄 𝑆))
2017, 19breqtrd 4609 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃(le‘𝐾)(𝑄 𝑆))
214, 5, 6hlatlej2 33680 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
221, 11, 12, 21syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑃 𝑄))
2322, 9breqtrd 4609 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑄(le‘𝐾)(𝑅 𝑆))
244, 5, 6hlatexch2 33700 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑆𝐴) ∧ 𝑄𝑆) → (𝑄(le‘𝐾)(𝑅 𝑆) → 𝑅(le‘𝐾)(𝑄 𝑆)))
251, 12, 2, 3, 13, 24syl131anc 1331 . . . 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 33668 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
281, 27syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝐾 ∈ Lat)
29 eqid 2610 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
3029, 6atbase 33594 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3111, 30syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃 ∈ (Base‘𝐾))
3229, 6atbase 33594 . . . . 5 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
332, 32syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑅 ∈ (Base‘𝐾))
3429, 5, 6hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ (Base‘𝐾))
351, 12, 3, 34syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑄 𝑆) ∈ (Base‘𝐾))
3629, 4, 5latjle12 16885 . . . 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 1320 . . 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 958 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → (𝑃 𝑅)(le‘𝐾)(𝑄 𝑆))
39 simp31 1090 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → 𝑃𝑅)
404, 5, 6ps-1 33781 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑅𝐴𝑃𝑅) ∧ (𝑄𝐴𝑆𝐴)) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
411, 11, 2, 39, 12, 3, 40syl132anc 1336 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴𝑆𝐴) ∧ (𝑃𝑅𝑄𝑆 ∧ (𝑃 𝑄) = (𝑅 𝑆))) → ((𝑃 𝑅)(le‘𝐾)(𝑄 𝑆) ↔ (𝑃 𝑅) = (𝑄 𝑆)))
4238, 41mpbid 221 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  cdlemg18a  34984
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