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Theorem hlatexch3N 38953
Description: Rearrange join of atoms in an equality. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
hlatexch4.j = (join‘𝐾)
hlatexch4.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlatexch3N ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑄 𝑅))

Proof of Theorem hlatexch3N
StepHypRef Expression
1 simp1 1134 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝐾 ∈ HL)
2 simp21 1204 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑃𝐴)
3 simp22 1205 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄𝐴)
4 eqid 2728 . . . . . 6 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . 6 = (join‘𝐾)
6 hlatexch4.a . . . . . 6 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 38848 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
81, 2, 3, 7syl3anc 1369 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄(le‘𝐾)(𝑃 𝑄))
9 simp23 1206 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅𝐴)
104, 5, 6hlatlej2 38848 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → 𝑅(le‘𝐾)(𝑃 𝑅))
111, 2, 9, 10syl3anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅(le‘𝐾)(𝑃 𝑅))
12 simp3r 1200 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
1311, 12breqtrrd 5176 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅(le‘𝐾)(𝑃 𝑄))
14 hllat 38835 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
15143ad2ant1 1131 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝐾 ∈ Lat)
16 eqid 2728 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1716, 6atbase 38761 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
183, 17syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄 ∈ (Base‘𝐾))
1916, 6atbase 38761 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
209, 19syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅 ∈ (Base‘𝐾))
2116, 5, 6hlatjcl 38839 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
221, 2, 3, 21syl3anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2316, 4, 5latjle12 18442 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑄(le‘𝐾)(𝑃 𝑄) ∧ 𝑅(le‘𝐾)(𝑃 𝑄)) ↔ (𝑄 𝑅)(le‘𝐾)(𝑃 𝑄)))
2415, 18, 20, 22, 23syl13anc 1370 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → ((𝑄(le‘𝐾)(𝑃 𝑄) ∧ 𝑅(le‘𝐾)(𝑃 𝑄)) ↔ (𝑄 𝑅)(le‘𝐾)(𝑃 𝑄)))
258, 13, 24mpbi2and 711 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑄 𝑅)(le‘𝐾)(𝑃 𝑄))
26 simp3l 1199 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄𝑅)
274, 5, 6ps-1 38950 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑄𝑅) ∧ (𝑃𝐴𝑄𝐴)) → ((𝑄 𝑅)(le‘𝐾)(𝑃 𝑄) ↔ (𝑄 𝑅) = (𝑃 𝑄)))
281, 3, 9, 26, 2, 3, 27syl132anc 1386 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → ((𝑄 𝑅)(le‘𝐾)(𝑃 𝑄) ↔ (𝑄 𝑅) = (𝑃 𝑄)))
2925, 28mpbid 231 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑄 𝑅) = (𝑃 𝑄))
3029eqcomd 2734 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑄 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937   class class class wbr 5148  cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  joincjn 18303  Latclat 18423  Atomscatm 38735  HLchlt 38822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-lat 18424  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823
This theorem is referenced by: (None)
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