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Theorem hlatexch3N 33582
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 1053 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝐾 ∈ HL)
2 simp21 1086 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑃𝐴)
3 simp22 1087 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄𝐴)
4 eqid 2604 . . . . . 6 (le‘𝐾) = (le‘𝐾)
5 hlatexch4.j . . . . . 6 = (join‘𝐾)
6 hlatexch4.a . . . . . 6 𝐴 = (Atoms‘𝐾)
74, 5, 6hlatlej2 33478 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄(le‘𝐾)(𝑃 𝑄))
81, 2, 3, 7syl3anc 1317 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄(le‘𝐾)(𝑃 𝑄))
9 simp23 1088 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅𝐴)
104, 5, 6hlatlej2 33478 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → 𝑅(le‘𝐾)(𝑃 𝑅))
111, 2, 9, 10syl3anc 1317 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅(le‘𝐾)(𝑃 𝑅))
12 simp3r 1082 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑃 𝑅))
1311, 12breqtrrd 4600 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅(le‘𝐾)(𝑃 𝑄))
14 hllat 33466 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
15143ad2ant1 1074 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝐾 ∈ Lat)
16 eqid 2604 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
1716, 6atbase 33392 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
183, 17syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄 ∈ (Base‘𝐾))
1916, 6atbase 33392 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
209, 19syl 17 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑅 ∈ (Base‘𝐾))
2116, 5, 6hlatjcl 33469 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
221, 2, 3, 21syl3anc 1317 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) ∈ (Base‘𝐾))
2316, 4, 5latjle12 16826 . . . . 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 1319 . . . 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 957 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑄 𝑅)(le‘𝐾)(𝑃 𝑄))
26 simp3l 1081 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → 𝑄𝑅)
274, 5, 6ps-1 33579 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑄𝑅) ∧ (𝑃𝐴𝑄𝐴)) → ((𝑄 𝑅)(le‘𝐾)(𝑃 𝑄) ↔ (𝑄 𝑅) = (𝑃 𝑄)))
281, 3, 9, 26, 2, 3, 27syl132anc 1335 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → ((𝑄 𝑅)(le‘𝐾)(𝑃 𝑄) ↔ (𝑄 𝑅) = (𝑃 𝑄)))
2925, 28mpbid 220 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑄 𝑅) = (𝑃 𝑄))
3029eqcomd 2610 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅 ∧ (𝑃 𝑄) = (𝑃 𝑅))) → (𝑃 𝑄) = (𝑄 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774   class class class wbr 4572  cfv 5785  (class class class)co 6522  Basecbs 15636  lecple 15716  joincjn 16708  Latclat 16809  Atomscatm 33366  HLchlt 33453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-riota 6484  df-ov 6525  df-oprab 6526  df-preset 16692  df-poset 16710  df-plt 16722  df-lub 16738  df-glb 16739  df-join 16740  df-meet 16741  df-p0 16803  df-lat 16810  df-covers 33369  df-ats 33370  df-atl 33401  df-cvlat 33425  df-hlat 33454
This theorem is referenced by: (None)
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