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Theorem hlatj12 36543
Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 17686 for atoms. (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
hlatjcom.j = (join‘𝐾)
hlatjcom.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
hlatj12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))

Proof of Theorem hlatj12
StepHypRef Expression
1 hlatjcom.j . . . . 5 = (join‘𝐾)
2 hlatjcom.a . . . . 5 𝐴 = (Atoms‘𝐾)
31, 2hlatjcom 36540 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
433adant3r3 1180 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
54oveq1d 7148 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑃) 𝑅))
61, 2hlatjass 36542 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))
7 simpl 485 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ HL)
8 simpr2 1191 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
9 simpr1 1190 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
10 simpr3 1192 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
111, 2hlatjass 36542 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑃𝐴𝑅𝐴)) → ((𝑄 𝑃) 𝑅) = (𝑄 (𝑃 𝑅)))
127, 8, 9, 10, 11syl13anc 1368 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄 𝑃) 𝑅) = (𝑄 (𝑃 𝑅)))
135, 6, 123eqtr3d 2863 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wcel 2114  cfv 6331  (class class class)co 7133  joincjn 17533  Atomscatm 36435  HLchlt 36522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-lat 17635  df-ats 36439  df-atl 36470  df-cvlat 36494  df-hlat 36523
This theorem is referenced by:  3atlem1  36655  3atlem2  36656  dalawlem12  37054  cdleme35b  37622
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