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Theorem hlatj12 34172
Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 17029 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 34169 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) = (𝑄 𝑃))
433adant3r3 1273 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) = (𝑄 𝑃))
54oveq1d 6625 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = ((𝑄 𝑃) 𝑅))
61, 2hlatjass 34171 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) = (𝑃 (𝑄 𝑅)))
7 simpl 473 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ HL)
8 simpr2 1066 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
9 simpr1 1065 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
10 simpr3 1067 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
111, 2hlatjass 34171 . . 3 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑃𝐴𝑅𝐴)) → ((𝑄 𝑃) 𝑅) = (𝑄 (𝑃 𝑅)))
127, 8, 9, 10, 11syl13anc 1325 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑄 𝑃) 𝑅) = (𝑄 (𝑃 𝑅)))
135, 6, 123eqtr3d 2663 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 (𝑄 𝑅)) = (𝑄 (𝑃 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cfv 5852  (class class class)co 6610  joincjn 16876  Atomscatm 34065  HLchlt 34152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-lat 16978  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153
This theorem is referenced by:  3atlem1  34284  3atlem2  34285  dalawlem12  34683  cdleme35b  35253
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