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Theorem latj13 17014
Description: Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
latjass.b 𝐵 = (Base‘𝐾)
latjass.j = (join‘𝐾)
Assertion
Ref Expression
latj13 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑍 (𝑌 𝑋)))

Proof of Theorem latj13
StepHypRef Expression
1 simpl 473 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
2 simpr2 1066 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
3 simpr3 1067 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
4 simpr1 1065 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
5 latjass.b . . . 4 𝐵 = (Base‘𝐾)
6 latjass.j . . . 4 = (join‘𝐾)
75, 6latj32 17013 . . 3 ((𝐾 ∈ Lat ∧ (𝑌𝐵𝑍𝐵𝑋𝐵)) → ((𝑌 𝑍) 𝑋) = ((𝑌 𝑋) 𝑍))
81, 2, 3, 4, 7syl13anc 1325 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) 𝑋) = ((𝑌 𝑋) 𝑍))
95, 6latjcl 16967 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
1093adant3r1 1271 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
115, 6latjcom 16975 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
121, 4, 10, 11syl3anc 1323 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
135, 6latjcl 16967 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑋𝐵) → (𝑌 𝑋) ∈ 𝐵)
141, 2, 4, 13syl3anc 1323 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑋) ∈ 𝐵)
155, 6latjcom 16975 . . 3 ((𝐾 ∈ Lat ∧ 𝑍𝐵 ∧ (𝑌 𝑋) ∈ 𝐵) → (𝑍 (𝑌 𝑋)) = ((𝑌 𝑋) 𝑍))
161, 3, 14, 15syl3anc 1323 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑍 (𝑌 𝑋)) = ((𝑌 𝑋) 𝑍))
178, 12, 163eqtr4d 2670 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = (𝑍 (𝑌 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  cfv 5850  (class class class)co 6605  Basecbs 15776  joincjn 16860  Latclat 16961
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  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 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-preset 16844  df-poset 16862  df-lub 16890  df-glb 16891  df-join 16892  df-meet 16893  df-lat 16962
This theorem is referenced by:  3atlem1  34235  dalawlem3  34625  dalawlem6  34628  cdleme1  34980  cdleme11g  35018
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