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Theorem latjass 16918
Description: Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 27776 analog.) (Contributed by NM, 17-Sep-2011.)
Hypotheses
Ref Expression
latjass.b 𝐵 = (Base‘𝐾)
latjass.j = (join‘𝐾)
Assertion
Ref Expression
latjass ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))

Proof of Theorem latjass
StepHypRef Expression
1 latjass.b . 2 𝐵 = (Base‘𝐾)
2 eqid 2610 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 472 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
4 latjass.j . . . . 5 = (join‘𝐾)
51, 4latjcl 16874 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
653adant3r3 1268 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
7 simpr3 1062 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
81, 4latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
93, 6, 7, 8syl3anc 1318 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
10 simpr1 1060 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 4latjcl 16874 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
12113adant3r1 1266 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
131, 4latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
143, 10, 12, 13syl3anc 1318 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
151, 2, 4latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)))
163, 10, 12, 15syl3anc 1318 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)))
17 simpr2 1061 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
181, 2, 4latlej1 16883 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → 𝑌(le‘𝐾)(𝑌 𝑍))
19183adant3r1 1266 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑌 𝑍))
201, 2, 4latlej2 16884 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑌 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
213, 10, 12, 20syl3anc 1318 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
221, 2, 3, 17, 12, 14, 19, 21lattrd 16881 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑋 (𝑌 𝑍)))
231, 2, 4latjle12 16885 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑋 (𝑌 𝑍)) ∈ 𝐵)) → ((𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ 𝑌(le‘𝐾)(𝑋 (𝑌 𝑍))) ↔ (𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍))))
243, 10, 17, 14, 23syl13anc 1320 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ 𝑌(le‘𝐾)(𝑋 (𝑌 𝑍))) ↔ (𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍))))
2516, 22, 24mpbi2and 958 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍)))
261, 2, 4latlej2 16884 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → 𝑍(le‘𝐾)(𝑌 𝑍))
27263adant3r1 1266 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)(𝑌 𝑍))
281, 2, 3, 7, 12, 14, 27, 21lattrd 16881 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)(𝑋 (𝑌 𝑍)))
291, 2, 4latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ 𝐵𝑍𝐵 ∧ (𝑋 (𝑌 𝑍)) ∈ 𝐵)) → (((𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ 𝑍(le‘𝐾)(𝑋 (𝑌 𝑍))) ↔ ((𝑋 𝑌) 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍))))
303, 6, 7, 14, 29syl13anc 1320 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ 𝑍(le‘𝐾)(𝑋 (𝑌 𝑍))) ↔ ((𝑋 𝑌) 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍))))
3125, 28, 30mpbi2and 958 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
321, 2, 4latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
33323adant3r3 1268 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
341, 2, 4latlej1 16883 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝑋 𝑌)(le‘𝐾)((𝑋 𝑌) 𝑍))
353, 6, 7, 34syl3anc 1318 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌)(le‘𝐾)((𝑋 𝑌) 𝑍))
361, 2, 3, 10, 6, 9, 33, 35lattrd 16881 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)((𝑋 𝑌) 𝑍))
371, 2, 4latlej2 16884 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
38373adant3r3 1268 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
391, 2, 3, 17, 6, 9, 38, 35lattrd 16881 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)((𝑋 𝑌) 𝑍))
401, 2, 4latlej2 16884 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍))
413, 6, 7, 40syl3anc 1318 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍))
421, 2, 4latjle12 16885 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑌𝐵𝑍𝐵 ∧ ((𝑋 𝑌) 𝑍) ∈ 𝐵)) → ((𝑌(le‘𝐾)((𝑋 𝑌) 𝑍) ∧ 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍)) ↔ (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍)))
433, 17, 7, 9, 42syl13anc 1320 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌(le‘𝐾)((𝑋 𝑌) 𝑍) ∧ 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍)) ↔ (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍)))
4439, 41, 43mpbi2and 958 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍))
451, 2, 4latjle12 16885 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵 ∧ ((𝑋 𝑌) 𝑍) ∈ 𝐵)) → ((𝑋(le‘𝐾)((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍))(le‘𝐾)((𝑋 𝑌) 𝑍)))
463, 10, 12, 9, 45syl13anc 1320 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋(le‘𝐾)((𝑋 𝑌) 𝑍) ∧ (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍)) ↔ (𝑋 (𝑌 𝑍))(le‘𝐾)((𝑋 𝑌) 𝑍)))
4736, 44, 46mpbi2and 958 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍))(le‘𝐾)((𝑋 𝑌) 𝑍))
481, 2, 3, 9, 14, 31, 47latasymd 16880 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-lat 16869
This theorem is referenced by:  latj12  16919  latj32  16920  latj4  16924  latmass  17011  latmassOLD  33534  hlatjass  33674  cvrexchlem  33723  cvrat3  33746  2atmat  33865  4atlem3  33900  4atlem3a  33901  4atlem4a  33903  4atlem4d  33906  4at2  33918  2lplnja  33923  pmapjlln1  34159  dalawlem3  34177  dalawlem12  34186  cdleme30a  34684  trlcolem  35032  cdlemh1  35121  cdlemkid1  35228  doca2N  35433  djajN  35444
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