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Theorem latjass 17142
Description: Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 28520 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 2651 . 2 (le‘𝐾) = (le‘𝐾)
3 simpl 472 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
4 latjass.j . . . . 5 = (join‘𝐾)
51, 4latjcl 17098 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
653adant3r3 1297 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
7 simpr3 1089 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
81, 4latjcl 17098 . . 3 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
93, 6, 7, 8syl3anc 1366 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) ∈ 𝐵)
10 simpr1 1087 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
111, 4latjcl 17098 . . . 4 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
12113adant3r1 1295 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
131, 4latjcl 17098 . . 3 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
143, 10, 12, 13syl3anc 1366 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
151, 2, 4latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → 𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)))
163, 10, 12, 15syl3anc 1366 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)(𝑋 (𝑌 𝑍)))
17 simpr2 1088 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
181, 2, 4latlej1 17107 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → 𝑌(le‘𝐾)(𝑌 𝑍))
19183adant3r1 1295 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑌 𝑍))
201, 2, 4latlej2 17108 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑌 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
213, 10, 12, 20syl3anc 1366 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
221, 2, 3, 17, 12, 14, 19, 21lattrd 17105 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑋 (𝑌 𝑍)))
231, 2, 4latjle12 17109 . . . . 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 1368 . . . 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 976 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌)(le‘𝐾)(𝑋 (𝑌 𝑍)))
261, 2, 4latlej2 17108 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → 𝑍(le‘𝐾)(𝑌 𝑍))
27263adant3r1 1295 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)(𝑌 𝑍))
281, 2, 3, 7, 12, 14, 27, 21lattrd 17105 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)(𝑋 (𝑌 𝑍)))
291, 2, 4latjle12 17109 . . . 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 1368 . . 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 976 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍)(le‘𝐾)(𝑋 (𝑌 𝑍)))
321, 2, 4latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑋(le‘𝐾)(𝑋 𝑌))
33323adant3r3 1297 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)(𝑋 𝑌))
341, 2, 4latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → (𝑋 𝑌)(le‘𝐾)((𝑋 𝑌) 𝑍))
353, 6, 7, 34syl3anc 1366 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌)(le‘𝐾)((𝑋 𝑌) 𝑍))
361, 2, 3, 10, 6, 9, 33, 35lattrd 17105 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋(le‘𝐾)((𝑋 𝑌) 𝑍))
371, 2, 4latlej2 17108 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(𝑋 𝑌))
38373adant3r3 1297 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)(𝑋 𝑌))
391, 2, 3, 17, 6, 9, 38, 35lattrd 17105 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌(le‘𝐾)((𝑋 𝑌) 𝑍))
401, 2, 4latlej2 17108 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵𝑍𝐵) → 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍))
413, 6, 7, 40syl3anc 1366 . . . 4 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍(le‘𝐾)((𝑋 𝑌) 𝑍))
421, 2, 4latjle12 17109 . . . . 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 1368 . . . 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 976 . . 3 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍)(le‘𝐾)((𝑋 𝑌) 𝑍))
451, 2, 4latjle12 17109 . . . 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 1368 . . 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 976 . 2 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍))(le‘𝐾)((𝑋 𝑌) 𝑍))
481, 2, 3, 9, 14, 31, 47latasymd 17104 1 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093
This theorem is referenced by:  latj12  17143  latj32  17144  latj4  17148  latmass  17235  latmassOLD  34834  hlatjass  34974  cvrexchlem  35023  cvrat3  35046  2atmat  35165  4atlem3  35200  4atlem3a  35201  4atlem4a  35203  4atlem4d  35206  4at2  35218  2lplnja  35223  pmapjlln1  35459  dalawlem3  35477  dalawlem12  35486  cdleme30a  35983  trlcolem  36331  cdlemh1  36420  cdlemkid1  36527  doca2N  36732  djajN  36743
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