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Theorem latmmdiN 36250
Description: Lattice meet distributes over itself. (inindi 4200 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
olmass.b 𝐵 = (Base‘𝐾)
olmass.m = (meet‘𝐾)
Assertion
Ref Expression
latmmdiN ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Proof of Theorem latmmdiN
StepHypRef Expression
1 ollat 36229 . . . . 5 (𝐾 ∈ OL → 𝐾 ∈ Lat)
21adantr 481 . . . 4 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
3 simpr1 1186 . . . 4 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
4 olmass.b . . . . 5 𝐵 = (Base‘𝐾)
5 olmass.m . . . . 5 = (meet‘𝐾)
64, 5latmidm 17684 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
72, 3, 6syl2anc 584 . . 3 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑋) = 𝑋)
87oveq1d 7160 . 2 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑋) (𝑌 𝑍)) = (𝑋 (𝑌 𝑍)))
9 simpl 483 . . 3 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
10 simpr2 1187 . . 3 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
11 simpr3 1188 . . 3 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
124, 5latm4 36249 . . 3 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑋𝐵) ∧ (𝑌𝐵𝑍𝐵)) → ((𝑋 𝑋) (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
139, 3, 3, 10, 11, 12syl122anc 1371 . 2 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑋) (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
148, 13eqtr3d 2855 1 ((𝐾 ∈ OL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  meetcmee 17543  Latclat 17643  OLcol 36190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-lat 17644  df-oposet 36192  df-ol 36194
This theorem is referenced by:  omlfh1N  36274
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