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Theorem meetcom 17758
Description: The meet of a poset commutes. (The antecedent 𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom i.e. "the meets exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 17-Sep-2011.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetcom.b 𝐵 = (Base‘𝐾)
meetcom.m = (meet‘𝐾)
Assertion
Ref Expression
meetcom (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom )) → (𝑋 𝑌) = (𝑌 𝑋))

Proof of Theorem meetcom
StepHypRef Expression
1 meetcom.b . . 3 𝐵 = (Base‘𝐾)
2 meetcom.m . . 3 = (meet‘𝐾)
31, 2meetcomALT 17757 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
43adantr 484 1 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom )) → (𝑋 𝑌) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  cop 4522  dom cdm 5525  cfv 6339  (class class class)co 7170  Basecbs 16586  Posetcpo 17666  meetcmee 17671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-glb 17701  df-meet 17703
This theorem is referenced by:  latmcom  17801
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