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Theorem joincom 18357
Description: The join of a poset is commutative. (The antecedent 𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom i.e., "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
joincom.b 𝐵 = (Base‘𝐾)
joincom.j = (join‘𝐾)
Assertion
Ref Expression
joincom (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ⟨𝑌, 𝑋⟩ ∈ dom )) → (𝑋 𝑌) = (𝑌 𝑋))

Proof of Theorem joincom
StepHypRef Expression
1 joincom.b . . 3 𝐵 = (Base‘𝐾)
2 joincom.j . . 3 = (join‘𝐾)
31, 2joincomALT 18356 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
43adantr 480 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 395  w3a 1084   = wceq 1533  wcel 2098  cop 4626  dom cdm 5666  cfv 6533  (class class class)co 7401  Basecbs 17143  Posetcpo 18262  joincjn 18266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-lub 18301  df-join 18303
This theorem is referenced by:  latjcom  18402
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