Theorem meetcom 18585

Description: The meet of a poset is commutative. (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 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ∧ ⟨𝑌, 𝑋⟩ ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))

Proof of Theorem meetcom

Step	Hyp	Ref	Expression
1		meetcom.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		meetcom.m	. . 3 ⊢ ∧ = (meet‘𝐾)
3	1, 2	meetcomALT 18584	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
4	3	adantr 486	1 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (⟨𝑋, 𝑌⟩ ∈ dom ∧ ∧ ⟨𝑌, 𝑋⟩ ∈ dom ∧ )) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ⟨cop 4834  dom cdm 5651  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  Posetcpo 18490  meetcmee 18495

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-glb 18528  df-meet 18530

This theorem is used by:  latmcom  18646