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Theorem joincom 18399
Description: The join of a poset is commutative. (The antecedent βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ 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 𝐡 = (Baseβ€˜πΎ)
joincom.j ∨ = (joinβ€˜πΎ)
Assertion
Ref Expression
joincom (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ dom ∨ )) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))

Proof of Theorem joincom
StepHypRef Expression
1 joincom.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 joincom.j . . 3 ∨ = (joinβ€˜πΎ)
31, 2joincomALT 18398 . 2 ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
43adantr 479 1 (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (βŸ¨π‘‹, π‘ŒβŸ© ∈ dom ∨ ∧ βŸ¨π‘Œ, π‘‹βŸ© ∈ dom ∨ )) β†’ (𝑋 ∨ π‘Œ) = (π‘Œ ∨ 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4636  dom cdm 5680  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  Posetcpo 18304  joincjn 18308
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-lub 18343  df-join 18345
This theorem is referenced by:  latjcom  18444
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