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Theorem joincomALT 17010
Description: The join of a poset commutes. (This may not be a theorem under other definitions of meet.) (Contributed by NM, 16-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
joincom.b 𝐵 = (Base‘𝐾)
joincom.j = (join‘𝐾)
Assertion
Ref Expression
joincomALT ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))

Proof of Theorem joincomALT
StepHypRef Expression
1 prcom 4258 . . . 4 {𝑌, 𝑋} = {𝑋, 𝑌}
21fveq2i 6181 . . 3 ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌})
32a1i 11 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌}))
4 eqid 2620 . . 3 (lub‘𝐾) = (lub‘𝐾)
5 joincom.j . . 3 = (join‘𝐾)
6 simp1 1059 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝐾𝑉)
7 simp3 1061 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 simp2 1060 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑋𝐵)
94, 5, 6, 7, 8joinval 16986 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = ((lub‘𝐾)‘{𝑌, 𝑋}))
104, 5, 6, 8, 7joinval 16986 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
113, 9, 103eqtr4rd 2665 1 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  {cpr 4170  cfv 5876  (class class class)co 6635  Basecbs 15838  lubclub 16923  joincjn 16925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-lub 16955  df-join 16957
This theorem is referenced by:  joincom  17011
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