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Theorem joincomALT 18341
Description: The join of a poset is commutative. (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 4732 . . . 4 {𝑌, 𝑋} = {𝑋, 𝑌}
21fveq2i 6884 . . 3 ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌})
32a1i 11 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → ((lub‘𝐾)‘{𝑌, 𝑋}) = ((lub‘𝐾)‘{𝑋, 𝑌}))
4 eqid 2733 . . 3 (lub‘𝐾) = (lub‘𝐾)
5 joincom.j . . 3 = (join‘𝐾)
6 simp1 1137 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝐾𝑉)
7 simp3 1139 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑌𝐵)
8 simp2 1138 . . 3 ((𝐾𝑉𝑋𝐵𝑌𝐵) → 𝑋𝐵)
94, 5, 6, 7, 8joinval 18317 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑌 𝑋) = ((lub‘𝐾)‘{𝑌, 𝑋}))
104, 5, 6, 8, 7joinval 18317 . 2 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = ((lub‘𝐾)‘{𝑋, 𝑌}))
113, 9, 103eqtr4rd 2784 1 ((𝐾𝑉𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑌 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  {cpr 4626  cfv 6535  (class class class)co 7396  Basecbs 17131  lubclub 18249  joincjn 18251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-lub 18286  df-join 18288
This theorem is referenced by:  joincom  18342
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