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Theorem posjidm 48768
Description: Poset join is idempotent. latjidm 18519 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.)
Hypotheses
Ref Expression
posjidm.b 𝐵 = (Base‘𝐾)
posjidm.j = (join‘𝐾)
Assertion
Ref Expression
posjidm ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)

Proof of Theorem posjidm
StepHypRef Expression
1 eqid 2734 . . 3 (lub‘𝐾) = (lub‘𝐾)
2 posjidm.j . . 3 = (join‘𝐾)
3 simpl 482 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝐾 ∈ Poset)
4 simpr 484 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋𝐵)
51, 2, 3, 4, 4joinval 18434 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 𝑋) = ((lub‘𝐾)‘{𝑋, 𝑋}))
6 posjidm.b . . 3 𝐵 = (Base‘𝐾)
7 eqid 2734 . . 3 (le‘𝐾) = (le‘𝐾)
86, 7posref 18375 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → 𝑋(le‘𝐾)𝑋)
9 eqidd 2735 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → {𝑋, 𝑋} = {𝑋, 𝑋})
103, 6, 4, 4, 7, 8, 9, 1lubpr 48760 . 2 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → ((lub‘𝐾)‘{𝑋, 𝑋}) = 𝑋)
115, 10eqtrd 2774 1 ((𝐾 ∈ Poset ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  {cpr 4632  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  Posetcpo 18364  lubclub 18366  joincjn 18368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-lub 18403  df-join 18405
This theorem is referenced by: (None)
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