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Theorem p0le 17935
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
p0le.b 𝐵 = (Base‘𝐾)
p0le.g 𝐺 = (glb‘𝐾)
p0le.l = (le‘𝐾)
p0le.0 0 = (0.‘𝐾)
p0le.k (𝜑𝐾𝑉)
p0le.x (𝜑𝑋𝐵)
p0le.d (𝜑𝐵 ∈ dom 𝐺)
Assertion
Ref Expression
p0le (𝜑0 𝑋)

Proof of Theorem p0le
StepHypRef Expression
1 p0le.k . . 3 (𝜑𝐾𝑉)
2 p0le.b . . . 4 𝐵 = (Base‘𝐾)
3 p0le.g . . . 4 𝐺 = (glb‘𝐾)
4 p0le.0 . . . 4 0 = (0.‘𝐾)
52, 3, 4p0val 17933 . . 3 (𝐾𝑉0 = (𝐺𝐵))
61, 5syl 17 . 2 (𝜑0 = (𝐺𝐵))
7 p0le.l . . 3 = (le‘𝐾)
8 p0le.d . . 3 (𝜑𝐵 ∈ dom 𝐺)
9 p0le.x . . 3 (𝜑𝑋𝐵)
102, 7, 3, 1, 8, 9glble 17878 . 2 (𝜑 → (𝐺𝐵) 𝑋)
116, 10eqbrtrd 5075 1 (𝜑0 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110   class class class wbr 5053  dom cdm 5551  cfv 6380  Basecbs 16760  lecple 16809  glbcglb 17817  0.cp0 17929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-glb 17853  df-p0 17931
This theorem is referenced by:  op0le  36937  atl0le  37055
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