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Theorem p0le 17647
 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 17645 . . 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 17604 . 2 (𝜑 → (𝐺𝐵) 𝑋)
116, 10eqbrtrd 5052 1 (𝜑0 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  Basecbs 16477  lecple 16566  glbcglb 17547  0.cp0 17641 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-glb 17579  df-p0 17643 This theorem is referenced by:  op0le  36498  atl0le  36616
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