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Theorem glble 17878
Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
glbprop.b 𝐵 = (Base‘𝐾)
glbprop.l = (le‘𝐾)
glbprop.u 𝑈 = (glb‘𝐾)
glbprop.k (𝜑𝐾𝑉)
glbprop.s (𝜑𝑆 ∈ dom 𝑈)
glble.x (𝜑𝑋𝑆)
Assertion
Ref Expression
glble (𝜑 → (𝑈𝑆) 𝑋)

Proof of Theorem glble
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 5057 . 2 (𝑦 = 𝑋 → ((𝑈𝑆) 𝑦 ↔ (𝑈𝑆) 𝑋))
2 glbprop.b . . . 4 𝐵 = (Base‘𝐾)
3 glbprop.l . . . 4 = (le‘𝐾)
4 glbprop.u . . . 4 𝑈 = (glb‘𝐾)
5 glbprop.k . . . 4 (𝜑𝐾𝑉)
6 glbprop.s . . . 4 (𝜑𝑆 ∈ dom 𝑈)
72, 3, 4, 5, 6glbprop 17877 . . 3 (𝜑 → (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
87simpld 498 . 2 (𝜑 → ∀𝑦𝑆 (𝑈𝑆) 𝑦)
9 glble.x . 2 (𝜑𝑋𝑆)
101, 8, 9rspcdva 3539 1 (𝜑 → (𝑈𝑆) 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  wral 3061   class class class wbr 5053  dom cdm 5551  cfv 6380  Basecbs 16760  lecple 16809  glbcglb 17817
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
This theorem is referenced by:  p0le  17935  clatglble  18023  glbsscl  45928
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