Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  glble Structured version   Visualization version   GIF version

Theorem glble 17604
 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 5034 . 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 17603 . . 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 3573 1 (𝜑 → (𝑈𝑆) 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  Basecbs 16477  lecple 16566  glbcglb 17547 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 This theorem is referenced by:  p0le  17647  clatglble  17729
 Copyright terms: Public domain W3C validator