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Theorem luble 16927
Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)
Hypotheses
Ref Expression
lubprop.b 𝐵 = (Base‘𝐾)
lubprop.l = (le‘𝐾)
lubprop.u 𝑈 = (lub‘𝐾)
lubprop.k (𝜑𝐾𝑉)
lubprop.s (𝜑𝑆 ∈ dom 𝑈)
luble.x (𝜑𝑋𝑆)
Assertion
Ref Expression
luble (𝜑𝑋 (𝑈𝑆))

Proof of Theorem luble
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lubprop.b . . . 4 𝐵 = (Base‘𝐾)
2 lubprop.l . . . 4 = (le‘𝐾)
3 lubprop.u . . . 4 𝑈 = (lub‘𝐾)
4 lubprop.k . . . 4 (𝜑𝐾𝑉)
5 lubprop.s . . . 4 (𝜑𝑆 ∈ dom 𝑈)
61, 2, 3, 4, 5lubprop 16926 . . 3 (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
76simpld 475 . 2 (𝜑 → ∀𝑦𝑆 𝑦 (𝑈𝑆))
8 luble.x . 2 (𝜑𝑋𝑆)
9 breq1 4626 . . 3 (𝑦 = 𝑋 → (𝑦 (𝑈𝑆) ↔ 𝑋 (𝑈𝑆)))
109rspccva 3298 . 2 ((∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ 𝑋𝑆) → 𝑋 (𝑈𝑆))
117, 8, 10syl2anc 692 1 (𝜑𝑋 (𝑈𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908   class class class wbr 4623  dom cdm 5084  cfv 5857  Basecbs 15800  lecple 15888  lubclub 16882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-lub 16914
This theorem is referenced by:  ple1  16984
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