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Theorem luble 18354
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 breq1 5152 . 2 (𝑦 = 𝑋 → (𝑦 (𝑈𝑆) ↔ 𝑋 (𝑈𝑆)))
2 lubprop.b . . . 4 𝐵 = (Base‘𝐾)
3 lubprop.l . . . 4 = (le‘𝐾)
4 lubprop.u . . . 4 𝑈 = (lub‘𝐾)
5 lubprop.k . . . 4 (𝜑𝐾𝑉)
6 lubprop.s . . . 4 (𝜑𝑆 ∈ dom 𝑈)
72, 3, 4, 5, 6lubprop 18353 . . 3 (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
87simpld 493 . 2 (𝜑 → ∀𝑦𝑆 𝑦 (𝑈𝑆))
9 luble.x . 2 (𝜑𝑋𝑆)
101, 8, 9rspcdva 3607 1 (𝜑𝑋 (𝑈𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3050   class class class wbr 5149  dom cdm 5678  cfv 6549  Basecbs 17183  lecple 17243  lubclub 18304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-lub 18341
This theorem is referenced by:  ple1  18425  lubsscl  48165
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