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Theorem lubprop 16967
Description: Properties of greatest lower bound of a poset. (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)
Assertion
Ref Expression
lubprop (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
Distinct variable groups:   𝑧,𝐵   𝑦,𝑧,𝐾   𝑦,𝑆,𝑧   𝑦,   𝑦,𝑈,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑦)   (𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem lubprop
Dummy variable 𝑥 is 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 biid 251 . . . 4 ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
5 lubprop.k . . . 4 (𝜑𝐾𝑉)
6 lubprop.s . . . . 5 (𝜑𝑆 ∈ dom 𝑈)
71, 2, 3, 5, 6lubelss 16963 . . . 4 (𝜑𝑆𝐵)
81, 2, 3, 4, 5, 7lubval 16965 . . 3 (𝜑 → (𝑈𝑆) = (𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))))
98eqcomd 2626 . 2 (𝜑 → (𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))) = (𝑈𝑆))
101, 3, 5, 6lubcl 16966 . . 3 (𝜑 → (𝑈𝑆) ∈ 𝐵)
111, 2, 3, 4, 5, 6lubeu 16964 . . 3 (𝜑 → ∃!𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
12 breq2 4648 . . . . . 6 (𝑥 = (𝑈𝑆) → (𝑦 𝑥𝑦 (𝑈𝑆)))
1312ralbidv 2983 . . . . 5 (𝑥 = (𝑈𝑆) → (∀𝑦𝑆 𝑦 𝑥 ↔ ∀𝑦𝑆 𝑦 (𝑈𝑆)))
14 breq1 4647 . . . . . . 7 (𝑥 = (𝑈𝑆) → (𝑥 𝑧 ↔ (𝑈𝑆) 𝑧))
1514imbi2d 330 . . . . . 6 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
1615ralbidv 2983 . . . . 5 (𝑥 = (𝑈𝑆) → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
1713, 16anbi12d 746 . . . 4 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧))))
1817riota2 6618 . . 3 (((𝑈𝑆) ∈ 𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))) → ((∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)) ↔ (𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))) = (𝑈𝑆)))
1910, 11, 18syl2anc 692 . 2 (𝜑 → ((∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)) ↔ (𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))) = (𝑈𝑆)))
209, 19mpbird 247 1 (𝜑 → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  ∃!wreu 2911   class class class wbr 4644  dom cdm 5104  cfv 5876  crio 6595  Basecbs 15838  lecple 15929  lubclub 16923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-lub 16955
This theorem is referenced by:  luble  16968  lublem  17099
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