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Theorem glbprop 17609
Description: Properties of greatest lower bound of a poset. (Contributed 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)
Assertion
Ref Expression
glbprop (𝜑 → (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
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 glbprop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 glbprop.b . . . 4 𝐵 = (Base‘𝐾)
2 glbprop.l . . . 4 = (le‘𝐾)
3 glbprop.u . . . 4 𝑈 = (glb‘𝐾)
4 biid 264 . . . 4 ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
5 glbprop.k . . . 4 (𝜑𝐾𝑉)
6 glbprop.s . . . . 5 (𝜑𝑆 ∈ dom 𝑈)
71, 2, 3, 5, 6glbelss 17605 . . . 4 (𝜑𝑆𝐵)
81, 2, 3, 4, 5, 7glbval 17607 . . 3 (𝜑 → (𝑈𝑆) = (𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))))
98eqcomd 2830 . 2 (𝜑 → (𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))) = (𝑈𝑆))
101, 3, 5, 6glbcl 17608 . . 3 (𝜑 → (𝑈𝑆) ∈ 𝐵)
111, 2, 3, 4, 5, 6glbeu 17606 . . 3 (𝜑 → ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
12 breq1 5055 . . . . . 6 (𝑥 = (𝑈𝑆) → (𝑥 𝑦 ↔ (𝑈𝑆) 𝑦))
1312ralbidv 3192 . . . . 5 (𝑥 = (𝑈𝑆) → (∀𝑦𝑆 𝑥 𝑦 ↔ ∀𝑦𝑆 (𝑈𝑆) 𝑦))
14 breq2 5056 . . . . . . 7 (𝑥 = (𝑈𝑆) → (𝑧 𝑥𝑧 (𝑈𝑆)))
1514imbi2d 344 . . . . . 6 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
1615ralbidv 3192 . . . . 5 (𝑥 = (𝑈𝑆) → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
1713, 16anbi12d 633 . . . 4 (𝑥 = (𝑈𝑆) → ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆)))))
1817riota2 7132 . . 3 (((𝑈𝑆) ∈ 𝐵 ∧ ∃!𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))) → ((∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))) ↔ (𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))) = (𝑈𝑆)))
1910, 11, 18syl2anc 587 . 2 (𝜑 → ((∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))) ↔ (𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))) = (𝑈𝑆)))
209, 19mpbird 260 1 (𝜑 → (∀𝑦𝑆 (𝑈𝑆) 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 (𝑈𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  ∃!wreu 3135   class class class wbr 5052  dom cdm 5542  cfv 6343  crio 7106  Basecbs 16483  lecple 16572  glbcglb 17553
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-glb 17585
This theorem is referenced by:  glble  17610  clatglb  17734
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