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Theorem lubl 17718
Description: The LUB of a complete lattice subset is the least bound. (Contributed by NM, 19-Oct-2011.)
Hypotheses
Ref Expression
lublem.b 𝐵 = (Base‘𝐾)
lublem.l = (le‘𝐾)
lublem.u 𝑈 = (lub‘𝐾)
Assertion
Ref Expression
lubl ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝐵) → (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋))
Distinct variable groups:   𝑦,𝐾   𝑦,𝑆   𝑦,𝑈   𝑦,   𝑦,𝑋
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem lubl
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 lublem.b . . . 4 𝐵 = (Base‘𝐾)
2 lublem.l . . . 4 = (le‘𝐾)
3 lublem.u . . . 4 𝑈 = (lub‘𝐾)
41, 2, 3lublem 17716 . . 3 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → (∀𝑦𝑆 𝑦 (𝑈𝑆) ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧)))
54simprd 496 . 2 ((𝐾 ∈ CLat ∧ 𝑆𝐵) → ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧))
6 breq2 5061 . . . . 5 (𝑧 = 𝑋 → (𝑦 𝑧𝑦 𝑋))
76ralbidv 3194 . . . 4 (𝑧 = 𝑋 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑋))
8 breq2 5061 . . . 4 (𝑧 = 𝑋 → ((𝑈𝑆) 𝑧 ↔ (𝑈𝑆) 𝑋))
97, 8imbi12d 346 . . 3 (𝑧 = 𝑋 → ((∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋)))
109rspccva 3619 . 2 ((∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧 → (𝑈𝑆) 𝑧) ∧ 𝑋𝐵) → (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋))
115, 10stoic3 1768 1 ((𝐾 ∈ CLat ∧ 𝑆𝐵𝑋𝐵) → (∀𝑦𝑆 𝑦 𝑋 → (𝑈𝑆) 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  lubclub 17540  CLatccla 17705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-lub 17572  df-clat 17706
This theorem is referenced by:  lubss  17719  lubun  17721
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