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Theorem toslub 31251
Description: In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
Hypotheses
Ref Expression
toslub.b 𝐵 = (Base‘𝐾)
toslub.l < = (lt‘𝐾)
toslub.1 (𝜑𝐾 ∈ Toset)
toslub.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
toslub (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Proof of Theorem toslub
Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toslub.b . . . 4 𝐵 = (Base‘𝐾)
2 toslub.l . . . 4 < = (lt‘𝐾)
3 toslub.1 . . . 4 (𝜑𝐾 ∈ Toset)
4 toslub.2 . . . 4 (𝜑𝐴𝐵)
5 eqid 2738 . . . 4 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5toslublem 31250 . . 3 ((𝜑𝑎𝐵) → ((∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐)) ↔ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
76riotabidva 7252 . 2 (𝜑 → (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
8 eqid 2738 . . 3 (lub‘𝐾) = (lub‘𝐾)
9 biid 260 . . 3 ((∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐)) ↔ (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐)))
101, 5, 8, 9, 3, 4lubval 18074 . 2 (𝜑 → ((lub‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))))
111, 5, 2tosso 18137 . . . . 5 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 266 . . . 4 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 495 . . 3 (𝐾 ∈ Toset → < Or 𝐵)
14 id 22 . . . 4 ( < Or 𝐵< Or 𝐵)
1514supval2 9214 . . 3 ( < Or 𝐵 → sup(𝐴, 𝐵, < ) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
163, 13, 153syl 18 . 2 (𝜑 → sup(𝐴, 𝐵, < ) = (𝑎𝐵 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏𝐵 (𝑏 < 𝑎 → ∃𝑑𝐴 𝑏 < 𝑑))))
177, 10, 163eqtr4d 2788 1 (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106  wral 3064  wrex 3065  wss 3887   class class class wbr 5074   I cid 5488   Or wor 5502  cres 5591  cfv 6433  crio 7231  supcsup 9199  Basecbs 16912  lecple 16969  ltcplt 18026  lubclub 18027  Tosetctos 18134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-sup 9201  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-toset 18135
This theorem is referenced by:  xrsp1  31291
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