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Theorem toslub 29642
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 2620 . . . 4 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5toslublem 29641 . . 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 6612 . 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 2620 . . 3 (lub‘𝐾) = (lub‘𝐾)
9 biid 251 . . 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 16965 . 2 (𝜑 → ((lub‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))))
111, 5, 2tosso 17017 . . . . 5 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 256 . . . 4 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 475 . . 3 (𝐾 ∈ Toset → < Or 𝐵)
14 id 22 . . . 4 ( < Or 𝐵< Or 𝐵)
1514supval2 8346 . . 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 2664 1 (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wcel 1988  wral 2909  wrex 2910  wss 3567   class class class wbr 4644   I cid 5013   Or wor 5024  cres 5106  cfv 5876  crio 6595  supcsup 8331  Basecbs 15838  lecple 15929  ltcplt 16922  lubclub 16923  Tosetctos 17014
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-3or 1037  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-rmo 2917  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-po 5025  df-so 5026  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-sup 8333  df-preset 16909  df-poset 16927  df-plt 16939  df-lub 16955  df-toset 17015
This theorem is referenced by:  xrsp1  29656
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