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Theorem toslub 30582
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 2818 . . . 4 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5toslublem 30581 . . 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 7122 . 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 2818 . . 3 (lub‘𝐾) = (lub‘𝐾)
9 biid 262 . . 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 17582 . 2 (𝜑 → ((lub‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))))
111, 5, 2tosso 17634 . . . . 5 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 268 . . . 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 8907 . . 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 2863 1 (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057   I cid 5452   Or wor 5466  cres 5550  cfv 6348  crio 7102  supcsup 8892  Basecbs 16471  lecple 16560  ltcplt 17539  lubclub 17540  Tosetctos 17631
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-3or 1080  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-rmo 3143  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-po 5467  df-so 5468  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-sup 8894  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-toset 17632
This theorem is referenced by:  xrsp1  30596
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