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Theorem toslub 30828
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 30827 . . 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 7147 . 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 264 . . 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 17710 . 2 (𝜑 → ((lub‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))))
111, 5, 2tosso 17762 . . . . 5 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 270 . . . 4 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 498 . . 3 (𝐾 ∈ Toset → < Or 𝐵)
14 id 22 . . . 4 ( < Or 𝐵< Or 𝐵)
1514supval2 8992 . . 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 2783 1 (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3053  wrex 3054  wss 3843   class class class wbr 5030   I cid 5428   Or wor 5441  cres 5527  cfv 6339  crio 7126  supcsup 8977  Basecbs 16586  lecple 16675  ltcplt 17667  lubclub 17668  Tosetctos 17759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-po 5442  df-so 5443  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-sup 8979  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-toset 17760
This theorem is referenced by:  xrsp1  30868
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