Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  toslub Structured version   Visualization version   GIF version

Theorem toslub 30648
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 2819 . . . 4 (le‘𝐾) = (le‘𝐾)
61, 2, 3, 4, 5toslublem 30647 . . 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 7125 . 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 2819 . . 3 (lub‘𝐾) = (lub‘𝐾)
9 biid 263 . . 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 17586 . 2 (𝜑 → ((lub‘𝐾)‘𝐴) = (𝑎𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐𝐵 (∀𝑏𝐴 𝑏(le‘𝐾)𝑐𝑎(le‘𝐾)𝑐))))
111, 5, 2tosso 17638 . . . . 5 (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
1211ibi 269 . . . 4 (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
1312simpld 497 . . 3 (𝐾 ∈ Toset → < Or 𝐵)
14 id 22 . . . 4 ( < Or 𝐵< Or 𝐵)
1514supval2 8911 . . 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 2864 1 (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1531  wcel 2108  wral 3136  wrex 3137  wss 3934   class class class wbr 5057   I cid 5452   Or wor 5466  cres 5550  cfv 6348  crio 7105  supcsup 8896  Basecbs 16475  lecple 16564  ltcplt 17543  lubclub 17544  Tosetctos 17635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7106  df-sup 8898  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-toset 17636
This theorem is referenced by:  xrsp1  30662
  Copyright terms: Public domain W3C validator