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

Theorem toslub 27304
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  |-  ( ph  ->  K  e. Toset )
toslub.2  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
toslub  |-  ( ph  ->  ( ( 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  |-  ( ph  ->  K  e. Toset )
4 toslub.2 . . . 4  |-  ( ph  ->  A  C_  B )
5 eqid 2460 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
61, 2, 3, 4, 5toslublem 27303 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (
( A. b  e.  A  b ( le
`  K ) a  /\  A. c  e.  B  ( A. b  e.  A  b ( le `  K ) c  ->  a ( le
`  K ) c ) )  <->  ( A. b  e.  A  -.  a  .<  b  /\  A. b  e.  B  (
b  .<  a  ->  E. d  e.  A  b  .<  d ) ) ) )
76riotabidva 6253 . 2  |-  ( ph  ->  ( iota_ a  e.  B  ( A. b  e.  A  b ( le `  K ) a  /\  A. c  e.  B  ( A. b  e.  A  b ( le `  K ) c  -> 
a ( le `  K ) c ) ) )  =  (
iota_ a  e.  B  ( A. b  e.  A  -.  a  .<  b  /\  A. b  e.  B  ( b  .<  a  ->  E. d  e.  A  b 
.<  d ) ) ) )
8 eqid 2460 . . 3  |-  ( lub `  K )  =  ( lub `  K )
9 biid 236 . . 3  |-  ( ( A. b  e.  A  b ( le `  K ) a  /\  A. c  e.  B  ( A. b  e.  A  b ( le `  K ) c  -> 
a ( le `  K ) c ) )  <->  ( A. b  e.  A  b ( le `  K ) a  /\  A. c  e.  B  ( A. b  e.  A  b ( le `  K ) c  ->  a ( le
`  K ) c ) ) )
101, 5, 8, 9, 3, 4lubval 15460 . 2  |-  ( ph  ->  ( ( lub `  K
) `  A )  =  ( iota_ a  e.  B  ( A. b  e.  A  b ( le `  K ) a  /\  A. c  e.  B  ( A. b  e.  A  b ( le `  K ) c  ->  a ( le
`  K ) c ) ) ) )
111, 5, 2tosso 15512 . . . . 5  |-  ( K  e. Toset  ->  ( K  e. Toset  <->  ( 
.<  Or  B  /\  (  _I  |`  B )  C_  ( le `  K ) ) ) )
1211ibi 241 . . . 4  |-  ( K  e. Toset  ->  (  .<  Or  B  /\  (  _I  |`  B ) 
C_  ( le `  K ) ) )
1312simpld 459 . . 3  |-  ( K  e. Toset  ->  .<  Or  B )
14 id 22 . . . 4  |-  (  .<  Or  B  ->  .<  Or  B
)
1514supval2 7904 . . 3  |-  (  .<  Or  B  ->  sup ( A ,  B ,  .<  )  =  ( iota_ a  e.  B  ( A. b  e.  A  -.  a  .<  b  /\  A. b  e.  B  (
b  .<  a  ->  E. d  e.  A  b  .<  d ) ) ) )
163, 13, 153syl 20 . 2  |-  ( ph  ->  sup ( A ,  B ,  .<  )  =  ( iota_ a  e.  B  ( A. b  e.  A  -.  a  .<  b  /\  A. b  e.  B  ( b  .<  a  ->  E. d  e.  A  b 
.<  d ) ) ) )
177, 10, 163eqtr4d 2511 1  |-  ( ph  ->  ( ( lub `  K
) `  A )  =  sup ( A ,  B ,  .<  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   class class class wbr 4440    _I cid 4783    Or wor 4792    |` cres 4994   ` cfv 5579   iota_crio 6235   supcsup 7889   Basecbs 14479   lecple 14551   ltcplt 15417   lubclub 15418  Tosetctos 15509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-sup 7890  df-poset 15422  df-plt 15434  df-lub 15450  df-toset 15510
This theorem is referenced by:  xrsp1  27318
  Copyright terms: Public domain W3C validator