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Theorem toslub 28108
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 2402 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
61, 2, 3, 4, 5toslublem 28107 . . 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 6256 . 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 2402 . . 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 15938 . 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 15990 . . . . 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 457 . . 3  |-  ( K  e. Toset  ->  .<  Or  B )
14 id 22 . . . 4  |-  (  .<  Or  B  ->  .<  Or  B
)
1514supval2 7948 . . 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 18 . 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 2453 1  |-  ( ph  ->  ( ( lub `  K
) `  A )  =  sup ( A ,  B ,  .<  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755    C_ wss 3414   class class class wbr 4395    _I cid 4733    Or wor 4743    |` cres 4825   ` cfv 5569   iota_crio 6239   supcsup 7934   Basecbs 14841   lecple 14916   ltcplt 15894   lubclub 15895  Tosetctos 15987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-sup 7935  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-toset 15988
This theorem is referenced by:  xrsp1  28122
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