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Theorem toslub 26275
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 2454 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
61, 2, 3, 4, 5toslublem 26274 . . 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 6179 . 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 2454 . . 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 15274 . 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 15326 . . . . 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 7817 . . 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 2505 1  |-  ( ph  ->  ( ( lub `  K
) `  A )  =  sup ( A ,  B ,  .<  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799   E.wrex 2800    C_ wss 3437   class class class wbr 4401    _I cid 4740    Or wor 4749    |` cres 4951   ` cfv 5527   iota_crio 6161   supcsup 7802   Basecbs 14293   lecple 14365   ltcplt 15231   lubclub 15232  Tosetctos 15323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-sup 7803  df-poset 15236  df-plt 15248  df-lub 15264  df-toset 15324
This theorem is referenced by:  xrsp1  26289
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