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Theorem tosglb 27170
Description: Same theorem as toslub 27168, for infinimum. (Contributed by Thierry Arnoux, 17-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.)
Hypotheses
Ref Expression
tosglb.b  |-  B  =  ( Base `  K
)
tosglb.l  |-  .<  =  ( lt `  K )
tosglb.1  |-  ( ph  ->  K  e. Toset )
tosglb.2  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
tosglb  |-  ( ph  ->  ( ( glb `  K
) `  A )  =  sup ( A ,  B ,  `'  .<  ) )

Proof of Theorem tosglb
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tosglb.b . . . 4  |-  B  =  ( Base `  K
)
2 tosglb.l . . . 4  |-  .<  =  ( lt `  K )
3 tosglb.1 . . . 4  |-  ( ph  ->  K  e. Toset )
4 tosglb.2 . . . 4  |-  ( ph  ->  A  C_  B )
5 eqid 2460 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
61, 2, 3, 4, 5tosglblem 27169 . . 3  |-  ( (
ph  /\  a  e.  B )  ->  (
( A. b  e.  A  a ( le
`  K ) b  /\  A. c  e.  B  ( A. b  e.  A  c ( le `  K ) b  ->  c ( le
`  K ) a ) )  <->  ( 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  a ( le `  K ) b  /\  A. c  e.  B  ( A. b  e.  A  c ( le `  K ) b  -> 
c ( le `  K ) a ) ) )  =  (
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  |-  ( glb `  K )  =  ( glb `  K )
9 biid 236 . . 3  |-  ( ( A. b  e.  A  a ( le `  K ) b  /\  A. c  e.  B  ( A. b  e.  A  c ( le `  K ) b  -> 
c ( le `  K ) a ) )  <->  ( A. b  e.  A  a ( le `  K ) b  /\  A. c  e.  B  ( A. b  e.  A  c ( le `  K ) b  ->  c ( le
`  K ) a ) ) )
101, 5, 8, 9, 3, 4glbval 15473 . 2  |-  ( ph  ->  ( ( glb `  K
) `  A )  =  ( iota_ a  e.  B  ( A. b  e.  A  a ( le `  K ) b  /\  A. c  e.  B  ( A. b  e.  A  c ( le `  K ) b  ->  c ( le
`  K ) a ) ) ) )
111, 5, 2tosso 15512 . . . . . 6  |-  ( K  e. Toset  ->  ( K  e. Toset  <->  ( 
.<  Or  B  /\  (  _I  |`  B )  C_  ( le `  K ) ) ) )
1211ibi 241 . . . . 5  |-  ( K  e. Toset  ->  (  .<  Or  B  /\  (  _I  |`  B ) 
C_  ( le `  K ) ) )
1312simpld 459 . . . 4  |-  ( K  e. Toset  ->  .<  Or  B )
14 cnvso 5537 . . . 4  |-  (  .<  Or  B  <->  `'  .<  Or  B
)
1513, 14sylib 196 . . 3  |-  ( K  e. Toset  ->  `'  .<  Or  B
)
16 id 22 . . . 4  |-  ( `' 
.<  Or  B  ->  `'  .<  Or  B )
1716supval2 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 ) ) ) )
183, 15, 173syl 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 ) ) ) )
197, 10, 183eqtr4d 2511 1  |-  ( ph  ->  ( ( glb `  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   `'ccnv 4991    |` cres 4994   ` cfv 5579   iota_crio 6235   supcsup 7889   Basecbs 14479   lecple 14551   ltcplt 15417   glbcglb 15419  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-glb 15451  df-toset 15510
This theorem is referenced by:  xrsp0  27181
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