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Theorem tosglb 26267
Description: Same theorem as toslub 26265, 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 2451 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
61, 2, 3, 4, 5tosglblem 26266 . . 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 6170 . 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 2451 . . 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 15271 . 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 15310 . . . . . 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 5476 . . . 4  |-  (  .<  Or  B  <->  `'  .<  Or  B
)
1513, 14sylib 196 . . 3  |-  ( K  e. Toset  ->  `'  .<  Or  B
)
16 id 22 . . . 4  |-  ( `' 
.<  Or  B  ->  `'  .<  Or  B )
1716supval2 7808 . . 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 2502 1  |-  ( ph  ->  ( ( glb `  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 2795   E.wrex 2796    C_ wss 3428   class class class wbr 4392    _I cid 4731    Or wor 4740   `'ccnv 4939    |` cres 4942   ` cfv 5518   iota_crio 6152   supcsup 7793   Basecbs 14278   lecple 14349   ltcplt 15215   glbcglb 15217  Tosetctos 15307
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 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-sup 7794  df-poset 15220  df-plt 15232  df-glb 15249  df-toset 15308
This theorem is referenced by:  xrsp0  26278
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