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Theorem glble 14372
Description: The greatest lower bound is the least element. (Contributed by NM, 12-Oct-2011.)
Hypotheses
Ref Expression
glbval.b  |-  B  =  ( Base `  K
)
glbval.l  |-  .<_  =  ( le `  K )
glbval.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
glble  |-  ( ( ( K  e.  A  /\  S  C_  B )  /\  ( ( G `
 S )  e.  B  /\  X  e.  S ) )  -> 
( G `  S
)  .<_  X )

Proof of Theorem glble
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 glbval.b . . . . . 6  |-  B  =  ( Base `  K
)
2 glbval.l . . . . . 6  |-  .<_  =  ( le `  K )
3 glbval.g . . . . . 6  |-  G  =  ( glb `  K
)
41, 2, 3glbprop 14371 . . . . 5  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( G `  S )  e.  B )  ->  ( A. y  e.  S  ( G `  S ) 
.<_  y  /\  A. z  e.  B  ( A. y  e.  S  z  .<_  y  ->  z  .<_  ( G `  S ) ) ) )
54simpld 446 . . . 4  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( G `  S )  e.  B )  ->  A. y  e.  S  ( G `  S )  .<_  y )
6 breq2 4159 . . . . 5  |-  ( y  =  X  ->  (
( G `  S
)  .<_  y  <->  ( G `  S )  .<_  X ) )
76rspccv 2994 . . . 4  |-  ( A. y  e.  S  ( G `  S )  .<_  y  ->  ( X  e.  S  ->  ( G `
 S )  .<_  X ) )
85, 7syl 16 . . 3  |-  ( ( K  e.  A  /\  S  C_  B  /\  ( G `  S )  e.  B )  ->  ( X  e.  S  ->  ( G `  S ) 
.<_  X ) )
983exp 1152 . 2  |-  ( K  e.  A  ->  ( S  C_  B  ->  (
( G `  S
)  e.  B  -> 
( X  e.  S  ->  ( G `  S
)  .<_  X ) ) ) )
109imp43 579 1  |-  ( ( ( K  e.  A  /\  S  C_  B )  /\  ( ( G `
 S )  e.  B  /\  X  e.  S ) )  -> 
( G `  S
)  .<_  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651    C_ wss 3265   class class class wbr 4155   ` cfv 5396   Basecbs 13398   lecple 13465   glbcglb 14329
This theorem is referenced by:  p0le  14401  clatglble  14481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-undef 6481  df-riota 6487  df-glb 14361
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