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Theorem clatglbss 14554
Description: Subset law for greatest lower bound. (Contributed by Mario Carneiro, 16-Apr-2014.)
Hypotheses
Ref Expression
clatglb.b  |-  B  =  ( Base `  K
)
clatglb.l  |-  .<_  =  ( le `  K )
clatglb.g  |-  G  =  ( glb `  K
)
Assertion
Ref Expression
clatglbss  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  .<_  ( G `  S
) )

Proof of Theorem clatglbss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
2 simpl2 961 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
3 simp3 959 . . . . 5  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  T )
43sselda 3348 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  e.  T )
5 clatglb.b . . . . 5  |-  B  =  ( Base `  K
)
6 clatglb.l . . . . 5  |-  .<_  =  ( le `  K )
7 clatglb.g . . . . 5  |-  G  =  ( glb `  K
)
85, 6, 7clatglble 14552 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  ( G `  T )  .<_  y )
91, 2, 4, 8syl3anc 1184 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  ( G `  T
)  .<_  y )
109ralrimiva 2789 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  ( G `  T )  .<_  y )
11 simp1 957 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
125, 7clatglbcl 14541 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( G `  T )  e.  B )
13123adant3 977 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  e.  B )
14 sstr 3356 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  B )  ->  S  C_  B )
1514ancoms 440 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
16153adant1 975 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  B )
175, 6, 7clatleglb 14553 . . 3  |-  ( ( K  e.  CLat  /\  ( G `  T )  e.  B  /\  S  C_  B )  ->  (
( G `  T
)  .<_  ( G `  S )  <->  A. y  e.  S  ( G `  T )  .<_  y ) )
1811, 13, 16, 17syl3anc 1184 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  (
( G `  T
)  .<_  ( G `  S )  <->  A. y  e.  S  ( G `  T )  .<_  y ) )
1910, 18mpbird 224 1  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  .<_  ( G `  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   glbcglb 14400   CLatccla 14536
This theorem is referenced by:  dochss  32163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-glb 14432  df-join 14433  df-meet 14434  df-lat 14475  df-clat 14537
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