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Theorem clatglbss 15959
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 997 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
2 simpl2 998 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
3 simp3 996 . . . . 5  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  T )
43sselda 3489 . . . 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 15957 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  ( G `  T )  .<_  y )
91, 2, 4, 8syl3anc 1226 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  ( G `  T
)  .<_  y )
109ralrimiva 2868 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  ( G `  T )  .<_  y )
11 simp1 994 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
125, 7clatglbcl 15946 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( G `  T )  e.  B )
13123adant3 1014 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  e.  B )
14 sstr 3497 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  B )  ->  S  C_  B )
1514ancoms 451 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
16153adant1 1012 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  B )
175, 6, 7clatleglb 15958 . . 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 1226 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  (
( G `  T
)  .<_  ( G `  S )  <->  A. y  e.  S  ( G `  T )  .<_  y ) )
1910, 18mpbird 232 1  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  .<_  ( G `  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   class class class wbr 4439   ` cfv 5570   Basecbs 14719   lecple 14794   glbcglb 15774   CLatccla 15939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-oprab 6274  df-poset 15777  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-lat 15878  df-clat 15940
This theorem is referenced by:  dochss  37508
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