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Theorem clatglbss 15399
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 991 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
2 simpl2 992 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
3 simp3 990 . . . . 5  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  T )
43sselda 3454 . . . 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 15397 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  ( G `  T )  .<_  y )
91, 2, 4, 8syl3anc 1219 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  ( G `  T
)  .<_  y )
109ralrimiva 2822 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  ( G `  T )  .<_  y )
11 simp1 988 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
125, 7clatglbcl 15386 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( G `  T )  e.  B )
13123adant3 1008 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  e.  B )
14 sstr 3462 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  B )  ->  S  C_  B )
1514ancoms 453 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
16153adant1 1006 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  B )
175, 6, 7clatleglb 15398 . . 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 1219 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3426   class class class wbr 4390   ` cfv 5516   Basecbs 14276   lecple 14347   glbcglb 15215   CLatccla 15379
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 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-oprab 6194  df-poset 15218  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-lat 15318  df-clat 15380
This theorem is referenced by:  dochss  35316
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