MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clatglbss Unicode version

Theorem clatglbss 14509
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 3308 . . . 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 14507 . . . 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 2749 . 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 14496 . . . 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 3316 . . . . 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 14508 . . 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 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   glbcglb 14355   CLatccla 14491
This theorem is referenced by:  dochss  31848
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-glb 14387  df-join 14388  df-meet 14389  df-lat 14430  df-clat 14492
  Copyright terms: Public domain W3C validator