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Theorem clatglbss 14231
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 958 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
2 simpl2 959 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
3 simp3 957 . . . . 5  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  T )
43sselda 3180 . . . 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 14229 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  ( G `  T )  .<_  y )
91, 2, 4, 8syl3anc 1182 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  ( G `  T
)  .<_  y )
109ralrimiva 2626 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  ( G `  T )  .<_  y )
11 simp1 955 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
125, 7clatglbcl 14218 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( G `  T )  e.  B )
13123adant3 975 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( G `  T )  e.  B )
14 sstr 3187 . . . . 5  |-  ( ( S  C_  T  /\  T  C_  B )  ->  S  C_  B )
1514ancoms 439 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
16153adant1 973 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  B )
175, 6, 7clatleglb 14230 . . 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 1182 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  (
( G `  T
)  .<_  ( G `  S )  <->  A. y  e.  S  ( G `  T )  .<_  y ) )
1910, 18mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   glbcglb 14077   CLatccla 14213
This theorem is referenced by:  dochss  31555
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152  df-clat 14214
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