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Theorem lubss 15395
Description: Subset law for least upper bounds. (chsupss 24882 analog.) (Contributed by NM, 20-Oct-2011.)
Hypotheses
Ref Expression
lublem.b  |-  B  =  ( Base `  K
)
lublem.l  |-  .<_  =  ( le `  K )
lublem.u  |-  U  =  ( lub `  K
)
Assertion
Ref Expression
lubss  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  S )  .<_  ( U `  T
) )

Proof of Theorem lubss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
2 sstr2 3463 . . . . 5  |-  ( S 
C_  T  ->  ( T  C_  B  ->  S  C_  B ) )
32impcom 430 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
433adant1 1006 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  S  C_  B )
5 lublem.b . . . . 5  |-  B  =  ( Base `  K
)
6 lublem.u . . . . 5  |-  U  =  ( lub `  K
)
75, 6clatlubcl 15386 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( U `  T )  e.  B )
873adant3 1008 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  T )  e.  B )
91, 4, 83jca 1168 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( K  e.  CLat  /\  S  C_  B  /\  ( U `
 T )  e.  B ) )
10 simpl1 991 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
11 simpl2 992 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
12 ssel2 3451 . . . . 5  |-  ( ( S  C_  T  /\  y  e.  S )  ->  y  e.  T )
13123ad2antl3 1152 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  e.  T )
14 lublem.l . . . . 5  |-  .<_  =  ( le `  K )
155, 14, 6lubub 15393 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  y  .<_  ( U `  T
) )
1610, 11, 13, 15syl3anc 1219 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  .<_  ( U `  T ) )
1716ralrimiva 2822 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  y  .<_  ( U `  T ) )
185, 14, 6lubl 15394 . 2  |-  ( ( K  e.  CLat  /\  S  C_  B  /\  ( U `
 T )  e.  B )  ->  ( A. y  e.  S  y  .<_  ( U `  T )  ->  ( U `  S )  .<_  ( U `  T
) ) )
199, 17, 18sylc 60 1  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  S )  .<_  ( U `  T
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3428   class class class wbr 4392   ` cfv 5518   Basecbs 14278   lecple 14349   lubclub 15216   CLatccla 15381
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
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 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-lub 15248  df-glb 15249  df-clat 15382
This theorem is referenced by:  lubel  15396  atlatmstc  33272  atlatle  33273  pmaple  33713  paddunN  33879  poml4N  33905
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