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Theorem lubss 15604
Description: Subset law for least upper bounds. (chsupss 25936 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 996 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
2 sstr2 3511 . . . . 5  |-  ( S 
C_  T  ->  ( T  C_  B  ->  S  C_  B ) )
32impcom 430 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
433adant1 1014 . . 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 15595 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( U `  T )  e.  B )
873adant3 1016 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  T )  e.  B )
91, 4, 83jca 1176 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( K  e.  CLat  /\  S  C_  B  /\  ( U `
 T )  e.  B ) )
10 simpl1 999 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
11 simpl2 1000 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
12 ssel2 3499 . . . . 5  |-  ( ( S  C_  T  /\  y  e.  S )  ->  y  e.  T )
13123ad2antl3 1160 . . . 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 15602 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  y  .<_  ( U `  T
) )
1610, 11, 13, 15syl3anc 1228 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  .<_  ( U `  T ) )
1716ralrimiva 2878 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  y  .<_  ( U `  T ) )
185, 14, 6lubl 15603 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   class class class wbr 4447   ` cfv 5586   Basecbs 14486   lecple 14558   lubclub 15425   CLatccla 15590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-lub 15457  df-glb 15458  df-clat 15591
This theorem is referenced by:  lubel  15605  atlatmstc  34116  atlatle  34117  pmaple  34557  paddunN  34723  poml4N  34749
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