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Theorem lubss 14241
Description: Subset law for least upper bounds. (chsupss 21937 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 955 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
2 sstr2 3199 . . . . 5  |-  ( S 
C_  T  ->  ( T  C_  B  ->  S  C_  B ) )
32impcom 419 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
433adant1 973 . . 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 14233 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( U `  T )  e.  B )
873adant3 975 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  T )  e.  B )
91, 4, 83jca 1132 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( K  e.  CLat  /\  S  C_  B  /\  ( U `
 T )  e.  B ) )
10 simpl1 958 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
11 simpl2 959 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
12 ssel2 3188 . . . . 5  |-  ( ( S  C_  T  /\  y  e.  S )  ->  y  e.  T )
13123ad2antl3 1119 . . . 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 14239 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  y  .<_  ( U `  T
) )
1610, 11, 13, 15syl3anc 1182 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  .<_  ( U `  T ) )
1716ralrimiva 2639 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  y  .<_  ( U `  T ) )
185, 14, 6lubl 14240 . 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 56 1  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  S )  .<_  ( U `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039   ` cfv 5271   Basecbs 13164   lecple 13231   lubclub 14092   CLatccla 14229
This theorem is referenced by:  lubel  14242  atlatmstc  30131  atlatle  30132  pmaple  30572  paddunN  30738  poml4N  30764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-undef 6314  df-riota 6320  df-lub 14124  df-clat 14230
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