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Theorem lubss 16309
Description: Subset law for least upper bounds. (chsupss 26821 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 1005 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  K  e.  CLat )
2 sstr2 3477 . . . . 5  |-  ( S 
C_  T  ->  ( T  C_  B  ->  S  C_  B ) )
32impcom 431 . . . 4  |-  ( ( T  C_  B  /\  S  C_  T )  ->  S  C_  B )
433adant1 1023 . . 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 16300 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B )  ->  ( U `  T )  e.  B )
873adant3 1025 . . 3  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( U `  T )  e.  B )
91, 4, 83jca 1185 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  ( K  e.  CLat  /\  S  C_  B  /\  ( U `
 T )  e.  B ) )
10 simpl1 1008 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  K  e.  CLat )
11 simpl2 1009 . . . 4  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  T  C_  B )
12 ssel2 3465 . . . . 5  |-  ( ( S  C_  T  /\  y  e.  S )  ->  y  e.  T )
13123ad2antl3 1169 . . . 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 16307 . . . 4  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  y  e.  T )  ->  y  .<_  ( U `  T
) )
1610, 11, 13, 15syl3anc 1264 . . 3  |-  ( ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  /\  y  e.  S )  ->  y  .<_  ( U `  T ) )
1716ralrimiva 2846 . 2  |-  ( ( K  e.  CLat  /\  T  C_  B  /\  S  C_  T )  ->  A. y  e.  S  y  .<_  ( U `  T ) )
185, 14, 6lubl 16308 . 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 62 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   class class class wbr 4426   ` cfv 5601   Basecbs 15075   lecple 15150   lubclub 16129   CLatccla 16295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-lub 16162  df-glb 16163  df-clat 16296
This theorem is referenced by:  lubel  16310  atlatmstc  32584  atlatle  32585  pmaple  33025  paddunN  33191  poml4N  33217
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