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Theorem prsss 28231
Description: Relation of a subpreset. (Contributed by Thierry Arnoux, 13-Sep-2018.)
Hypotheses
Ref Expression
ordtNEW.b  |-  B  =  ( Base `  K
)
ordtNEW.l  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
Assertion
Ref Expression
prsss  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )

Proof of Theorem prsss
StepHypRef Expression
1 ordtNEW.l . . . . 5  |-  .<_  =  ( ( le `  K
)  i^i  ( B  X.  B ) )
21ineq1i 3636 . . . 4  |-  (  .<_  i^i  ( A  X.  A
) )  =  ( ( ( le `  K )  i^i  ( B  X.  B ) )  i^i  ( A  X.  A ) )
3 inass 3648 . . . 4  |-  ( ( ( le `  K
)  i^i  ( B  X.  B ) )  i^i  ( A  X.  A
) )  =  ( ( le `  K
)  i^i  ( ( B  X.  B )  i^i  ( A  X.  A
) ) )
42, 3eqtri 2431 . . 3  |-  (  .<_  i^i  ( A  X.  A
) )  =  ( ( le `  K
)  i^i  ( ( B  X.  B )  i^i  ( A  X.  A
) ) )
5 xpss12 5050 . . . . . 6  |-  ( ( A  C_  B  /\  A  C_  B )  -> 
( A  X.  A
)  C_  ( B  X.  B ) )
65anidms 643 . . . . 5  |-  ( A 
C_  B  ->  ( A  X.  A )  C_  ( B  X.  B
) )
7 dfss1 3643 . . . . 5  |-  ( ( A  X.  A ) 
C_  ( B  X.  B )  <->  ( ( B  X.  B )  i^i  ( A  X.  A
) )  =  ( A  X.  A ) )
86, 7sylib 196 . . . 4  |-  ( A 
C_  B  ->  (
( B  X.  B
)  i^i  ( A  X.  A ) )  =  ( A  X.  A
) )
98ineq2d 3640 . . 3  |-  ( A 
C_  B  ->  (
( le `  K
)  i^i  ( ( B  X.  B )  i^i  ( A  X.  A
) ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
104, 9syl5eq 2455 . 2  |-  ( A 
C_  B  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
1110adantl 464 1  |-  ( ( K  e.  Preset  /\  A  C_  B )  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413    X. cxp 4940   ` cfv 5525   Basecbs 14733   lecple 14808    Preset cpreset 15771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3060  df-in 3420  df-ss 3427  df-opab 4453  df-xp 4948
This theorem is referenced by:  prsssdm  28232  ordtrestNEW  28236  ordtrest2NEW  28238
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