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Theorem prsss 27534
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 3696 . . . 4  |-  (  .<_  i^i  ( A  X.  A
) )  =  ( ( ( le `  K )  i^i  ( B  X.  B ) )  i^i  ( A  X.  A ) )
3 inass 3708 . . . 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 2496 . . 3  |-  (  .<_  i^i  ( A  X.  A
) )  =  ( ( le `  K
)  i^i  ( ( B  X.  B )  i^i  ( A  X.  A
) ) )
5 xpss12 5106 . . . . . 6  |-  ( ( A  C_  B  /\  A  C_  B )  -> 
( A  X.  A
)  C_  ( B  X.  B ) )
65anidms 645 . . . . 5  |-  ( A 
C_  B  ->  ( A  X.  A )  C_  ( B  X.  B
) )
7 dfss1 3703 . . . . 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 3700 . . 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 2520 . 2  |-  ( A 
C_  B  ->  (  .<_  i^i  ( A  X.  A ) )  =  ( ( le `  K )  i^i  ( A  X.  A ) ) )
1110adantl 466 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 369    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476    X. cxp 4997   ` cfv 5586   Basecbs 14486   lecple 14558    Preset cpreset 15409
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-opab 4506  df-xp 5005
This theorem is referenced by:  prsssdm  27535  ordtrest2NEW  27541
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