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Theorem xpdom2g 7133
Description: Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom2g  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )

Proof of Theorem xpdom2g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xpeq1 4825 . . . . 5  |-  ( x  =  C  ->  (
x  X.  A )  =  ( C  X.  A ) )
2 xpeq1 4825 . . . . 5  |-  ( x  =  C  ->  (
x  X.  B )  =  ( C  X.  B ) )
31, 2breq12d 4159 . . . 4  |-  ( x  =  C  ->  (
( x  X.  A
)  ~<_  ( x  X.  B )  <->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
43imbi2d 308 . . 3  |-  ( x  =  C  ->  (
( A  ~<_  B  -> 
( x  X.  A
)  ~<_  ( x  X.  B ) )  <->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) ) )
5 vex 2895 . . . 4  |-  x  e. 
_V
65xpdom2 7132 . . 3  |-  ( A  ~<_  B  ->  ( x  X.  A )  ~<_  ( x  X.  B ) )
74, 6vtoclg 2947 . 2  |-  ( C  e.  V  ->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
87imp 419 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4146    X. cxp 4809    ~<_ cdom 7036
This theorem is referenced by:  xpdom1g  7134  xpen  7199  infcdaabs  8012  infxpdom  8017  fin56  8199  unirnfdomd  8368  pwcdandom  8468  gchxpidm  8470  gchhar  8472  fnct  23939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fv 5395  df-dom 7040
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