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Theorem xpdom2g 7196
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 4884 . . . . 5  |-  ( x  =  C  ->  (
x  X.  A )  =  ( C  X.  A ) )
2 xpeq1 4884 . . . . 5  |-  ( x  =  C  ->  (
x  X.  B )  =  ( C  X.  B ) )
31, 2breq12d 4217 . . . 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 2951 . . . 4  |-  x  e. 
_V
65xpdom2 7195 . . 3  |-  ( A  ~<_  B  ->  ( x  X.  A )  ~<_  ( x  X.  B ) )
74, 6vtoclg 3003 . 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 1652    e. wcel 1725   class class class wbr 4204    X. cxp 4868    ~<_ cdom 7099
This theorem is referenced by:  xpdom1g  7197  xpen  7262  infcdaabs  8078  infxpdom  8083  fin56  8265  unirnfdomd  8434  pwcdandom  8534  gchxpidm  8536  gchhar  8538  fnct  24097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fv 5454  df-dom 7103
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