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Theorem xpdom2g 6954
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 4702 . . . . 5  |-  ( x  =  C  ->  (
x  X.  A )  =  ( C  X.  A ) )
2 xpeq1 4702 . . . . 5  |-  ( x  =  C  ->  (
x  X.  B )  =  ( C  X.  B ) )
31, 2breq12d 4037 . . . 4  |-  ( x  =  C  ->  (
( x  X.  A
)  ~<_  ( x  X.  B )  <->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
43imbi2d 307 . . 3  |-  ( x  =  C  ->  (
( A  ~<_  B  -> 
( x  X.  A
)  ~<_  ( x  X.  B ) )  <->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) ) )
5 vex 2792 . . . 4  |-  x  e. 
_V
65xpdom2 6953 . . 3  |-  ( A  ~<_  B  ->  ( x  X.  A )  ~<_  ( x  X.  B ) )
74, 6vtoclg 2844 . 2  |-  ( C  e.  V  ->  ( A  ~<_  B  ->  ( C  X.  A )  ~<_  ( C  X.  B ) ) )
87imp 418 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1685   class class class wbr 4024    X. cxp 4686    ~<_ cdom 6857
This theorem is referenced by:  xpdom1g  6955  xpen  7020  infcdaabs  7828  infxpdom  7833  fin56  8015  unirnfdomd  8185  pwcdandom  8285  gchxpidm  8287  gchhar  8289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fv 5229  df-dom 6861
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