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Theorem xpdom1g 6959
Description: Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpdom1g  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )

Proof of Theorem xpdom1g
StepHypRef Expression
1 reldom 6869 . . . 4  |-  Rel  ~<_
21brrelexi 4729 . . 3  |-  ( A  ~<_  B  ->  A  e.  _V )
3 xpcomeng 6954 . . . 4  |-  ( ( A  e.  _V  /\  C  e.  V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
43ancoms 439 . . 3  |-  ( ( C  e.  V  /\  A  e.  _V )  ->  ( A  X.  C
)  ~~  ( C  X.  A ) )
52, 4sylan2 460 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~~  ( C  X.  A ) )
6 xpdom2g 6958 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( C  X.  B ) )
71brrelex2i 4730 . . . 4  |-  ( A  ~<_  B  ->  B  e.  _V )
8 xpcomeng 6954 . . . 4  |-  ( ( C  e.  V  /\  B  e.  _V )  ->  ( C  X.  B
)  ~~  ( B  X.  C ) )
97, 8sylan2 460 . . 3  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  B
)  ~~  ( B  X.  C ) )
10 domentr 6920 . . 3  |-  ( ( ( C  X.  A
)  ~<_  ( C  X.  B )  /\  ( C  X.  B )  ~~  ( B  X.  C
) )  ->  ( C  X.  A )  ~<_  ( B  X.  C ) )
116, 9, 10syl2anc 642 . 2  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( C  X.  A
)  ~<_  ( B  X.  C ) )
12 endomtr 6919 . 2  |-  ( ( ( A  X.  C
)  ~~  ( C  X.  A )  /\  ( C  X.  A )  ~<_  ( B  X.  C ) )  ->  ( A  X.  C )  ~<_  ( B  X.  C ) )
135, 11, 12syl2anc 642 1  |-  ( ( C  e.  V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    X. cxp 4687    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  xpdom1  6961  xpen  7024  infpwfien  7689  iunctb  8196  canthp1lem1  8274  gchxpidm  8291  xpct  23338  fnct  23341
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-en 6864  df-dom 6865
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