MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpen Unicode version

Theorem xpen 7040
Description: Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
xpen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )

Proof of Theorem xpen
StepHypRef Expression
1 relen 6884 . . . . 5  |-  Rel  ~~
21brrelexi 4745 . . . 4  |-  ( C 
~~  D  ->  C  e.  _V )
3 endom 6904 . . . 4  |-  ( A 
~~  B  ->  A  ~<_  B )
4 xpdom1g 6975 . . . 4  |-  ( ( C  e.  _V  /\  A  ~<_  B )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
52, 3, 4syl2anr 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  C ) )
61brrelex2i 4746 . . . 4  |-  ( A 
~~  B  ->  B  e.  _V )
7 endom 6904 . . . 4  |-  ( C 
~~  D  ->  C  ~<_  D )
8 xpdom2g 6974 . . . 4  |-  ( ( B  e.  _V  /\  C  ~<_  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
96, 7, 8syl2an 463 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  C
)  ~<_  ( B  X.  D ) )
10 domtr 6930 . . 3  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  C )  /\  ( B  X.  C )  ~<_  ( B  X.  D ) )  ->  ( A  X.  C )  ~<_  ( B  X.  D ) )
115, 9, 10syl2anc 642 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~<_  ( B  X.  D ) )
121brrelex2i 4746 . . . 4  |-  ( C 
~~  D  ->  D  e.  _V )
13 ensym 6926 . . . . 5  |-  ( A 
~~  B  ->  B  ~~  A )
14 endom 6904 . . . . 5  |-  ( B 
~~  A  ->  B  ~<_  A )
1513, 14syl 15 . . . 4  |-  ( A 
~~  B  ->  B  ~<_  A )
16 xpdom1g 6975 . . . 4  |-  ( ( D  e.  _V  /\  B  ~<_  A )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
1712, 15, 16syl2anr 464 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  D ) )
181brrelexi 4745 . . . 4  |-  ( A 
~~  B  ->  A  e.  _V )
19 ensym 6926 . . . . 5  |-  ( C 
~~  D  ->  D  ~~  C )
20 endom 6904 . . . . 5  |-  ( D 
~~  C  ->  D  ~<_  C )
2119, 20syl 15 . . . 4  |-  ( C 
~~  D  ->  D  ~<_  C )
22 xpdom2g 6974 . . . 4  |-  ( ( A  e.  _V  /\  D  ~<_  C )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
2318, 21, 22syl2an 463 . . 3  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  D
)  ~<_  ( A  X.  C ) )
24 domtr 6930 . . 3  |-  ( ( ( B  X.  D
)  ~<_  ( A  X.  D )  /\  ( A  X.  D )  ~<_  ( A  X.  C ) )  ->  ( B  X.  D )  ~<_  ( A  X.  C ) )
2517, 23, 24syl2anc 642 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( B  X.  D
)  ~<_  ( A  X.  C ) )
26 sbth 6997 . 2  |-  ( ( ( A  X.  C
)  ~<_  ( B  X.  D )  /\  ( B  X.  D )  ~<_  ( A  X.  C ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2711, 25, 26syl2anc 642 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    X. cxp 4703    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  map2xp  7047  unxpdom2  7087  sucxpdom  7088  xpnum  7600  infxpenlem  7657  infxpidm2  7660  xpcdaen  7825  mapcdaen  7826  pwcdaen  7827  cdaxpdom  7831  ackbij1lem5  7866  canthp1lem1  8290  xpnnen  12503  xpomenOLD  12505  qnnen  12508  rexpen  12522  met2ndci  18084  re2ndc  18323  dyadmbl  18971  opnmblALT  18974  mbfimaopnlem  19026  xpengOLD  26478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881
  Copyright terms: Public domain W3C validator