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Theorem map2xp 7047
Description: A cardinal power with exponent 2 is equivalent to a cross product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
map2xp  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )

Proof of Theorem map2xp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df2o3 6508 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 3660 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2316 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 5885 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4232 . . . . 5  |-  { (/) }  e.  _V
65a1i 10 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4232 . . . . 5  |-  { 1o }  e.  _V
87a1i 10 . . . 4  |-  ( A  e.  V  ->  { 1o }  e.  _V )
9 id 19 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
10 1n0 6510 . . . . . . . 8  |-  1o  =/=  (/)
11 df-ne 2461 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1210, 11mpbi 199 . . . . . . 7  |-  -.  1o  =  (/)
13 elsni 3677 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1412, 13mto 167 . . . . . 6  |-  -.  1o  e.  { (/) }
15 disjsn 3706 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1614, 15mpbir 200 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1716a1i 10 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
18 mapunen 7046 . . . 4  |-  ( ( ( { (/) }  e.  _V  /\  { 1o }  e.  _V  /\  A  e.  V )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( A  ^m  ( { (/) }  u.  { 1o } ) )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
196, 8, 9, 17, 18syl31anc 1185 . . 3  |-  ( A  e.  V  ->  ( A  ^m  ( { (/) }  u.  { 1o }
) )  ~~  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) ) )
204, 19syl5eqbr 4072 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
21 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
22 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2321, 22breq12d 4052 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
24 vex 2804 . . . . 5  |-  x  e. 
_V
25 0ex 4166 . . . . 5  |-  (/)  e.  _V
2624, 25mapsnen 6954 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2723, 26vtoclg 2856 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
28 oveq1 5881 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2928, 22breq12d 4052 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
30 df1o2 6507 . . . . . 6  |-  1o  =  { (/) }
3130, 5eqeltri 2366 . . . . 5  |-  1o  e.  _V
3224, 31mapsnen 6954 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3329, 32vtoclg 2856 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
34 xpen 7040 . . 3  |-  ( ( ( A  ^m  { (/)
} )  ~~  A  /\  ( A  ^m  { 1o } )  ~~  A
)  ->  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )
3527, 33, 34syl2anc 642 . 2  |-  ( A  e.  V  ->  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) )  ~~  ( A  X.  A ) )
36 entr 6929 . 2  |-  ( ( ( A  ^m  2o )  ~~  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) )  /\  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )  -> 
( A  ^m  2o )  ~~  ( A  X.  A ) )
3720, 35, 36syl2anc 642 1  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   {cpr 3654   class class class wbr 4039    X. cxp 4703  (class class class)co 5874   1oc1o 6488   2oc2o 6489    ^m cmap 6788    ~~ cen 6876
This theorem is referenced by:  pwxpndom2  8303
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-reu 2563  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-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-2o 6496  df-er 6676  df-map 6790  df-en 6880  df-dom 6881
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