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Theorem map2xp 7031
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 6492 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 3647 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2303 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 5869 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4216 . . . . 5  |-  { (/) }  e.  _V
65a1i 10 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4216 . . . . 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 6494 . . . . . . . 8  |-  1o  =/=  (/)
11 df-ne 2448 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1210, 11mpbi 199 . . . . . . 7  |-  -.  1o  =  (/)
13 elsni 3664 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1412, 13mto 167 . . . . . 6  |-  -.  1o  e.  { (/) }
15 disjsn 3693 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1614, 15mpbir 200 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1716a1i 10 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
18 mapunen 7030 . . . 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 4056 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
21 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
22 id 19 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2321, 22breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
24 vex 2791 . . . . 5  |-  x  e. 
_V
25 0ex 4150 . . . . 5  |-  (/)  e.  _V
2624, 25mapsnen 6938 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2723, 26vtoclg 2843 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
28 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2928, 22breq12d 4036 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
30 df1o2 6491 . . . . . 6  |-  1o  =  { (/) }
3130, 5eqeltri 2353 . . . . 5  |-  1o  e.  _V
3224, 31mapsnen 6938 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3329, 32vtoclg 2843 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
34 xpen 7024 . . 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 6913 . 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 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   {cpr 3641   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   1oc1o 6472   2oc2o 6473    ^m cmap 6772    ~~ cen 6860
This theorem is referenced by:  pwxpndom2  8287
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-reu 2550  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-2o 6480  df-er 6660  df-map 6774  df-en 6864  df-dom 6865
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