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Theorem map2xp 7269
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 6729 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 3813 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2455 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 6084 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4397 . . . . 5  |-  { (/) }  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4397 . . . . 5  |-  { 1o }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { 1o }  e.  _V )
9 id 20 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
10 1n0 6731 . . . . . . . 8  |-  1o  =/=  (/)
11 df-ne 2600 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1210, 11mpbi 200 . . . . . . 7  |-  -.  1o  =  (/)
13 elsni 3830 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1412, 13mto 169 . . . . . 6  |-  -.  1o  e.  { (/) }
15 disjsn 3860 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1614, 15mpbir 201 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1716a1i 11 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
18 mapunen 7268 . . . 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 1187 . . 3  |-  ( A  e.  V  ->  ( A  ^m  ( { (/) }  u.  { 1o }
) )  ~~  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) ) )
204, 19syl5eqbr 4237 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
21 oveq1 6080 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
22 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2321, 22breq12d 4217 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
24 vex 2951 . . . . 5  |-  x  e. 
_V
25 0ex 4331 . . . . 5  |-  (/)  e.  _V
2624, 25mapsnen 7176 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2723, 26vtoclg 3003 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
28 oveq1 6080 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2928, 22breq12d 4217 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
30 df1o2 6728 . . . . . 6  |-  1o  =  { (/) }
3130, 5eqeltri 2505 . . . . 5  |-  1o  e.  _V
3224, 31mapsnen 7176 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3329, 32vtoclg 3003 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
34 xpen 7262 . . 3  |-  ( ( ( A  ^m  { (/)
} )  ~~  A  /\  ( A  ^m  { 1o } )  ~~  A
)  ->  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )
3527, 33, 34syl2anc 643 . 2  |-  ( A  e.  V  ->  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) )  ~~  ( A  X.  A ) )
36 entr 7151 . 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 643 1  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   {cpr 3807   class class class wbr 4204    X. cxp 4868  (class class class)co 6073   1oc1o 6709   2oc2o 6710    ^m cmap 7010    ~~ cen 7098
This theorem is referenced by:  pwxpndom2  8532
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-2o 6717  df-er 6897  df-map 7012  df-en 7102  df-dom 7103
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