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Theorem pwcdaen 8029
Description: Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
pwcdaen  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  +c  B )  ~~  ( ~P A  X.  ~P B
) )

Proof of Theorem pwcdaen
StepHypRef Expression
1 ovex 6073 . . 3  |-  ( A  +c  B )  e. 
_V
21pw2en 7182 . 2  |-  ~P ( A  +c  B )  ~~  ( 2o  ^m  ( A  +c  B ) )
3 2on 6699 . . . 4  |-  2o  e.  On
4 mapcdaen 8028 . . . 4  |-  ( ( 2o  e.  On  /\  A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
53, 4mp3an1 1266 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
6 pw2eng 7181 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
7 pw2eng 7181 . . . . 5  |-  ( B  e.  W  ->  ~P B  ~~  ( 2o  ^m  B ) )
8 xpen 7237 . . . . 5  |-  ( ( ~P A  ~~  ( 2o  ^m  A )  /\  ~P B  ~~  ( 2o 
^m  B ) )  ->  ( ~P A  X.  ~P B )  ~~  ( ( 2o  ^m  A )  X.  ( 2o  ^m  B ) ) )
96, 7, 8syl2an 464 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  X.  ~P B )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) )
10 enen2 7215 . . . 4  |-  ( ( ~P A  X.  ~P B )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) )  ->  ( ( 2o 
^m  ( A  +c  B ) )  ~~  ( ~P A  X.  ~P B )  <->  ( 2o  ^m  ( A  +c  B
) )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) ) )
119, 10syl 16 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( 2o  ^m  ( A  +c  B
) )  ~~  ( ~P A  X.  ~P B
)  <->  ( 2o  ^m  ( A  +c  B
) )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) ) )
125, 11mpbird 224 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ~P A  X.  ~P B ) )
13 entr 7126 . 2  |-  ( ( ~P ( A  +c  B )  ~~  ( 2o  ^m  ( A  +c  B ) )  /\  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ~P A  X.  ~P B ) )  ->  ~P ( A  +c  B )  ~~  ( ~P A  X.  ~P B ) )
142, 12, 13sylancr 645 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  +c  B )  ~~  ( ~P A  X.  ~P B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   ~Pcpw 3767   class class class wbr 4180   Oncon0 4549    X. cxp 4843  (class class class)co 6048   2oc2o 6685    ^m cmap 6985    ~~ cen 7073    +c ccda 8011
This theorem is referenced by:  pwcda1  8038  pwcdadom  8060  canthp1lem1  8491  gchxpidm  8508  gchhar  8510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-1o 6691  df-2o 6692  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-cda 8012
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