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Theorem pwcdaen 8577
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 6320 . . 3  |-  ( A  +c  B )  e. 
_V
21pw2en 7636 . 2  |-  ~P ( A  +c  B )  ~~  ( 2o  ^m  ( A  +c  B ) )
3 2on 7150 . . . 4  |-  2o  e.  On
4 mapcdaen 8576 . . . 4  |-  ( ( 2o  e.  On  /\  A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
53, 4mp3an1 1311 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
6 pw2eng 7635 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
7 pw2eng 7635 . . . . 5  |-  ( B  e.  W  ->  ~P B  ~~  ( 2o  ^m  B ) )
8 xpen 7692 . . . . 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 477 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  X.  ~P B )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) )
10 enen2 7670 . . . 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 232 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ~P A  X.  ~P B ) )
13 entr 7579 . 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 663 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ~P ( A  +c  B )  ~~  ( ~P A  X.  ~P B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767   ~Pcpw 4016   class class class wbr 4453   Oncon0 4884    X. cxp 5003  (class class class)co 6295   2oc2o 7136    ^m cmap 7432    ~~ cen 7525    +c ccda 8559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-1o 7142  df-2o 7143  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-cda 8560
This theorem is referenced by:  pwcda1  8586  pwcdadom  8608  canthp1lem1  9042  gchxpidm  9059  gchhar  9069
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