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Theorem pwcdaen 7811
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 5883 . . 3  |-  ( A  +c  B )  e. 
_V
21pw2en 6969 . 2  |-  ~P ( A  +c  B )  ~~  ( 2o  ^m  ( A  +c  B ) )
3 2on 6487 . . . 4  |-  2o  e.  On
4 mapcdaen 7810 . . . 4  |-  ( ( 2o  e.  On  /\  A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
53, 4mp3an1 1264 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
6 pw2eng 6968 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
7 pw2eng 6968 . . . . 5  |-  ( B  e.  W  ->  ~P B  ~~  ( 2o  ^m  B ) )
8 xpen 7024 . . . . 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 463 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  X.  ~P B )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) )
10 enen2 7002 . . . 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 15 . . 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 223 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ~P A  X.  ~P B ) )
13 entr 6913 . 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 644 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 176    /\ wa 358    e. wcel 1684   ~Pcpw 3625   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  pwcda1  7820  pwcdadom  7842  canthp1lem1  8274  gchxpidm  8291  gchhar  8293
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-3or 935  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  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  df-cda 7794
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