MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pwcdaen Structured version   Unicode version

Theorem pwcdaen 8070
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 6109 . . 3  |-  ( A  +c  B )  e. 
_V
21pw2en 7218 . 2  |-  ~P ( A  +c  B )  ~~  ( 2o  ^m  ( A  +c  B ) )
3 2on 6735 . . . 4  |-  2o  e.  On
4 mapcdaen 8069 . . . 4  |-  ( ( 2o  e.  On  /\  A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
53, 4mp3an1 1267 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ( 2o 
^m  A )  X.  ( 2o  ^m  B
) ) )
6 pw2eng 7217 . . . . 5  |-  ( A  e.  V  ->  ~P A  ~~  ( 2o  ^m  A ) )
7 pw2eng 7217 . . . . 5  |-  ( B  e.  W  ->  ~P B  ~~  ( 2o  ^m  B ) )
8 xpen 7273 . . . . 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 465 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ~P A  X.  ~P B )  ~~  (
( 2o  ^m  A
)  X.  ( 2o 
^m  B ) ) )
10 enen2 7251 . . . 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 225 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2o  ^m  ( A  +c  B ) ) 
~~  ( ~P A  X.  ~P B ) )
13 entr 7162 . 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 646 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 178    /\ wa 360    e. wcel 1726   ~Pcpw 3801   class class class wbr 4215   Oncon0 4584    X. cxp 4879  (class class class)co 6084   2oc2o 6721    ^m cmap 7021    ~~ cen 7109    +c ccda 8052
This theorem is referenced by:  pwcda1  8079  pwcdadom  8101  canthp1lem1  8532  gchxpidm  8549  gchhar  8559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-1o 6727  df-2o 6728  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-cda 8053
  Copyright terms: Public domain W3C validator