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Metamath Proof Explorer

Theorem pwdjuen

Description: Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion pwdjuen 
|- ( ( A e. V /\ B e. W ) -> ~P ( A |_| B ) ~~ ( ~P A X. ~P B ) )

Proof

Step Hyp Ref Expression
1 djuex 
 |-  ( ( A e. V /\ B e. W ) -> ( A |_| B ) e. _V )
2 pw2eng 
 |-  ( ( A |_| B ) e. _V -> ~P ( A |_| B ) ~~ ( 2o ^m ( A |_| B ) ) )
3 1 2 syl 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( A |_| B ) ~~ ( 2o ^m ( A |_| B ) ) )
4 2on 
 |-  2o e. On
5 mapdjuen 
 |-  ( ( 2o e. On /\ A e. V /\ B e. W ) -> ( 2o ^m ( A |_| B ) ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) )
6 4 5 mp3an1 
 |-  ( ( A e. V /\ B e. W ) -> ( 2o ^m ( A |_| B ) ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) )
7 pw2eng 
 |-  ( A e. V -> ~P A ~~ ( 2o ^m A ) )
8 pw2eng 
 |-  ( B e. W -> ~P B ~~ ( 2o ^m B ) )
9 xpen 
 |-  ( ( ~P A ~~ ( 2o ^m A ) /\ ~P B ~~ ( 2o ^m B ) ) -> ( ~P A X. ~P B ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) )
10 7 8 9 syl2an 
 |-  ( ( A e. V /\ B e. W ) -> ( ~P A X. ~P B ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) )
11 enen2 
 |-  ( ( ~P A X. ~P B ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) -> ( ( 2o ^m ( A |_| B ) ) ~~ ( ~P A X. ~P B ) <-> ( 2o ^m ( A |_| B ) ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) ) )
12 10 11 syl 
 |-  ( ( A e. V /\ B e. W ) -> ( ( 2o ^m ( A |_| B ) ) ~~ ( ~P A X. ~P B ) <-> ( 2o ^m ( A |_| B ) ) ~~ ( ( 2o ^m A ) X. ( 2o ^m B ) ) ) )
13 6 12 mpbird 
 |-  ( ( A e. V /\ B e. W ) -> ( 2o ^m ( A |_| B ) ) ~~ ( ~P A X. ~P B ) )
14 entr 
 |-  ( ( ~P ( A |_| B ) ~~ ( 2o ^m ( A |_| B ) ) /\ ( 2o ^m ( A |_| B ) ) ~~ ( ~P A X. ~P B ) ) -> ~P ( A |_| B ) ~~ ( ~P A X. ~P B ) )
15 3 13 14 syl2anc 
 |-  ( ( A e. V /\ B e. W ) -> ~P ( A |_| B ) ~~ ( ~P A X. ~P B ) )