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Theorem mapcdaen 7806
Description: Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
mapcdaen  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  +c  C ) ) 
~~  ( ( A  ^m  B )  X.  ( A  ^m  C
) ) )

Proof of Theorem mapcdaen
StepHypRef Expression
1 cdaval 7792 . . . . 5  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
213adant1 973 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
32oveq2d 5836 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  +c  C ) )  =  ( A  ^m  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) ) )
4 simp2 956 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
5 snex 4215 . . . . 5  |-  { (/) }  e.  _V
6 xpexg 4799 . . . . 5  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
74, 5, 6sylancl 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
8 simp3 957 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
9 snex 4215 . . . . 5  |-  { 1o }  e.  _V
10 xpexg 4799 . . . . 5  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
118, 9, 10sylancl 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
12 simp1 955 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
13 xp01disj 6491 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1413a1i 10 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
15 mapunen 7026 . . . 4  |-  ( ( ( ( B  X.  { (/) } )  e. 
_V  /\  ( C  X.  { 1o } )  e.  _V  /\  A  e.  V )  /\  (
( B  X.  { (/)
} )  i^i  ( C  X.  { 1o }
) )  =  (/) )  ->  ( A  ^m  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) ) 
~~  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) ) )
167, 11, 12, 14, 15syl31anc 1185 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  (
( B  X.  { (/)
} )  u.  ( C  X.  { 1o }
) ) )  ~~  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) ) )
173, 16eqbrtrd 4044 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  +c  C ) ) 
~~  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) ) )
18 enrefg 6889 . . . . 5  |-  ( A  e.  V  ->  A  ~~  A )
1912, 18syl 15 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
20 0ex 4151 . . . . 5  |-  (/)  e.  _V
21 xpsneng 6943 . . . . 5  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
224, 20, 21sylancl 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
23 mapen 7021 . . . 4  |-  ( ( A  ~~  A  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
2419, 22, 23syl2anc 642 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
25 1on 6482 . . . . 5  |-  1o  e.  On
26 xpsneng 6943 . . . . 5  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
278, 25, 26sylancl 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
28 mapen 7021 . . . 4  |-  ( ( A  ~~  A  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( A  ^m  ( C  X.  { 1o } ) )  ~~  ( A  ^m  C ) )
2919, 27, 28syl2anc 642 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( C  X.  { 1o }
) )  ~~  ( A  ^m  C ) )
30 xpen 7020 . . 3  |-  ( ( ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B )  /\  ( A  ^m  ( C  X.  { 1o }
) )  ~~  ( A  ^m  C ) )  ->  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) ) 
~~  ( ( A  ^m  B )  X.  ( A  ^m  C
) ) )
3124, 29, 30syl2anc 642 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) )  ~~  ( ( A  ^m  B )  X.  ( A  ^m  C ) ) )
32 entr 6909 . 2  |-  ( ( ( A  ^m  ( B  +c  C ) ) 
~~  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) )  /\  ( ( A  ^m  ( B  X.  { (/) } ) )  X.  ( A  ^m  ( C  X.  { 1o } ) ) ) 
~~  ( ( A  ^m  B )  X.  ( A  ^m  C
) ) )  -> 
( A  ^m  ( B  +c  C ) ) 
~~  ( ( A  ^m  B )  X.  ( A  ^m  C
) ) )
3317, 31, 32syl2anc 642 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  +c  C ) ) 
~~  ( ( A  ^m  B )  X.  ( A  ^m  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1685   _Vcvv 2789    u. cun 3151    i^i cin 3152   (/)c0 3456   {csn 3641   class class class wbr 4024   Oncon0 4391    X. cxp 4686  (class class class)co 5820   1oc1o 6468    ^m cmap 6768    ~~ cen 6856    +c ccda 7789
This theorem is referenced by:  pwcdaen  7807
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-1o 6475  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-cda 7790
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