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Theorem mapcdaen 7810
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 7796 . . . . 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 5874 . . 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 4216 . . . . 5  |-  { (/) }  e.  _V
6 xpexg 4800 . . . . 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 4216 . . . . 5  |-  { 1o }  e.  _V
10 xpexg 4800 . . . . 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 6495 . . . . 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 7030 . . . 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 4043 . 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 6893 . . . . 5  |-  ( A  e.  V  ->  A  ~~  A )
1912, 18syl 15 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
20 0ex 4150 . . . . 5  |-  (/)  e.  _V
21 xpsneng 6947 . . . . 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 7025 . . . 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 6486 . . . . 5  |-  1o  e.  On
26 xpsneng 6947 . . . . 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 7025 . . . 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 7024 . . 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 6913 . 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 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   class class class wbr 4023   Oncon0 4392    X. cxp 4687  (class class class)co 5858   1oc1o 6472    ^m cmap 6772    ~~ cen 6860    +c ccda 7793
This theorem is referenced by:  pwcdaen  7811
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-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-cda 7794
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