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

Theorem mapcdaen 8064
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 8050 . . . . 5  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
213adant1 975 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
32oveq2d 6097 . . 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 958 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
5 snex 4405 . . . . 5  |-  { (/) }  e.  _V
6 xpexg 4989 . . . . 5  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
74, 5, 6sylancl 644 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
8 simp3 959 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
9 snex 4405 . . . . 5  |-  { 1o }  e.  _V
10 xpexg 4989 . . . . 5  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
118, 9, 10sylancl 644 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
12 simp1 957 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
13 xp01disj 6740 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1413a1i 11 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
15 mapunen 7276 . . . 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 1187 . . 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 4232 . 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 7139 . . . . 5  |-  ( A  e.  V  ->  A  ~~  A )
1912, 18syl 16 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
20 0ex 4339 . . . . 5  |-  (/)  e.  _V
21 xpsneng 7193 . . . . 5  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
224, 20, 21sylancl 644 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
23 mapen 7271 . . . 4  |-  ( ( A  ~~  A  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
2419, 22, 23syl2anc 643 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
25 1on 6731 . . . . 5  |-  1o  e.  On
26 xpsneng 7193 . . . . 5  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
278, 25, 26sylancl 644 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
28 mapen 7271 . . . 4  |-  ( ( A  ~~  A  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( A  ^m  ( C  X.  { 1o } ) )  ~~  ( A  ^m  C ) )
2919, 27, 28syl2anc 643 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( C  X.  { 1o }
) )  ~~  ( A  ^m  C ) )
30 xpen 7270 . . 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 643 . 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 7159 . 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 643 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 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814   class class class wbr 4212   Oncon0 4581    X. cxp 4876  (class class class)co 6081   1oc1o 6717    ^m cmap 7018    ~~ cen 7106    +c ccda 8047
This theorem is referenced by:  pwcdaen  8065
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-1o 6724  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-cda 8048
  Copyright terms: Public domain W3C validator