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

Theorem mapcdaen 7990
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 7976 . . . . 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 6029 . . 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 4339 . . . . 5  |-  { (/) }  e.  _V
6 xpexg 4922 . . . . 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 4339 . . . . 5  |-  { 1o }  e.  _V
10 xpexg 4922 . . . . 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 6669 . . . . 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 7205 . . . 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 4166 . 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 7068 . . . . 5  |-  ( A  e.  V  ->  A  ~~  A )
1912, 18syl 16 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
20 0ex 4273 . . . . 5  |-  (/)  e.  _V
21 xpsneng 7122 . . . . 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 7200 . . . 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 6660 . . . . 5  |-  1o  e.  On
26 xpsneng 7122 . . . . 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 7200 . . . 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 7199 . . 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 7088 . 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 1649    e. wcel 1717   _Vcvv 2892    u. cun 3254    i^i cin 3255   (/)c0 3564   {csn 3750   class class class wbr 4146   Oncon0 4515    X. cxp 4809  (class class class)co 6013   1oc1o 6646    ^m cmap 6947    ~~ cen 7035    +c ccda 7973
This theorem is referenced by:  pwcdaen  7991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-suc 4521  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-1o 6653  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-cda 7974
  Copyright terms: Public domain W3C validator