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

Theorem mapcdaen 7743
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 7729 . . . . 5  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
213adant1 978 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
32oveq2d 5773 . . 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 961 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
5 snex 4154 . . . . 5  |-  { (/) }  e.  _V
6 xpexg 4753 . . . . 5  |-  ( ( B  e.  W  /\  {
(/) }  e.  _V )  ->  ( B  X.  { (/) } )  e. 
_V )
74, 5, 6sylancl 646 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  e.  _V )
8 simp3 962 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
9 snex 4154 . . . . 5  |-  { 1o }  e.  _V
10 xpexg 4753 . . . . 5  |-  ( ( C  e.  X  /\  { 1o }  e.  _V )  ->  ( C  X.  { 1o } )  e. 
_V )
118, 9, 10sylancl 646 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  e.  _V )
12 simp1 960 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  e.  V )
13 xp01disj 6428 . . . . 5  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1413a1i 12 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) )  =  (/) )
15 mapunen 6963 . . . 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 1190 . . 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 3983 . 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 6826 . . . . 5  |-  ( A  e.  V  ->  A  ~~  A )
1912, 18syl 17 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
20 0ex 4090 . . . . 5  |-  (/)  e.  _V
21 xpsneng 6880 . . . . 5  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
224, 20, 21sylancl 646 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
23 mapen 6958 . . . 4  |-  ( ( A  ~~  A  /\  ( B  X.  { (/) } )  ~~  B )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
2419, 22, 23syl2anc 645 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( B  X.  { (/) } ) )  ~~  ( A  ^m  B ) )
25 1on 6419 . . . . 5  |-  1o  e.  On
26 xpsneng 6880 . . . . 5  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
278, 25, 26sylancl 646 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
28 mapen 6958 . . . 4  |-  ( ( A  ~~  A  /\  ( C  X.  { 1o } )  ~~  C
)  ->  ( A  ^m  ( C  X.  { 1o } ) )  ~~  ( A  ^m  C ) )
2919, 27, 28syl2anc 645 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  ^m  ( C  X.  { 1o }
) )  ~~  ( A  ^m  C ) )
30 xpen 6957 . . 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 645 . 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 6846 . 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 645 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 6    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2740    u. cun 3092    i^i cin 3093   (/)c0 3397   {csn 3581   class class class wbr 3963   Oncon0 4329    X. cxp 4624  (class class class)co 5757   1oc1o 6405    ^m cmap 6705    ~~ cen 6793    +c ccda 7726
This theorem is referenced by:  pwcdaen  7744
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-suc 4335  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-1o 6412  df-er 6593  df-map 6707  df-en 6797  df-dom 6798  df-cda 7727
  Copyright terms: Public domain W3C validator