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Theorem xpcdaen 8602
Description: Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
xpcdaen  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  ( B  +c  C ) ) 
~~  ( ( A  X.  B )  +c  ( A  X.  C
) ) )

Proof of Theorem xpcdaen
StepHypRef Expression
1 enrefg 7599 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  A )
213ad2ant1 1026 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
3 simp2 1006 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
4 0ex 4548 . . . . . . 7  |-  (/)  e.  _V
5 xpsneng 7654 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
63, 4, 5sylancl 666 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
76ensymd 7618 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( B  X.  { (/) } ) )
8 xpen 7732 . . . . 5  |-  ( ( A  ~~  A  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  B )  ~~  ( A  X.  ( B  X.  { (/) } ) ) )
92, 7, 8syl2anc 665 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) ) )
10 simp3 1007 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
11 1on 7188 . . . . . . 7  |-  1o  e.  On
12 xpsneng 7654 . . . . . . 7  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1310, 11, 12sylancl 666 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
1413ensymd 7618 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( C  X.  { 1o }
) )
15 xpen 7732 . . . . 5  |-  ( ( A  ~~  A  /\  C  ~~  ( C  X.  { 1o } ) )  ->  ( A  X.  C )  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
162, 14, 15syl2anc 665 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
17 xp01disj 7197 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1817xpeq2i 4866 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( A  X.  (/) )
19 xpindi 4979 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )
20 xp0 5266 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
2118, 19, 203eqtr3i 2457 . . . . 5  |-  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/)
2221a1i 11 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/) )
23 cdaenun 8593 . . . 4  |-  ( ( ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) )  /\  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) )  /\  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/) )  -> 
( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) ) )
249, 16, 22, 23syl3anc 1264 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) ) )
25 cdaval 8589 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
26253adant1 1023 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
2726xpeq2d 4869 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  ( B  +c  C ) )  =  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) ) )
28 xpundi 4898 . . . 4  |-  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) )
2927, 28syl6eq 2477 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  ( B  +c  C ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) ) )
3024, 29breqtrrd 4443 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) ) )
3130ensymd 7618 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  ( B  +c  C ) ) 
~~  ( ( A  X.  B )  +c  ( A  X.  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    i^i cin 3432   (/)c0 3758   {csn 3993   class class class wbr 4417    X. cxp 4843   Oncon0 5433  (class class class)co 6296   1oc1o 7174    ~~ cen 7565    +c ccda 8586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-1o 7181  df-er 7362  df-en 7569  df-dom 7570  df-cda 8587
This theorem is referenced by: (None)
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