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Theorem xpcdaen 8564
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 7548 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  A )
213ad2ant1 1017 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
3 simp2 997 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
4 0ex 4577 . . . . . . 7  |-  (/)  e.  _V
5 xpsneng 7603 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
63, 4, 5sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
76ensymd 7567 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( B  X.  { (/) } ) )
8 xpen 7681 . . . . 5  |-  ( ( A  ~~  A  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  B )  ~~  ( A  X.  ( B  X.  { (/) } ) ) )
92, 7, 8syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) ) )
10 simp3 998 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
11 1on 7138 . . . . . . 7  |-  1o  e.  On
12 xpsneng 7603 . . . . . . 7  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1310, 11, 12sylancl 662 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
1413ensymd 7567 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( C  X.  { 1o }
) )
15 xpen 7681 . . . . 5  |-  ( ( A  ~~  A  /\  C  ~~  ( C  X.  { 1o } ) )  ->  ( A  X.  C )  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
162, 14, 15syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
17 xp01disj 7147 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1817xpeq2i 5020 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( A  X.  (/) )
19 xpindi 5136 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )
20 xp0 5425 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
2118, 19, 203eqtr3i 2504 . . . . 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 8555 . . . 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 1228 . . 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 8551 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
26253adant1 1014 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
2726xpeq2d 5023 . . . 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 5052 . . . 4  |-  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) )
2927, 28syl6eq 2524 . . 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 4473 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) ) )
3130ensymd 7567 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   class class class wbr 4447   Oncon0 4878    X. cxp 4997  (class class class)co 6285   1oc1o 7124    ~~ cen 7514    +c ccda 8548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-cda 8549
This theorem is referenced by: (None)
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