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Theorem xpcdaen 7693
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 6779 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  A )
213ad2ant1 981 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
3 simp2 961 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
4 0ex 4047 . . . . . . 7  |-  (/)  e.  _V
5 xpsneng 6832 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
63, 4, 5sylancl 646 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
7 ensym 6796 . . . . . 6  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
86, 7syl 17 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( B  X.  { (/) } ) )
9 xpen 6909 . . . . 5  |-  ( ( A  ~~  A  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  B )  ~~  ( A  X.  ( B  X.  { (/) } ) ) )
102, 8, 9syl2anc 645 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) ) )
11 simp3 962 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
12 1on 6372 . . . . . . 7  |-  1o  e.  On
13 xpsneng 6832 . . . . . . 7  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1411, 12, 13sylancl 646 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
15 ensym 6796 . . . . . 6  |-  ( ( C  X.  { 1o } )  ~~  C  ->  C  ~~  ( C  X.  { 1o }
) )
1614, 15syl 17 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( C  X.  { 1o }
) )
17 xpen 6909 . . . . 5  |-  ( ( A  ~~  A  /\  C  ~~  ( C  X.  { 1o } ) )  ->  ( A  X.  C )  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
182, 16, 17syl2anc 645 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
19 xp01disj 6381 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
2019xpeq2i 4617 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( A  X.  (/) )
21 xpindi 4726 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )
22 xp0 5005 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
2320, 21, 223eqtr3i 2281 . . . . 5  |-  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/)
2423a1i 12 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/) )
25 cdaenun 7684 . . . 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 } ) ) ) )
2610, 18, 24, 25syl3anc 1187 . . 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 } ) ) ) )
27 cdaval 7680 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
28273adant1 978 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
2928xpeq2d 4620 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  ( B  +c  C ) )  =  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) ) )
30 xpundi 4648 . . . 4  |-  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) )
3129, 30syl6eq 2301 . . 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 } ) ) ) )
3226, 31breqtrrd 3946 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) ) )
33 ensym 6796 . 2  |-  ( ( ( A  X.  B
)  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) )  ->  ( A  X.  ( B  +c  C ) )  ~~  ( ( A  X.  B )  +c  ( A  X.  C ) ) )
3432, 33syl 17 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 set class
Syntax hints:    -> wi 6    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2727    u. cun 3076    i^i cin 3077   (/)c0 3362   {csn 3544   class class class wbr 3920   Oncon0 4285    X. cxp 4578  (class class class)co 5710   1oc1o 6358    ~~ cen 6746    +c ccda 7677
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-suc 4291  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-1o 6365  df-er 6546  df-en 6750  df-dom 6751  df-cda 7678
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