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Theorem xpcdaen 7825
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 6909 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  A )
213ad2ant1 976 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
3 simp2 956 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
4 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
5 xpsneng 6963 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
63, 4, 5sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
7 ensym 6926 . . . . . 6  |-  ( ( B  X.  { (/) } )  ~~  B  ->  B  ~~  ( B  X.  { (/) } ) )
86, 7syl 15 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( B  X.  { (/) } ) )
9 xpen 7040 . . . . 5  |-  ( ( A  ~~  A  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  B )  ~~  ( A  X.  ( B  X.  { (/) } ) ) )
102, 8, 9syl2anc 642 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) ) )
11 simp3 957 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
12 1on 6502 . . . . . . 7  |-  1o  e.  On
13 xpsneng 6963 . . . . . . 7  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1411, 12, 13sylancl 643 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
15 ensym 6926 . . . . . 6  |-  ( ( C  X.  { 1o } )  ~~  C  ->  C  ~~  ( C  X.  { 1o }
) )
1614, 15syl 15 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( C  X.  { 1o }
) )
17 xpen 7040 . . . . 5  |-  ( ( A  ~~  A  /\  C  ~~  ( C  X.  { 1o } ) )  ->  ( A  X.  C )  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
182, 16, 17syl2anc 642 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
19 xp01disj 6511 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
2019xpeq2i 4726 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( A  X.  (/) )
21 xpindi 4835 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )
22 xp0 5114 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
2320, 21, 223eqtr3i 2324 . . . . 5  |-  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/)
2423a1i 10 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )  =  (/) )
25 cdaenun 7816 . . . 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 1182 . . 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 7812 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
28273adant1 973 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
2928xpeq2d 4729 . . . 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 4757 . . . 4  |-  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) )
3129, 30syl6eq 2344 . . 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 4065 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) ) )
33 ensym 6926 . 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 15 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 4    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   class class class wbr 4039   Oncon0 4408    X. cxp 4703  (class class class)co 5874   1oc1o 6488    ~~ cen 6876    +c ccda 7809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-cda 7810
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