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Theorem xpcdaen 8019
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 7098 . . . . . 6  |-  ( A  e.  V  ->  A  ~~  A )
213ad2ant1 978 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  A  ~~  A )
3 simp2 958 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  e.  W )
4 0ex 4299 . . . . . . 7  |-  (/)  e.  _V
5 xpsneng 7152 . . . . . . 7  |-  ( ( B  e.  W  /\  (/) 
e.  _V )  ->  ( B  X.  { (/) } ) 
~~  B )
63, 4, 5sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  X.  { (/)
} )  ~~  B
)
76ensymd 7117 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  B  ~~  ( B  X.  { (/) } ) )
8 xpen 7229 . . . . 5  |-  ( ( A  ~~  A  /\  B  ~~  ( B  X.  { (/) } ) )  ->  ( A  X.  B )  ~~  ( A  X.  ( B  X.  { (/) } ) ) )
92, 7, 8syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  B
)  ~~  ( A  X.  ( B  X.  { (/)
} ) ) )
10 simp3 959 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  e.  X )
11 1on 6690 . . . . . . 7  |-  1o  e.  On
12 xpsneng 7152 . . . . . . 7  |-  ( ( C  e.  X  /\  1o  e.  On )  -> 
( C  X.  { 1o } )  ~~  C
)
1310, 11, 12sylancl 644 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( C  X.  { 1o } )  ~~  C
)
1413ensymd 7117 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  C  ~~  ( C  X.  { 1o }
) )
15 xpen 7229 . . . . 5  |-  ( ( A  ~~  A  /\  C  ~~  ( C  X.  { 1o } ) )  ->  ( A  X.  C )  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
162, 14, 15syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( A  X.  C
)  ~~  ( A  X.  ( C  X.  { 1o } ) ) )
17 xp01disj 6699 . . . . . . 7  |-  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o }
) )  =  (/)
1817xpeq2i 4858 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( A  X.  (/) )
19 xpindi 4967 . . . . . 6  |-  ( A  X.  ( ( B  X.  { (/) } )  i^i  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  i^i  ( A  X.  ( C  X.  { 1o } ) ) )
20 xp0 5250 . . . . . 6  |-  ( A  X.  (/) )  =  (/)
2118, 19, 203eqtr3i 2432 . . . . 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 8010 . . . 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 1184 . . 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 8006 . . . . . 6  |-  ( ( B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
26253adant1 975 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( B  +c  C
)  =  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o }
) ) )
2726xpeq2d 4861 . . . 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 4889 . . . 4  |-  ( A  X.  ( ( B  X.  { (/) } )  u.  ( C  X.  { 1o } ) ) )  =  ( ( A  X.  ( B  X.  { (/) } ) )  u.  ( A  X.  ( C  X.  { 1o } ) ) )
2927, 28syl6eq 2452 . . 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 4198 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( ( A  X.  B )  +c  ( A  X.  C ) ) 
~~  ( A  X.  ( B  +c  C
) ) )
3130ensymd 7117 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 936    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279   (/)c0 3588   {csn 3774   class class class wbr 4172   Oncon0 4541    X. cxp 4835  (class class class)co 6040   1oc1o 6676    ~~ cen 7065    +c ccda 8003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-er 6864  df-en 7069  df-dom 7070  df-cda 8004
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