Theorem xpdjuen 7155

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
xpdjuen	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B ⊔ C ) ) ~~ ( ( A X. B ) ⊔ ( A X. C ) ) )

Proof of Theorem xpdjuen

Step	Hyp	Ref	Expression
1		enrefg 6711	. . . . . 6 |- ( A e. V -> A ~~ A )
2	1	3ad2ant1 1003	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A )
3		0ex 4093	. . . . . . 7 |- (/) e. _V
4		simp2 983	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
5		xpsnen2g 6776	. . . . . . 7 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	3, 4, 5	sylancr 411	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B )
7	6	ensymd 6730	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) )
8		xpen 6792	. . . . 5 |- ( ( A ~~ A /\ B ~~ ( { (/) } X. B ) ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
9	2, 7, 8	syl2anc 409	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
10		1on 6372	. . . . . . 7 |- 1o e. On
11		simp3 984	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
12		xpsnen2g 6776	. . . . . . 7 |- ( ( 1o e. On /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
13	10, 11, 12	sylancr 411	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
14	13	ensymd 6730	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) )
15		xpen 6792	. . . . 5 |- ( ( A ~~ A /\ C ~~ ( { 1o } X. C ) ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
16	2, 14, 15	syl2anc 409	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
17		xp01disjl 6383	. . . . . . 7 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
18	17	xpeq2i 4609	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) )
19		xpindi 4723	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) )
20		xp0 5007	. . . . . 6 |- ( A X. (/) ) = (/)
21	18, 19, 20	3eqtr3i 2186	. . . . 5 |- ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/)
22	21	a1i 9	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/) )
23		djuenun 7149	. . . 4 |- ( ( ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) /\ ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) /\ ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) ) = (/) ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) )
24	9, 16, 22, 23	syl3anc 1220	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) ) )
25		df-dju 6984	. . . . 5 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
26	25	xpeq2i 4609	. . . 4 |- ( A X. ( B ⊔ C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
27		xpundi 4644	. . . 4 |- ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
28	26, 27	eqtri 2178	. . 3 |- ( A X. ( B ⊔ C ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
29	24, 28	breqtrrdi 4008	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( A X. ( B ⊔ C ) ) )
30	29	ensymd 6730	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B ⊔ C ) ) ~~ ( ( A X. B ) ⊔ ( A X. C ) ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1335   e. wcel 2128   _Vcvv 2712   u. cun 3100   i^i cin 3101   (/)c0 3395   {csn 3561   class class class wbr 3967   Oncon0 4325   X. cxp 4586   1oc1o 6358   ~~ cen 6685   ⊔ cdju 6983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  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-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-1o 6365  df-er 6482  df-en 6688  df-dju 6984  df-inl 6993  df-inr 6994

This theorem is referenced by: (None)