Theorem xpdjuen 7527

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 7005	. . . . . 6 |- ( A e. V -> A ~~ A )
2	1	3ad2ant1 1045	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A )
3		0ex 4239	. . . . . . 7 |- (/) e. _V
4		simp2 1025	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
5		xpsnen2g 7082	. . . . . . 7 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	3, 4, 5	sylancr 414	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B )
7	6	ensymd 7025	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) )
8		xpen 7100	. . . . 5 |- ( ( A ~~ A /\ B ~~ ( { (/) } X. B ) ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
9	2, 7, 8	syl2anc 411	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
10		1on 6656	. . . . . . 7 |- 1o e. On
11		simp3 1026	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
12		xpsnen2g 7082	. . . . . . 7 |- ( ( 1o e. On /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
13	10, 11, 12	sylancr 414	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
14	13	ensymd 7025	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) )
15		xpen 7100	. . . . 5 |- ( ( A ~~ A /\ C ~~ ( { 1o } X. C ) ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
16	2, 14, 15	syl2anc 411	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
17		xp01disjl 6669	. . . . . . 7 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
18	17	xpeq2i 4772	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) )
19		xpindi 4892	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) )
20		xp0 5184	. . . . . 6 |- ( A X. (/) ) = (/)
21	18, 19, 20	3eqtr3i 2263	. . . . 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 7521	. . . 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 1274	. . 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 7331	. . . . 5 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
26	25	xpeq2i 4772	. . . 4 |- ( A X. ( B ⊔ C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
27		xpundi 4808	. . . 4 |- ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
28	26, 27	eqtri 2255	. . 3 |- ( A X. ( B ⊔ C ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
29	24, 28	breqtrrdi 4153	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( A X. ( B ⊔ C ) ) )
30	29	ensymd 7025	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 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   u. cun 3211   i^i cin 3212   (/)c0 3510   {csn 3691   class class class wbr 4111   Oncon0 4486   X. cxp 4749   1oc1o 6642   ~~ cen 6975   ⊔ cdju 7330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-er 6769  df-en 6978  df-dju 7331  df-inl 7340  df-inr 7341

This theorem is referenced by: (None)