Theorem xpdjuen 7432

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 6936	. . . . . 6 |- ( A e. V -> A ~~ A )
2	1	3ad2ant1 1044	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A )
3		0ex 4216	. . . . . . 7 |- (/) e. _V
4		simp2 1024	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
5		xpsnen2g 7012	. . . . . . 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 6956	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) )
8		xpen 7030	. . . . 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 6588	. . . . . . 7 |- 1o e. On
11		simp3 1025	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
12		xpsnen2g 7012	. . . . . . 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 6956	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) )
15		xpen 7030	. . . . 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 6601	. . . . . . 7 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
18	17	xpeq2i 4746	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) )
19		xpindi 4865	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) )
20		xp0 5156	. . . . . 6 |- ( A X. (/) ) = (/)
21	18, 19, 20	3eqtr3i 2260	. . . . 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 7426	. . . 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 1273	. . 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 7236	. . . . 5 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
26	25	xpeq2i 4746	. . . 4 |- ( A X. ( B ⊔ C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
27		xpundi 4782	. . . 4 |- ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
28	26, 27	eqtri 2252	. . 3 |- ( A X. ( B ⊔ C ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
29	24, 28	breqtrrdi 4130	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( A X. ( B ⊔ C ) ) )
30	29	ensymd 6956	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 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   i^i cin 3199   (/)c0 3494   {csn 3669   class class class wbr 4088   Oncon0 4460   X. cxp 4723   1oc1o 6574   ~~ cen 6906   ⊔ cdju 7235

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-er 6701  df-en 6909  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by: (None)