Theorem xpdjuen 7574

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 7050	. . . . . 6 |- ( A e. V -> A ~~ A )
2	1	3ad2ant1 1049	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A )
3		0ex 4260	. . . . . . 7 |- (/) e. _V
4		simp2 1029	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
5		xpsnen2g 7127	. . . . . . 7 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	3, 4, 5	sylancr 418	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B )
7	6	ensymd 7070	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) )
8		xpen 7145	. . . . 5 |- ( ( A ~~ A /\ B ~~ ( { (/) } X. B ) ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
9	2, 7, 8	syl2anc 415	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. B ) ~~ ( A X. ( { (/) } X. B ) ) )
10		1on 6694	. . . . . . 7 |- 1o e. On
11		simp3 1030	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
12		xpsnen2g 7127	. . . . . . 7 |- ( ( 1o e. On /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
13	10, 11, 12	sylancr 418	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
14	13	ensymd 7070	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) )
15		xpen 7145	. . . . 5 |- ( ( A ~~ A /\ C ~~ ( { 1o } X. C ) ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
16	2, 14, 15	syl2anc 415	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. C ) ~~ ( A X. ( { 1o } X. C ) ) )
17		xp01disjl 6707	. . . . . . 7 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
18	17	xpeq2i 4795	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) )
19		xpindi 4915	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) )
20		xp0 5207	. . . . . 6 |- ( A X. (/) ) = (/)
21	18, 19, 20	3eqtr3i 2267	. . . . 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 7568	. . . 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 1278	. . 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 7378	. . . . 5 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
26	25	xpeq2i 4795	. . . 4 |- ( A X. ( B ⊔ C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
27		xpundi 4831	. . . 4 |- ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
28	26, 27	eqtri 2259	. . 3 |- ( A X. ( B ⊔ C ) ) = ( ( A X. ( { (/) } X. B ) ) u. ( A X. ( { 1o } X. C ) ) )
29	24, 28	breqtrrdi 4172	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( A X. ( B ⊔ C ) ) )
30	29	ensymd 7070	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. ( B ⊔ C ) ) ~~ ( ( A X. B ) ⊔ ( A X. C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3709   class class class wbr 4130   Oncon0 4508   X. cxp 4772   1oc1o 6680   ~~ cen 7020   ⊔ cdju 7377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-er 6807  df-en 7023  df-dju 7378  df-inl 7387  df-inr 7388

This theorem is used by: (None)