Theorem xpdjuen 7564

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 7040	. . . . . 6 |- ( A e. V -> A ~~ A )
2	1	3ad2ant1 1049	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ A )
3		0ex 4255	. . . . . . 7 |- (/) e. _V
4		simp2 1029	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
5		xpsnen2g 7117	. . . . . . 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 7060	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { (/) } X. B ) )
8		xpen 7135	. . . . 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 6684	. . . . . . 7 |- 1o e. On
11		simp3 1030	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
12		xpsnen2g 7117	. . . . . . 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 7060	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. C ) )
15		xpen 7135	. . . . 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 6697	. . . . . . 7 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
18	17	xpeq2i 4790	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( A X. (/) )
19		xpindi 4910	. . . . . 6 |- ( A X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( A X. ( { (/) } X. B ) ) i^i ( A X. ( { 1o } X. C ) ) )
20		xp0 5202	. . . . . 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 7558	. . . 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 7368	. . . . 5 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
26	25	xpeq2i 4790	. . . 4 |- ( A X. ( B ⊔ C ) ) = ( A X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
27		xpundi 4826	. . . 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 4167	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. B ) ⊔ ( A X. C ) ) ~~ ( A X. ( B ⊔ C ) ) )
30	29	ensymd 7060	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 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3705   class class class wbr 4125   Oncon0 4503   X. cxp 4767   1oc1o 6670   ~~ cen 7010   ⊔ cdju 7367

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-er 6797  df-en 7013  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by: (None)