Theorem subdir 8373

Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)

Assertion

Ref	Expression
subdir	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )

Proof of Theorem subdir

Step	Hyp	Ref	Expression
1		subdi 8372	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
2	1	3coml 1212	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
3		subcl 8186	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
4		mulcom 7970	. . . 4 |- ( ( ( A - B ) e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
5	3, 4	sylan 283	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
6	5	3impa 1196	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
7		mulcom 7970	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 1018	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 7970	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 1017	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 5914	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) - ( B x. C ) ) = ( ( C x. A ) - ( C x. B ) ) )
12	2, 6, 11	3eqtr4d 2232	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5896   CCcc 7839   x. cmul 7846   - cmin 8158

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-sub 8160

This theorem is referenced by:  mul02  8374  mulneg1  8382  subdiri  8395  subdird  8402  dvds2sub  11865  cncongr1  12135  cncongr2  12136  eulerthlemth  12264  pythagtriplem1  12297