Theorem subdi 8283

Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)

Assertion

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

Proof of Theorem subdi

Step	Hyp	Ref	Expression
1		simp1 987	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp3 989	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
3		subcl 8097	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4	3	3adant1 1005	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
5	1, 2, 4	adddid 7923	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( ( A x. C ) + ( A x. ( B - C ) ) ) )
6		pncan3 8106	. . . . . . 7 |- ( ( C e. CC /\ B e. CC ) -> ( C + ( B - C ) ) = B )
7	6	ancoms 266	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
8	7	3adant1 1005	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
9	8	oveq2d 5858	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( A x. B ) )
10	5, 9	eqtr3d 2200	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( A x. ( B - C ) ) ) = ( A x. B ) )
11		mulcl 7880	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
12	11	3adant3 1007	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
13		mulcl 7880	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
14	13	3adant2 1006	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
15		mulcl 7880	. . . . . 6 |- ( ( A e. CC /\ ( B - C ) e. CC ) -> ( A x. ( B - C ) ) e. CC )
16	3, 15	sylan2 284	. . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( A x. ( B - C ) ) e. CC )
17	16	3impb 1189	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) e. CC )
18	12, 14, 17	subaddd 8227	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A x. B ) - ( A x. C ) ) = ( A x. ( B - C ) ) <-> ( ( A x. C ) + ( A x. ( B - C ) ) ) = ( A x. B ) ) )
19	10, 18	mpbird 166	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) - ( A x. C ) ) = ( A x. ( B - C ) ) )
20	19	eqcomd 2171	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) = ( ( A x. B ) - ( A x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   + caddc 7756   x. cmul 7758   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071

This theorem is referenced by:  subdir  8284  subdii  8305  subdid  8312  expubnd  10512  subsq  10561  cos01bnd  11699  modmulconst  11763  odd2np1  11810  omoe  11833  omeo  11835  phiprmpw  12154  pythagtriplem14  12209