Theorem subdi 7382

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 904	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp3 906	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
3		subcl 7210	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4	3	3adant1 922	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
5	1, 2, 4	adddid 7051	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( ( A x. C ) + ( A x. ( B - C ) ) ) )
6		pncan3 7219	. . . . . . 7 |- ( ( C e. CC /\ B e. CC ) -> ( C + ( B - C ) ) = B )
7	6	ancoms 255	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
8	7	3adant1 922	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
9	8	oveq2d 5528	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( A x. B ) )
10	5, 9	eqtr3d 2074	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( A x. ( B - C ) ) ) = ( A x. B ) )
11		mulcl 7008	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
12	11	3adant3 924	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
13		mulcl 7008	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
14	13	3adant2 923	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
15		mulcl 7008	. . . . . 6 |- ( ( A e. CC /\ ( B - C ) e. CC ) -> ( A x. ( B - C ) ) e. CC )
16	3, 15	sylan2 270	. . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( A x. ( B - C ) ) e. CC )
17	16	3impb 1100	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) e. CC )
18	12, 14, 17	subaddd 7340	. . 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 156	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) - ( A x. C ) ) = ( A x. ( B - C ) ) )
20	19	eqcomd 2045	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 97   /\ w3a 885   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   + caddc 6892   x. cmul 6894   - cmin 7182

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184

This theorem is referenced by:  subdir  7383  subdii  7404  subdid  7411  expubnd  9311  subsq  9358