Theorem subdi 8554

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 1021	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp3 1023	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
3		subcl 8368	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4	3	3adant1 1039	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
5	1, 2, 4	adddid 8194	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( ( A x. C ) + ( A x. ( B - C ) ) ) )
6		pncan3 8377	. . . . . . 7 |- ( ( C e. CC /\ B e. CC ) -> ( C + ( B - C ) ) = B )
7	6	ancoms 268	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
8	7	3adant1 1039	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
9	8	oveq2d 6029	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( A x. B ) )
10	5, 9	eqtr3d 2264	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( A x. ( B - C ) ) ) = ( A x. B ) )
11		mulcl 8149	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
12	11	3adant3 1041	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
13		mulcl 8149	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
14	13	3adant2 1040	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
15		mulcl 8149	. . . . . 6 |- ( ( A e. CC /\ ( B - C ) e. CC ) -> ( A x. ( B - C ) ) e. CC )
16	3, 15	sylan2 286	. . . . 5 |- ( ( A e. CC /\ ( B e. CC /\ C e. CC ) ) -> ( A x. ( B - C ) ) e. CC )
17	16	3impb 1223	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) e. CC )
18	12, 14, 17	subaddd 8498	. . 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 167	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) - ( A x. C ) ) = ( A x. ( B - C ) ) )
20	19	eqcomd 2235	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   + caddc 8025   x. cmul 8027   - cmin 8340

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-sub 8342

This theorem is referenced by:  subdir  8555  subdii  8576  subdid  8583  expubnd  10848  subsq  10898  cos01bnd  12309  modmulconst  12374  odd2np1  12424  omoe  12447  omeo  12449  phiprmpw  12784  pythagtriplem14  12840