Theorem subdi 8492

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 1000	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp3 1002	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
3		subcl 8306	. . . . . 6 |- ( ( B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
4	3	3adant1 1018	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B - C ) e. CC )
5	1, 2, 4	adddid 8132	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( ( A x. C ) + ( A x. ( B - C ) ) ) )
6		pncan3 8315	. . . . . . 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 1018	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C + ( B - C ) ) = B )
9	8	oveq2d 5983	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( C + ( B - C ) ) ) = ( A x. B ) )
10	5, 9	eqtr3d 2242	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( A x. ( B - C ) ) ) = ( A x. B ) )
11		mulcl 8087	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
12	11	3adant3 1020	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
13		mulcl 8087	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
14	13	3adant2 1019	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) e. CC )
15		mulcl 8087	. . . . . 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 1202	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B - C ) ) e. CC )
18	12, 14, 17	subaddd 8436	. . 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 2213	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 981   = wceq 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   + caddc 7963   x. cmul 7965   - cmin 8278

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280

This theorem is referenced by:  subdir  8493  subdii  8514  subdid  8521  expubnd  10778  subsq  10828  cos01bnd  12184  modmulconst  12249  odd2np1  12299  omoe  12322  omeo  12324  phiprmpw  12659  pythagtriplem14  12715