Theorem subdir 7961

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 7960	. . 3 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
2	1	3coml 1153	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A - B ) ) = ( ( C x. A ) - ( C x. B ) ) )
3		subcl 7778	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
4		mulcom 7568	. . . 4 |- ( ( ( A - B ) e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
5	3, 4	sylan 278	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
6	5	3impa 1141	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( C x. ( A - B ) ) )
7		mulcom 7568	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8	7	3adant2 965	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
9		mulcom 7568	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 964	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	8, 10	oveq12d 5708	. 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 2137	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 103   /\ w3a 927   = wceq 1296   e. wcel 1445  (class class class)co 5690   CCcc 7445   x. cmul 7452   - cmin 7750

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381  ax-resscn 7534  ax-1cn 7535  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-sub 7752

This theorem is referenced by:  mul02  7962  mulneg1  7970  subdiri  7983  subdird  7990  dvds2sub  11273  cncongr1  11527  cncongr2  11528