Theorem mulsub2 8031

Description: Swap the order of subtraction in a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

Assertion

Ref	Expression
mulsub2	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) )

Proof of Theorem mulsub2

Step	Hyp	Ref	Expression
1		subcl 7832	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2		subcl 7832	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
3		mul2neg 8027	. . 3 |- ( ( ( A - B ) e. CC /\ ( C - D ) e. CC ) -> ( -u ( A - B ) x. -u ( C - D ) ) = ( ( A - B ) x. ( C - D ) ) )
4	1, 2, 3	syl2an 285	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) x. -u ( C - D ) ) = ( ( A - B ) x. ( C - D ) ) )
5		negsubdi2 7892	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
6		negsubdi2 7892	. . 3 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
7	5, 6	oveqan12d 5725	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( -u ( A - B ) x. -u ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) )
8	4, 7	eqtr3d 2134	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A - B ) x. ( C - D ) ) = ( ( B - A ) x. ( D - C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448  (class class class)co 5706   CCcc 7498   x. cmul 7505   - cmin 7804   -ucneg 7805

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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-setind 4390  ax-resscn 7587  ax-1cn 7588  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-sub 7806  df-neg 7807

This theorem is referenced by: (None)