Theorem mulsub2 8308

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 8105	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC )
2		subcl 8105	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C - D ) e. CC )
3		mul2neg 8304	. . 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 287	. 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 8165	. . 3 |- ( ( A e. CC /\ B e. CC ) -> -u ( A - B ) = ( B - A ) )
6		negsubdi2 8165	. . 3 |- ( ( C e. CC /\ D e. CC ) -> -u ( C - D ) = ( D - C ) )
7	5, 6	oveqan12d 5869	. 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 2205	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 1348   e. wcel 2141  (class class class)co 5850   CCcc 7759   x. cmul 7766   - cmin 8077   -ucneg 8078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519  ax-resscn 7853  ax-1cn 7854  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-sub 8079  df-neg 8080

This theorem is referenced by: (None)