Theorem modnegd

Description: Negation property of the modulo operation. (Contributed by Mario Carneiro, 9-Sep-2016)

	Ref	Expression
Hypotheses	modnegd.1	|- ( ph -> A e. RR )
	modnegd.2	|- ( ph -> B e. RR )
	modnegd.3	|- ( ph -> C e. RR+ )
	modnegd.4	|- ( ph -> ( A mod C ) = ( B mod C ) )
Assertion	modnegd	|- ( ph -> ( -u A mod C ) = ( -u B mod C ) )

Proof

Step	Hyp	Ref	Expression
1		modnegd.1	|- ( ph -> A e. RR )
2		modnegd.2	|- ( ph -> B e. RR )
3		modnegd.3	|- ( ph -> C e. RR+ )
4		modnegd.4	|- ( ph -> ( A mod C ) = ( B mod C ) )
5		1zzd	|- ( ph -> 1 e. ZZ )
6	5	znegcld	|- ( ph -> -u 1 e. ZZ )
7		modmul1	|- ( ( ( A e. RR /\ B e. RR ) /\ ( -u 1 e. ZZ /\ C e. RR+ ) /\ ( A mod C ) = ( B mod C ) ) -> ( ( A x. -u 1 ) mod C ) = ( ( B x. -u 1 ) mod C ) )
8	1 2 6 3 4 7	syl221anc	|- ( ph -> ( ( A x. -u 1 ) mod C ) = ( ( B x. -u 1 ) mod C ) )
9	1	recnd	|- ( ph -> A e. CC )
10		1cnd	|- ( ph -> 1 e. CC )
11	10	negcld	|- ( ph -> -u 1 e. CC )
12	9 11	mulcomd	|- ( ph -> ( A x. -u 1 ) = ( -u 1 x. A ) )
13	9	mulm1d	|- ( ph -> ( -u 1 x. A ) = -u A )
14	12 13	eqtrd	|- ( ph -> ( A x. -u 1 ) = -u A )
15	14	oveq1d	|- ( ph -> ( ( A x. -u 1 ) mod C ) = ( -u A mod C ) )
16	2	recnd	|- ( ph -> B e. CC )
17	16 11	mulcomd	|- ( ph -> ( B x. -u 1 ) = ( -u 1 x. B ) )
18	16	mulm1d	|- ( ph -> ( -u 1 x. B ) = -u B )
19	17 18	eqtrd	|- ( ph -> ( B x. -u 1 ) = -u B )
20	19	oveq1d	|- ( ph -> ( ( B x. -u 1 ) mod C ) = ( -u B mod C ) )
21	8 15 20	3eqtr3d	|- ( ph -> ( -u A mod C ) = ( -u B mod C ) )