Theorem moddiffl

Description: Value of the modulo operation rewritten to give two ways of expressing the quotient when " A is divided by B using Euclidean division." Multiplying both sides by B , this implies that A mod B differs from A by an integer multiple of B . (Contributed by Jeff Madsen, 17-Jun-2010) (Revised by Mario Carneiro, 6-Sep-2016)

	Ref	Expression
Assertion	moddiffl	|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )

Proof

Step	Hyp	Ref	Expression
1		modval	|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
2	1	oveq2d	|- ( ( A e. RR /\ B e. RR+ ) -> ( A - ( A mod B ) ) = ( A - ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) )
3		simpl	|- ( ( A e. RR /\ B e. RR+ ) -> A e. RR )
4	3	recnd	|- ( ( A e. RR /\ B e. RR+ ) -> A e. CC )
5		rpcn	|- ( B e. RR+ -> B e. CC )
6	5	adantl	|- ( ( A e. RR /\ B e. RR+ ) -> B e. CC )
7		rerpdivcl	|- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
8	7	flcld	|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. ZZ )
9	8	zcnd	|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. CC )
10	6 9	mulcld	|- ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC )
11	4 10	nncand	|- ( ( A e. RR /\ B e. RR+ ) -> ( A - ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
12	2 11	eqtrd	|- ( ( A e. RR /\ B e. RR+ ) -> ( A - ( A mod B ) ) = ( B x. ( |_ ` ( A / B ) ) ) )
13	12	oveq1d	|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( A mod B ) ) / B ) = ( ( B x. ( |_ ` ( A / B ) ) ) / B ) )
14		rpne0	|- ( B e. RR+ -> B =/= 0 )
15	14	adantl	|- ( ( A e. RR /\ B e. RR+ ) -> B =/= 0 )
16	9 6 15	divcan3d	|- ( ( A e. RR /\ B e. RR+ ) -> ( ( B x. ( |_ ` ( A / B ) ) ) / B ) = ( |_ ` ( A / B ) ) )
17	13 16	eqtrd	|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( A mod B ) ) / B ) = ( |_ ` ( A / B ) ) )