Theorem renpncan3

Description: Cancellation law for real subtraction. Compare npncan3 . (Contributed by Steven Nguyen, 28-Jan-2023)

	Ref	Expression
Assertion	renpncan3	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A -R B ) + ( C -R A ) ) = ( C -R B ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2		rersubcl	|- ( ( C e. RR /\ A e. RR ) -> ( C -R A ) e. RR )
3	2	ancoms	|- ( ( A e. RR /\ C e. RR ) -> ( C -R A ) e. RR )
4	3	3adant2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C -R A ) e. RR )
5		simp2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6		readdsub	|- ( ( A e. RR /\ ( C -R A ) e. RR /\ B e. RR ) -> ( ( A + ( C -R A ) ) -R B ) = ( ( A -R B ) + ( C -R A ) ) )
7	1 4 5 6	syl3anc	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + ( C -R A ) ) -R B ) = ( ( A -R B ) + ( C -R A ) ) )
8		repncan3	|- ( ( A e. RR /\ C e. RR ) -> ( A + ( C -R A ) ) = C )
9	8	3adant2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + ( C -R A ) ) = C )
10	9	oveq1d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + ( C -R A ) ) -R B ) = ( C -R B ) )
11	7 10	eqtr3d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A -R B ) + ( C -R A ) ) = ( C -R B ) )