Metamath Proof Explorer


Theorem resubdi

Description: Distribution of multiplication over real subtraction. (Contributed by Steven Nguyen, 3-Jun-2023)

Ref Expression
Assertion resubdi A B C A B - C = A B - A C

Proof

Step Hyp Ref Expression
1 remulcl A C A C
2 1 3adant2 A B C A C
3 simp1 A B C A
4 rersubcl B C B - C
5 4 3adant1 A B C B - C
6 3 5 remulcld A B C A B - C
7 3 recnd A B C A
8 simp3 A B C C
9 8 recnd A B C C
10 5 recnd A B C B - C
11 7 9 10 adddid A B C A C + B - C = A C + A B - C
12 repncan3 C B C + B - C = B
13 12 ancoms B C C + B - C = B
14 13 3adant1 A B C C + B - C = B
15 14 oveq2d A B C A C + B - C = A B
16 11 15 eqtr3d A B C A C + A B - C = A B
17 2 6 16 reladdrsub A B C A B - C = A B - A C