Metamath Proof Explorer


Theorem sn-00idlem1

Description: Lemma for sn-00id . (Contributed by SN, 25-Dec-2023)

Ref Expression
Assertion sn-00idlem1 A A 0 - 0 = A - A

Proof

Step Hyp Ref Expression
1 1re 1
2 resubdi A 1 1 A 1 - 1 = A 1 - A 1
3 1 1 2 mp3an23 A A 1 - 1 = A 1 - A 1
4 re1m1e0m0 1 - 1 = 0 - 0
5 4 oveq2i A 1 - 1 = A 0 - 0
6 5 a1i A A 1 - 1 = A 0 - 0
7 ax-1rid A A 1 = A
8 7 7 oveq12d A A 1 - A 1 = A - A
9 3 6 8 3eqtr3d A A 0 - 0 = A - A