Metamath Proof Explorer


Theorem sn-00idlem2

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

Ref Expression
Assertion sn-00idlem2 0 - 0 0 0 - 0 = 1

Proof

Step Hyp Ref Expression
1 0re 0
2 rennncan2 0 0 0 0 - 0 - 0 - 0 = 0 - 0
3 1 1 1 2 mp3an 0 - 0 - 0 - 0 = 0 - 0
4 re1m1e0m0 1 - 1 = 0 - 0
5 3 4 eqtr4i 0 - 0 - 0 - 0 = 1 - 1
6 rernegcl 0 0 - 0
7 1 6 ax-mp 0 - 0
8 sn-00idlem1 0 - 0 0 - 0 0 - 0 = 0 - 0 - 0 - 0
9 7 8 ax-mp 0 - 0 0 - 0 = 0 - 0 - 0 - 0
10 1re 1
11 sn-00idlem1 1 1 0 - 0 = 1 - 1
12 10 11 ax-mp 1 0 - 0 = 1 - 1
13 5 9 12 3eqtr4i 0 - 0 0 - 0 = 1 0 - 0
14 7 a1i 0 - 0 0 0 - 0
15 1red 0 - 0 0 1
16 id 0 - 0 0 0 - 0 0
17 14 15 14 16 remulcan2d 0 - 0 0 0 - 0 0 - 0 = 1 0 - 0 0 - 0 = 1
18 13 17 mpbii 0 - 0 0 0 - 0 = 1