Metamath Proof Explorer


Theorem 0wlkonlem1

Description: Lemma 1 for 0wlkon and 0trlon . (Contributed by AV, 3-Jan-2021) (Revised by AV, 23-Mar-2021)

Ref Expression
Hypothesis 0wlk.v V = Vtx G
Assertion 0wlkonlem1 P : 0 0 V P 0 = N N V N V

Proof

Step Hyp Ref Expression
1 0wlk.v V = Vtx G
2 id P : 0 0 V P : 0 0 V
3 0nn0 0 0
4 0elfz 0 0 0 0 0
5 3 4 mp1i P : 0 0 V 0 0 0
6 2 5 ffvelrnd P : 0 0 V P 0 V
7 6 adantr P : 0 0 V P 0 = N P 0 V
8 eleq1 N = P 0 N V P 0 V
9 8 eqcoms P 0 = N N V P 0 V
10 9 adantl P : 0 0 V P 0 = N N V P 0 V
11 7 10 mpbird P : 0 0 V P 0 = N N V
12 id N V N V
13 11 12 jccir P : 0 0 V P 0 = N N V N V