Metamath Proof Explorer


Theorem 0subgALT

Description: A shorter proof of 0subg using df-od . (Contributed by SN, 31-Jan-2025) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis 0subgALT.z 0 ˙ = 0 G
Assertion 0subgALT G Grp 0 ˙ SubGrp G

Proof

Step Hyp Ref Expression
1 0subgALT.z 0 ˙ = 0 G
2 eqid od G = od G
3 id G Grp G Grp
4 grpmnd G Grp G Mnd
5 1 0subm G Mnd 0 ˙ SubMnd G
6 4 5 syl G Grp 0 ˙ SubMnd G
7 2 1 od1 G Grp od G 0 ˙ = 1
8 1nn 1
9 7 8 eqeltrdi G Grp od G 0 ˙
10 1 fvexi 0 ˙ V
11 fveq2 a = 0 ˙ od G a = od G 0 ˙
12 11 eleq1d a = 0 ˙ od G a od G 0 ˙
13 10 12 ralsn a 0 ˙ od G a od G 0 ˙
14 9 13 sylibr G Grp a 0 ˙ od G a
15 2 3 6 14 finodsubmsubg G Grp 0 ˙ SubGrp G