Metamath Proof Explorer


Theorem ablsimpnosubgd

Description: A subgroup of an abelian simple group containing a nonidentity element is the whole group. (Contributed by Rohan Ridenour, 3-Aug-2023)

Ref Expression
Hypotheses ablsimpnosubgd.1 B = Base G
ablsimpnosubgd.2 0 ˙ = 0 G
ablsimpnosubgd.3 φ G Abel
ablsimpnosubgd.4 φ G SimpGrp
ablsimpnosubgd.5 φ S SubGrp G
ablsimpnosubgd.6 φ A S
ablsimpnosubgd.7 φ ¬ A = 0 ˙
Assertion ablsimpnosubgd φ S = B

Proof

Step Hyp Ref Expression
1 ablsimpnosubgd.1 B = Base G
2 ablsimpnosubgd.2 0 ˙ = 0 G
3 ablsimpnosubgd.3 φ G Abel
4 ablsimpnosubgd.4 φ G SimpGrp
5 ablsimpnosubgd.5 φ S SubGrp G
6 ablsimpnosubgd.6 φ A S
7 ablsimpnosubgd.7 φ ¬ A = 0 ˙
8 elsni A 0 ˙ A = 0 ˙
9 7 8 nsyl φ ¬ A 0 ˙
10 eleq2 S = 0 ˙ A S A 0 ˙
11 6 10 syl5ibcom φ S = 0 ˙ A 0 ˙
12 9 11 mtod φ ¬ S = 0 ˙
13 12 pm2.21d φ S = 0 ˙ S = B
14 idd φ S = B S = B
15 ablnsg G Abel NrmSGrp G = SubGrp G
16 15 eqcomd G Abel SubGrp G = NrmSGrp G
17 3 16 syl φ SubGrp G = NrmSGrp G
18 5 17 eleqtrd φ S NrmSGrp G
19 1 2 4 18 simpgnsgeqd φ S = 0 ˙ S = B
20 13 14 19 mpjaod φ S = B