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 = ( 0g𝐺 )
ablsimpnosubgd.3 ( 𝜑𝐺 ∈ Abel )
ablsimpnosubgd.4 ( 𝜑𝐺 ∈ SimpGrp )
ablsimpnosubgd.5 ( 𝜑𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
ablsimpnosubgd.6 ( 𝜑𝐴𝑆 )
ablsimpnosubgd.7 ( 𝜑 → ¬ 𝐴 = 0 )
Assertion ablsimpnosubgd ( 𝜑𝑆 = 𝐵 )

Proof

Step Hyp Ref Expression
1 ablsimpnosubgd.1 𝐵 = ( Base ‘ 𝐺 )
2 ablsimpnosubgd.2 0 = ( 0g𝐺 )
3 ablsimpnosubgd.3 ( 𝜑𝐺 ∈ Abel )
4 ablsimpnosubgd.4 ( 𝜑𝐺 ∈ SimpGrp )
5 ablsimpnosubgd.5 ( 𝜑𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
6 ablsimpnosubgd.6 ( 𝜑𝐴𝑆 )
7 ablsimpnosubgd.7 ( 𝜑 → ¬ 𝐴 = 0 )
8 elsni ( 𝐴 ∈ { 0 } → 𝐴 = 0 )
9 7 8 nsyl ( 𝜑 → ¬ 𝐴 ∈ { 0 } )
10 eleq2 ( 𝑆 = { 0 } → ( 𝐴𝑆𝐴 ∈ { 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 = 𝐵𝑆 = 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 ( 𝜑𝑆 ∈ ( NrmSGrp ‘ 𝐺 ) )
19 1 2 4 18 simpgnsgeqd ( 𝜑 → ( 𝑆 = { 0 } ∨ 𝑆 = 𝐵 ) )
20 13 14 19 mpjaod ( 𝜑𝑆 = 𝐵 )