Metamath Proof Explorer


Theorem trivnsgd

Description: The only normal subgroup of a trivial group is itself. (Contributed by Rohan Ridenour, 3-Aug-2023)

Ref Expression
Hypotheses trivnsgd.1 𝐵 = ( Base ‘ 𝐺 )
trivnsgd.2 0 = ( 0g𝐺 )
trivnsgd.3 ( 𝜑𝐺 ∈ Grp )
trivnsgd.4 ( 𝜑𝐵 = { 0 } )
Assertion trivnsgd ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { 𝐵 } )

Proof

Step Hyp Ref Expression
1 trivnsgd.1 𝐵 = ( Base ‘ 𝐺 )
2 trivnsgd.2 0 = ( 0g𝐺 )
3 trivnsgd.3 ( 𝜑𝐺 ∈ Grp )
4 trivnsgd.4 ( 𝜑𝐵 = { 0 } )
5 nsgsubg ( 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) )
6 5 a1i ( 𝜑 → ( 𝑥 ∈ ( NrmSGrp ‘ 𝐺 ) → 𝑥 ∈ ( SubGrp ‘ 𝐺 ) ) )
7 6 ssrdv ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ⊆ ( SubGrp ‘ 𝐺 ) )
8 1 2 3 4 trivsubgsnd ( 𝜑 → ( SubGrp ‘ 𝐺 ) = { 𝐵 } )
9 7 8 sseqtrd ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) ⊆ { 𝐵 } )
10 1 nsgid ( 𝐺 ∈ Grp → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) )
11 3 10 syl ( 𝜑𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) )
12 11 snssd ( 𝜑 → { 𝐵 } ⊆ ( NrmSGrp ‘ 𝐺 ) )
13 9 12 eqssd ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { 𝐵 } )