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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 = ( 0g ‘ 𝐺 )
trivnsgd.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
trivnsgd.4 ⊢ ( 𝜑 → 𝐵 = { 0 } )
Assertion trivnsgd ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { 𝐵 } )

Proof

Step Hyp Ref Expression
1 trivnsgd.1 ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 trivnsgd.2 ⊢ 0 = ( 0g ‘ 𝐺 )
3 trivnsgd.3 ⊢ ( 𝜑 → 𝐺 ∈ Grp )
4 trivnsgd.4 ⊢ ( 𝜑 → 𝐵 = { 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 ⊢ ( 𝜑 → 𝐵 ∈ ( NrmSGrp ‘ 𝐺 ) )
12 11 snssd ⊢ ( 𝜑 → { 𝐵 } ⊆ ( NrmSGrp ‘ 𝐺 ) )
13 9 12 eqssd ⊢ ( 𝜑 → ( NrmSGrp ‘ 𝐺 ) = { 𝐵 } )