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Metamath Proof Explorer

Theorem nmrpcl

Description: The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015)

Ref Expression
Hypotheses nmf.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
nmf.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
nmeq0.z ⊢ 0 = ( 0g ‘ 𝐺 )
Assertion nmrpcl ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 nmf.x ⊢ 𝑋 = ( Base ‘ 𝐺 )
2 nmf.n ⊢ 𝑁 = ( norm ‘ 𝐺 )
3 nmeq0.z ⊢ 0 = ( 0g ‘ 𝐺 )
4 1 2 nmcl ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )
5 4 3adant3 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ )
6 1 2 nmge0 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → 0 ≤ ( 𝑁 ‘ 𝐴 ) )
7 6 3adant3 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → 0 ≤ ( 𝑁 ‘ 𝐴 ) )
8 1 2 3 nmne0 ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → ( 𝑁 ‘ 𝐴 ) ≠ 0 )
9 5 7 8 ne0gt0d ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → 0 < ( 𝑁 ‘ 𝐴 ) )
10 5 9 elrpd ⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → ( 𝑁 ‘ 𝐴 ) ∈ ℝ+ )