Theorem isnghm3

Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015)

	Ref	Expression
Hypothesis	nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
Assertion	isnghm3	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) < +∞ ) )

Proof

Step	Hyp	Ref	Expression
1		nmofval.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
2	1	isnghm2	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
3	1	nmocl	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝑁 ‘ 𝐹 ) ∈ ℝ* )
4	1	nmoge0	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → 0 ≤ ( 𝑁 ‘ 𝐹 ) )
5		ge0gtmnf	⊢ ( ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ* ∧ 0 ≤ ( 𝑁 ‘ 𝐹 ) ) → -∞ < ( 𝑁 ‘ 𝐹 ) )
6	3 4 5	syl2anc	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → -∞ < ( 𝑁 ‘ 𝐹 ) )
7		xrrebnd	⊢ ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ* → ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ ↔ ( -∞ < ( 𝑁 ‘ 𝐹 ) ∧ ( 𝑁 ‘ 𝐹 ) < +∞ ) ) )
8	7	baibd	⊢ ( ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ* ∧ -∞ < ( 𝑁 ‘ 𝐹 ) ) → ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ ↔ ( 𝑁 ‘ 𝐹 ) < +∞ ) )
9	3 6 8	syl2anc	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( ( 𝑁 ‘ 𝐹 ) ∈ ℝ ↔ ( 𝑁 ‘ 𝐹 ) < +∞ ) )
10	2 9	bitrd	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) < +∞ ) )