Theorem isnmhm2

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

	Ref	Expression
Hypothesis	isnmhm2.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
Assertion	isnmhm2	⊢ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )

Proof

Step	Hyp	Ref	Expression
1		isnmhm2.1	⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
2		isnmhm	⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) )
3	2	baib	⊢ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) )
4	3	baibd	⊢ ( ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) )
5		lmghm	⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )
6		nlmngp	⊢ ( 𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp )
7		nlmngp	⊢ ( 𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp )
8	1	isnghm	⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) ) )
9	8	baib	⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) ) )
10	6 7 9	syl2an	⊢ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) ) )
11	10	baibd	⊢ ( ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
12	5 11	sylan2	⊢ ( ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
13	4 12	bitrd	⊢ ( ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )
14	13	3impa	⊢ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝑁 ‘ 𝐹 ) ∈ ℝ ) )