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Metamath Proof Explorer

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 ) )