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Metamath Proof Explorer

Theorem nmof

Description: The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015) (Proof shortened by AV, 26-Sep-2020)

Ref Expression
Hypothesis nmofval.1 ⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
Assertion nmof ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → 𝑁 : ( 𝑆 GrpHom 𝑇 ) ⟶ ℝ* )

Proof

Step Hyp Ref Expression
1 nmofval.1 ⊢ 𝑁 = ( 𝑆 normOp 𝑇 )
2 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
3 eqid ⊢ ( norm ‘ 𝑆 ) = ( norm ‘ 𝑆 )
4 eqid ⊢ ( norm ‘ 𝑇 ) = ( norm ‘ 𝑇 )
5 1 2 3 4 nmofval ⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → 𝑁 = ( 𝑓 ∈ ( 𝑆 GrpHom 𝑇 ) ↦ inf ( { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } , ℝ* , < ) ) )
6 ssrab2 ⊢ { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } ⊆ ( 0 [,) +∞ )
7 icossxr ⊢ ( 0 [,) +∞ ) ⊆ ℝ*
8 6 7 sstri ⊢ { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } ⊆ ℝ*
9 infxrcl ⊢ ( { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } ⊆ ℝ* → inf ( { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } , ℝ* , < ) ∈ ℝ* )
10 8 9 mp1i ⊢ ( ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) ∧ 𝑓 ∈ ( 𝑆 GrpHom 𝑇 ) ) → inf ( { 𝑟 ∈ ( 0 [,) +∞ ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑆 ) ( ( norm ‘ 𝑇 ) ‘ ( 𝑓 ‘ 𝑥 ) ) ≤ ( 𝑟 · ( ( norm ‘ 𝑆 ) ‘ 𝑥 ) ) } , ℝ* , < ) ∈ ℝ* )
11 5 10 fmpt3d ⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → 𝑁 : ( 𝑆 GrpHom 𝑇 ) ⟶ ℝ* )