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* )