P. 250 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pi1inv 24901*	An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = 𝑌)    &   ⊢ (𝜑 → (𝐹‘1) = 𝑌)    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽))
Theorem	pi1xfrf 24902*	Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    ⇒   ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
Theorem	pi1xfrval 24903*	The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    &   ⊢ (𝜑 → 𝐴 ∈ ∪ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
Theorem	pi1xfr 24904*	Given a path 𝐹 and its inverse 𝐼 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
Theorem	pi1xfrcnvlem 24905*	Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)
Theorem	pi1xfrcnv 24906*	Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → (◡𝐺 = 𝐻 ∧ ◡𝐺 ∈ (𝑄 GrpHom 𝑃)))
Theorem	pi1xfrgim 24907*	The mapping 𝐺 between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpIso 𝑄))
Theorem	pi1cof 24908*	Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑄))
Theorem	pi1coval 24909*	The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑇 ∈ ∪ 𝑉) → (𝐺‘[𝑇]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑇)]( ≃ph‘𝐾))
Theorem	pi1coghm 24910*	The mapping 𝐺 between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
12.5  Metric subcomplex vector spaces
12.5.1  Subcomplex modules
Syntax	cclm 24911	Syntax for the class of subcomplex modules.
class ℂMod
Definition	df-clm 24912*	Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂfld, which allows to use the complex addition, multiplication, etc. in theorems about subcomplex modules. Since the field of complex numbers is commutative and so are its subrings (see subrgcrng 20467), left modules over such subrings are the same as right modules, see rmodislmod 20766. Therefore, we drop the word "left" from "subcomplex left module". (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
Theorem	isclm 24913	A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Theorem	clmsca 24914	The ring of scalars 𝐹 of a subcomplex module is the restriction of the field of complex numbers to the base set of 𝐹. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))
Theorem	clmsubrg 24915	The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))
Theorem	clmlmod 24916	A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Theorem	clmgrp 24917	A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
Theorem	clmabl 24918	A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
Theorem	clmring 24919	The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
Theorem	clmfgrp 24920	The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
Theorem	clm0 24921	The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘𝐹))
Theorem	clm1 24922	The identity of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹))
Theorem	clmadd 24923	The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + = (+g‘𝐹))
Theorem	clmmul 24924	The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹))
Theorem	clmcj 24925	The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹))
Theorem	isclmi 24926	Reverse direction of isclm 24913. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Theorem	clmzss 24927	The scalar ring of a subcomplex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → ℤ ⊆ 𝐾)
Theorem	clmsscn 24928	The scalar ring of a subcomplex module is a subset of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
Theorem	clmsub 24929	Subtraction in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘𝐹)𝐵))
Theorem	clmneg 24930	Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴))
Theorem	clmneg1 24931	Minus one is in the scalar ring of a subcomplex module. (Contributed by AV, 28-Sep-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → -1 ∈ 𝐾)
Theorem	clmabs 24932	Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
Theorem	clmacl 24933	Closure of ring addition for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Theorem	clmmcl 24934	Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
Theorem	clmsubcl 24935	Closure of ring subtraction for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 − 𝑌) ∈ 𝐾)
Theorem	lmhmclm 24936	The domain of a linear operator is a subcomplex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
Theorem	clmvscl 24937	Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20714. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
Theorem	clmvsass 24938	Associative law for scalar product. Analogue of lmodvsass 20723. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Theorem	clmvscom 24939	Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))
Theorem	clmvsdir 24940	Distributive law for scalar product (right-distributivity). (lmodvsdir 20722 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Theorem	clmvsdi 24941	Distributive law for scalar product (left-distributivity). (lmodvsdi 20721 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
Theorem	clmvs1 24942	Scalar product with ring unity. (lmodvs1 20726 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋)
Theorem	clmvs2 24943	A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))
Theorem	clm0vs 24944	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 20731 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (0 · 𝑋) = 0 )
Theorem	clmopfne 24945	The (functionalized) operations of addition and multiplication by a scalar of a subcomplex module cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 3-Oct-2021.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + ≠ · )
Theorem	isclmp 24946*	The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
Theorem	isclmi0 24947*	Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (ℂfld ↾s 𝐾)    &   ⊢ 𝑊 ∈ Grp    &   ⊢ 𝐾 ∈ (SubRing‘ℂfld)    &   ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))    ⇒   ⊢ 𝑊 ∈ ℂMod
Theorem	clmvneg1 24948	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20741 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋))
Theorem	clmvsneg 24949	Multiplication of a vector by a negated scalar. (lmodvsneg 20742 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))
Theorem	clmmulg 24950	The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ∙ = (.g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉) → (𝐴 ∙ 𝐵) = (𝐴 · 𝐵))
Theorem	clmsubdir 24951	Scalar multiplication distributive law for subtraction. (lmodsubdir 20756 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝑋) = ((𝐴 · 𝑋) − (𝐵 · 𝑋)))
Theorem	clmpm1dir 24952	Subtractive distributive law for the scalar product of a subcomplex module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐾 = (Base‘(Scalar‘𝑊))    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))
Theorem	clmnegneg 24953	Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)
Theorem	clmnegsubdi2 24954	Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))
Theorem	clmsub4 24955	Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (Revised by AV, 29-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))
Theorem	clmvsrinv 24956	A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )
Theorem	clmvslinv 24957	Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )
Theorem	clmvsubval 24958	Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 20753. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵)))
Theorem	clmvsubval2 24959	Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴))
Theorem	clmvz 24960	Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴))
Theorem	zlmclm 24961	The ℤ-module operation turns an arbitrary abelian group into a subcomplex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ ℂMod)
Theorem	clmzlmvsca 24962	The scalar product of a subcomplex module matches the scalar product of the derived ℤ-module, which implies, together with zlmbas 21373 and zlmplusg 21375, that any module over ℤ is structure-equivalent to the canonical ℤ-module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋)) → (𝐴( ·𝑠 ‘𝐺)𝐵) = (𝐴( ·𝑠 ‘𝑊)𝐵))
Theorem	nmoleub2lem 24963*	Lemma for nmoleub2a 24966 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝜓 → (𝐿‘𝑥) ≤ 𝑅))    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2lem3 24964*	Lemma for nmoleub2a 24966 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑆))    &   ⊢ (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   ⊢ (𝜑 → ¬ (𝑀‘(𝐹‘𝐵)) ≤ (𝐴 · (𝐿‘𝐵)))    ⇒   ⊢ ¬ 𝜑
Theorem	nmoleub2lem2 24965*	Lemma for nmoleub2a 24966 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    &   ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅))    &   ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅))    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2a 24966*	The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) ≤ 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2b 24967*	The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub3 24968*	The operator norm is the supremum of the value of a linear operator on the unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → ℝ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmhmcn 24969	A linear operator over a normed subcomplex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐾 = (TopOpen‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
Theorem	cmodscexp 24970	The powers of i belong to the scalar subring of a subcomplex module if i belongs to the scalar subring . (Contributed by AV, 18-Oct-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾)
Theorem	cmodscmulexp 24971	The scalar product of a vector with powers of i belongs to the base set of a subcomplex module if the scalar subring of th subcomplex module contains i. (Contributed by AV, 18-Oct-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋)
12.5.2  Subcomplex vector spaces Usually, "complex vector spaces" are vector spaces over the field of the complex numbers, see for example the definition in [Roman] p. 36. In the setting of set.mm, it is convenient to consider collectively vector spaces on subfields of the field of complex numbers. We call these, "subcomplex vector spaces" and collect them in the class ℂVec defined in df-cvs 24973 and characterized in iscvs 24976. These include rational vector spaces (qcvs 24997), real vector spaces (recvs 24995) and complex vector spaces (cncvs 24994). This definition is analogous to the definition of subcomplex modules (and their class ℂMod), which are modules over subrings of the field of complex numbers. Note that ZZ-modules (that are roughly the same thing as Abelian groups, see zlmclm 24961) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 24998), because the ring ZZ is not a division ring (see zringndrg 21323). Since the field of complex numbers is commutative, so are its subrings, so there is no need to explicitly state "left module" or "left vector space" for subcomplex modules or vector spaces.
Syntax	ccvs 24972	Syntax for the class of subcomplex vector spaces.
class ℂVec
Definition	df-cvs 24973	Define the class of subcomplex vector spaces, which are the subcomplex modules which are also vector spaces. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ ℂVec = (ℂMod ∩ LVec)
Theorem	cvslvec 24974	A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LVec)
Theorem	cvsclm 24975	A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ ℂMod)
Theorem	iscvs 24976	A subcomplex vector space is a subcomplex module over a division ring. For example, the subcomplex modules over the rational or real or complex numbers are subcomplex vector spaces. (Contributed by AV, 4-Oct-2021.)
⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
Theorem	iscvsp 24977*	The predicate "is a subcomplex vector space". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
Theorem	iscvsi 24978*	Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑊 ∈ Grp    &   ⊢ 𝑆 = (ℂfld ↾s 𝐾)    &   ⊢ 𝑆 ∈ DivRing    &   ⊢ 𝐾 ∈ (SubRing‘ℂfld)    &   ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))    ⇒   ⊢ 𝑊 ∈ ℂVec
Theorem	cvsi 24979*	The properties of a subcomplex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑋 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑆 = (Base‘(Scalar‘𝑊))    &   ⊢ ∙ = ( ·sf ‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ ∙ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑆 (∀𝑧 ∈ 𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥)))))))
Theorem	cvsunit 24980	Unit group of the scalar ring of a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹))
Theorem	cvsdiv 24981	Division of the scalar ring of a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r‘𝐹)𝐵))
Theorem	cvsdivcl 24982	The scalar field of a subcomplex vector space is closed under division. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂVec ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)
Theorem	cvsmuleqdivd 24983	An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌))    ⇒   ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌))
Theorem	cvsdiveqd 24984	An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂVec)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    &   ⊢ (𝜑 → 𝑋 = ((𝐴 / 𝐵) · 𝑌))    ⇒   ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌)
Theorem	cnlmodlem1 24985	Lemma 1 for cnlmod 24989. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (Base‘𝑊) = ℂ
Theorem	cnlmodlem2 24986	Lemma 2 for cnlmod 24989. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (+g‘𝑊) = +
Theorem	cnlmodlem3 24987	Lemma 3 for cnlmod 24989. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (Scalar‘𝑊) = ℂfld
Theorem	cnlmod4 24988	Lemma 4 for cnlmod 24989. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( ·𝑠 ‘𝑊) = ·
Theorem	cnlmod 24989	The set of complex numbers is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ 𝑊 ∈ LMod
Theorem	cnstrcvs 24990	The set of complex numbers is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ 𝑊 ∈ ℂVec
Theorem	cnrbas 24991	The set of complex numbers is the base set of the complex left module of complex numbers. (Contributed by AV, 21-Sep-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ (Base‘𝐶) = ℂ
Theorem	cnrlmod 24992	The complex left module of complex numbers is a left module. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ 𝐶 ∈ LMod
Theorem	cnrlvec 24993	The complex left module of complex numbers is a left vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 21-Sep-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ 𝐶 ∈ LVec
Theorem	cncvs 24994	The complex left module of complex numbers is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 21-Sep-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ 𝐶 ∈ ℂVec
Theorem	recvs 24995	The field of the real numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.) (Proof shortened by SN, 23-Nov-2024.)
⊢ 𝑅 = (ringLMod‘ℝfld)    ⇒   ⊢ 𝑅 ∈ ℂVec
Theorem	recvsOLD 24996	Obsolete version of recvs 24995 as of 23-Nov-2024. (Contributed by AV, 22-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝑅 = (ringLMod‘ℝfld)    ⇒   ⊢ 𝑅 ∈ ℂVec
Theorem	qcvs 24997	The field of rational numbers as left module over itself is a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
⊢ 𝑄 = (ringLMod‘(ℂfld ↾s ℚ))    ⇒   ⊢ 𝑄 ∈ ℂVec
Theorem	zclmncvs 24998	The ring of integers as left module over itself is a subcomplex module, but not a subcomplex vector space. The vector operation is +, and the scalar product is ·. (Contributed by AV, 22-Oct-2021.)
⊢ 𝑍 = (ringLMod‘ℤring)    ⇒   ⊢ (𝑍 ∈ ℂMod ∧ 𝑍 ∉ ℂVec)
12.5.3  Normed subcomplex vector spaces This section characterizes normed subcomplex vector spaces as subcomplex vector spaces which are also normed vector spaces (that is, normed groups with a positively homogeneous norm). For the moment, there is no need of a special token to represent their class, so we only use the characterization isncvsngp 24999. Most theorems for normed subcomplex vector spaces have a label containing "ncvs". The idiom 𝑊 ∈ (NrmVec ∩ ℂVec) is used in the following to say that 𝑊 is a normed subcomplex vector space, i.e., a subcomplex vector space which is also a normed vector space.
Theorem	isncvsngp 24999*	A normed subcomplex vector space is a subcomplex vector space which is a normed group with a positively homogeneous norm. (Contributed by NM, 5-Jun-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
Theorem	isncvsngpd 25000*	Properties that determine a normed subcomplex vector space. (Contributed by NM, 15-Apr-2007.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂVec)    &   ⊢ (𝜑 → 𝑊 ∈ NrmGrp)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑘 ∈ 𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))    ⇒   ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec))