P. 251 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 25001-25100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clmmcl 25001	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 25002	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 25003	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 25004	Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20799. (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 25005	Associative law for scalar product. Analogue of lmodvsass 20808. (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 25006	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 25007	Distributive law for scalar product (right-distributivity). (lmodvsdir 20807 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 25008	Distributive law for scalar product (left-distributivity). (lmodvsdi 20806 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 25009	Scalar product with ring unity. (lmodvs1 20811 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋)
Theorem	clmvs2 25010	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 25011	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 20816 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 25012	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 25013*	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 25014*	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 25015	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20826 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 25016	Multiplication of a vector by a negated scalar. (lmodvsneg 20827 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 25017	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 25018	Scalar multiplication distributive law for subtraction. (lmodsubdir 20841 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 25019	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 25020	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 25021	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 25022	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 25023	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 25024	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 25025	Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 20838. (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 25026	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 25027	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 25028	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 25029	The scalar product of a subcomplex module matches the scalar product of the derived ℤ-module, which implies, together with zlmbas 21442 and zlmplusg 21443, 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 25030*	Lemma for nmoleub2a 25033 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 25031*	Lemma for nmoleub2a 25033 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 25032*	Lemma for nmoleub2a 25033 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 25033*	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 25034*	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 25035*	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 25036	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 25037	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 25038	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 25040 and characterized in iscvs 25043. These include rational vector spaces (qcvs 25063), real vector spaces (recvs 25062) and complex vector spaces (cncvs 25061). 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 25028) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 25064), because the ring ZZ is not a division ring (see zringndrg 21393). 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 25039	Syntax for the class of subcomplex vector spaces.
class ℂVec
Definition	df-cvs 25040	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 25041	A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LVec)
Theorem	cvsclm 25042	A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ ℂMod)
Theorem	iscvs 25043	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 25044*	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 25045*	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 25046*	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 25047	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 25048	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 25049	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 25050	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 25051	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 25052	Lemma 1 for cnlmod 25056. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (Base‘𝑊) = ℂ
Theorem	cnlmodlem2 25053	Lemma 2 for cnlmod 25056. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (+g‘𝑊) = +
Theorem	cnlmodlem3 25054	Lemma 3 for cnlmod 25056. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ (Scalar‘𝑊) = ℂfld
Theorem	cnlmod4 25055	Lemma 4 for cnlmod 25056. (Contributed by AV, 20-Sep-2021.)
⊢ 𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})    ⇒   ⊢ ( ·𝑠 ‘𝑊) = ·
Theorem	cnlmod 25056	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 25057	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 25058	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 25059	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 25060	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 25061	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 25062	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	qcvs 25063	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 25064	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 25065. 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 25065*	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 25066*	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))
Theorem	ncvsi 25067*	The properties of a normed subcomplex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))
Theorem	ncvsprp 25068	Proportionality property of the norm of a scalar product in a normed subcomplex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵)))
Theorem	ncvsge0 25069	The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁‘𝐵)))
Theorem	ncvsm1 25070	The norm of the opposite of a vector. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴))
Theorem	ncvsdif 25071	The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴))))
Theorem	ncvspi 25072	The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))
Theorem	ncvs1 25073	From any nonzero vector of a normed subcomplex vector space, construct a collinear vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝑁 = (norm‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ · = ( ·𝑠 ‘𝐺)    &   ⊢ 𝐹 = (Scalar‘𝐺)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) ∧ (1 / (𝑁‘𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁‘𝐴)) · 𝐴)) = 1)
Theorem	cnrnvc 25074	The module of complex numbers (as a module over itself) is a normed vector space over itself. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by AV, 9-Oct-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ 𝐶 ∈ NrmVec
Theorem	cnncvs 25075	The module of complex numbers (as a module over itself) is a normed subcomplex vector space. The vector operation is +, the scalar product is ·, and the norm is abs (see cnnm 25076) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ 𝐶 ∈ (NrmVec ∩ ℂVec)
Theorem	cnnm 25076	The norm of the normed subcomplex vector space of complex numbers is the absolute value. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.)
⊢ 𝐶 = (ringLMod‘ℂfld)    ⇒   ⊢ (norm‘𝐶) = abs
Theorem	ncvspds 25077	Value of the distance function in terms of the norm of a normed subcomplex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.)
⊢ 𝑁 = (norm‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐷 = (dist‘𝐺)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵))))
Theorem	cnindmet 25078	The metric induced on the complex numbers. cnmet 24675 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 21291 and also cnmet 24675. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
⊢ 𝑇 = (ℂfld toNrmGrp abs)    ⇒   ⊢ (dist‘𝑇) = (abs ∘ − )
Theorem	cnncvsaddassdemo 25079	Derive the associative law for complex number addition addass 11115 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
Theorem	cnncvsmulassdemo 25080	Derive the associative law for complex number multiplication mulass 11116 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Theorem	cnncvsabsnegdemo 25081	Derive the absolute value of a negative complex number absneg 15202 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴))
12.5.4  Subcomplex pre-Hilbert spaces
Syntax	ccph 25082	Extend class notation with the class of subcomplex pre-Hilbert spaces.
class ℂPreHil
Syntax	ctcph 25083	Function to put a norm on a pre-Hilbert space.
class toℂPreHil
Definition	df-cph 25084*	Define the class of subcomplex pre-Hilbert spaces. By restricting the scalar field to a subfield of ℂfld closed under square roots of nonnegative reals, we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))}
Definition	df-tcph 25085*	Define a function to augment a pre-Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed subcomplex pre-Hilbert space (see tcphcph 25153). (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))))
Theorem	iscph 25086*	A subcomplex pre-Hilbert space is exactly a pre-Hilbert space over a subfield of the field of complex numbers closed under square roots of nonnegative reals equipped with a norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
Theorem	cphphl 25087	A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
Theorem	cphnlm 25088	A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
Theorem	cphngp 25089	A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
Theorem	cphlmod 25090	A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
Theorem	cphlvec 25091	A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
Theorem	cphnvc 25092	A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
Theorem	cphsubrglem 25093	Lemma for cphsubrg 25096. (Contributed by Mario Carneiro, 9-Oct-2015.)
⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴))    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Theorem	cphreccllem 25094	Lemma for cphreccl 25097. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴))    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)
Theorem	cphsca 25095	A subcomplex pre-Hilbert space is a vector space over a subfield of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾))
Theorem	cphsubrg 25096	The scalar field of a subcomplex pre-Hilbert space is a subring of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))
Theorem	cphreccl 25097	The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾)
Theorem	cphdivcl 25098	The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)
Theorem	cphcjcl 25099	The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (∗‘𝐴) ∈ 𝐾)
Theorem	cphsqrtcl 25100	The scalar field of a subcomplex pre-Hilbert space is closed under square roots of nonnegative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)