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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclmvs1 23701 Scalar product with ring unit. (lmodvs1 19662 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)

Theoremclmvs2 23702 A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))

Theoremclm0vs 23703 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 19667 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )

Theoremclmopfne 23704 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.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)       (𝑊 ∈ ℂMod → +· )

Theoremisclmp 23705* The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))

Theoremisclmi0 23706* Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (ℂflds 𝐾)    &   𝑊 ∈ Grp    &   𝐾 ∈ (SubRing‘ℂfld)    &   (𝑥𝑉 → (1 · 𝑥) = 𝑥)    &   ((𝑦𝐾𝑥𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ((𝑦𝐾𝑥𝑉𝑧𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))       𝑊 ∈ ℂMod

Theoremclmvneg1 23707 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 19677 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))

Theoremclmvsneg 23708 Multiplication of a vector by a negated scalar. (lmodvsneg 19678 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))

Theoremclmmulg 23709 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (.g𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵𝑉) → (𝐴 𝐵) = (𝐴 · 𝐵))

Theoremclmsubdir 23710 Scalar multiplication distributive law for subtraction. (lmodsubdir 19692 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))

Theoremclmpm1dir 23711 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)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐾 = (Base‘(Scalar‘𝑊))       ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝐵𝐾𝐶𝑉)) → ((𝐴𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))

Theoremclmnegneg 23712 Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)

Theoremclmnegsubdi2 23713 Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))

Theoremclmsub4 23714 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)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))

Theoremclmvsrinv 23715 A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )

Theoremclmvslinv 23716 Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )

Theoremclmvsubval 23717 Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 19689. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = (𝐴 + (-1 · 𝐵)))

Theoremclmvsubval2 23718 Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = ((-1 · 𝐵) + 𝐴))

Theoremclmvz 23719 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴𝑉) → ( 0 𝐴) = (-1 · 𝐴))

Theoremzlmclm 23720 The -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)

Theoremclmzlmvsca 23721 The scalar product of a subcomplex module matches the scalar product of the derived -module, which implies, together with zlmbas 20665 and zlmplusg 20666, that any module over is structure-equivalent to the canonical -module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵𝑋)) → (𝐴( ·𝑠𝐺)𝐵) = (𝐴( ·𝑠𝑊)𝐵))

Theoremnmoleub2lem 23722* Lemma for nmoleub2a 23725 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ((((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦𝑉𝑦 ≠ (0g𝑆))) → (𝑀‘(𝐹𝑦)) ≤ (𝐴 · (𝐿𝑦)))    &   ((𝜑𝑥𝑉) → (𝜓 → (𝐿𝑥) ≤ 𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2lem3 23723* Lemma for nmoleub2a 23725 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵 ≠ (0g𝑆))    &   (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   (𝜑 → ¬ (𝑀‘(𝐹𝐵)) ≤ (𝐴 · (𝐿𝐵)))        ¬ 𝜑

Theoremnmoleub2lem2 23724* Lemma for nmoleub2a 23725 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥)𝑂𝑅 → (𝐿𝑥) ≤ 𝑅))    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥) < 𝑅 → (𝐿𝑥)𝑂𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥)𝑂𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2a 23725* 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) ≤ 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub2b 23726* 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) < 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmoleub3 23727* 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)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → ℝ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) = 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))

Theoremnmhmcn 23728 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))))

Theoremcmodscexp 23729 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)

Theoremcmodscmulexp 23730 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)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾𝐵𝑋𝑁 ∈ 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 23732 and characterized in iscvs 23735. These include rational vector spaces (qcvs 23755), real vector spaces (recvs 23754) and complex vector spaces (cncvs 23753).

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 23720) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 23756), because the ring ZZ is not a division ring (see zringndrg 20637).

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.

Syntaxccvs 23731 Syntax for the class of subcomplex vector spaces.
class ℂVec

Definitiondf-cvs 23732 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)

Theoremcvslvec 23733 A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ LVec)

Theoremcvsclm 23734 A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ ℂMod)

Theoremiscvs 23735 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))

Theoremiscvsp 23736* The predicate "is a subcomplex vector space". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂflds 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))

Theoremiscvsi 23737* Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
· = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑊 ∈ Grp    &   𝑆 = (ℂflds 𝐾)    &   𝑆 ∈ DivRing    &   𝐾 ∈ (SubRing‘ℂfld)    &   (𝑥𝑉 → (1 · 𝑥) = 𝑥)    &   ((𝑦𝐾𝑥𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ((𝑦𝐾𝑥𝑉𝑧𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ((𝑦𝐾𝑧𝐾𝑥𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))       𝑊 ∈ ℂVec

Theoremcvsi 23738* 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)    &    + = (+g𝑊)    &   𝑆 = (Base‘(Scalar‘𝑊))    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦𝑆 (∀𝑧𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥)))))))

Theoremcvsunit 23739 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))

Theoremcvsdiv 23740 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 ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r𝐹)𝐵))

Theoremcvsdivcl 23741 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 ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcvsmuleqdivd 23742 An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌))       (𝜑𝑋 = ((𝐵 / 𝐴) · 𝑌))

Theoremcvsdiveqd 23743 An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑋 = ((𝐴 / 𝐵) · 𝑌))       (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌)

Theoremcnlmodlem1 23744 Lemma 1 for cnlmod 23748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Base‘𝑊) = ℂ

Theoremcnlmodlem2 23745 Lemma 2 for cnlmod 23748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (+g𝑊) = +

Theoremcnlmodlem3 23746 Lemma 3 for cnlmod 23748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Scalar‘𝑊) = ℂfld

Theoremcnlmod4 23747 Lemma 4 for cnlmod 23748. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ·𝑠𝑊) = ·

Theoremcnlmod 23748 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

Theoremcnstrcvs 23749 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

Theoremcnrbas 23750 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

Theoremcnrlmod 23751 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

Theoremcnrlvec 23752 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

Theoremcncvs 23753 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

Theoremrecvs 23754 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.)
𝑅 = (ringLMod‘ℝfld)       𝑅 ∈ ℂVec

Theoremqcvs 23755 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‘(ℂflds ℚ))       𝑄 ∈ ℂVec

Theoremzclmncvs 23756 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 23757. 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.

Theoremisncvsngp 23757* 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥𝑉𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥))))

Theoremisncvsngpd 23758* 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝑊 ∈ NrmGrp)    &   ((𝜑 ∧ (𝑥𝑉𝑘𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))       (𝜑𝑊 ∈ (NrmVec ∩ ℂVec))

Theoremncvsi 23759* 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥𝑉 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑉 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)) ∧ ∀𝑘𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁𝑥)))))

Theoremncvsprp 23760 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝐾𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))

Theoremncvsge0 23761 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁𝐵)))

Theoremncvsm1 23762 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)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁𝐴))

Theoremncvsdif 23763 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)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉𝐵𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴))))

Theoremncvspi 23764 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)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑉𝐵𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))

Theoremncvs1 23765 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 = (0g𝐺)    &    · = ( ·𝑠𝐺)    &   𝐹 = (Scalar‘𝐺)    &   𝐾 = (Base‘𝐹)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑋𝐴0 ) ∧ (1 / (𝑁𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁𝐴)) · 𝐴)) = 1)

Theoremcnrnvc 23766 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

Theoremcnncvs 23767 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 23768) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ (NrmVec ∩ ℂVec)

Theoremcnnm 23768 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

Theoremncvspds 23769 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)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)    &    · = ( ·𝑠𝐺)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵))))

Theoremcnindmet 23770 The metric induced on the complex numbers. cnmet 23380 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 20555 and also cnmet 23380. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
𝑇 = (ℂfld toNrmGrp abs)       (dist‘𝑇) = (abs ∘ − )

Theoremcnncvsaddassdemo 23771 Derive the associative law for complex number addition addass 10622 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)))

Theoremcnncvsmulassdemo 23772 Derive the associative law for complex number multiplication mulass 10623 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)))

Theoremcnncvsabsnegdemo 23773 Derive the absolute value of a negative complex number absneg 14637 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 space

Syntaxccph 23774 Extend class notation with the class of subcomplex pre-Hilbert spaces.
class ℂPreHil

Syntaxctcph 23775 Function to put a norm on a pre-Hilbert space.
class toℂPreHil

Definitiondf-cph 23776* Define the class of subcomplex pre-Hilbert spaces. By restricting the scalar field to a subfield of fld 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 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}

Definitiondf-tcph 23777* 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. (Contributed by Mario Carneiro, 7-Oct-2015.)
toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))

Theoremiscph 23778* 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)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))

Theoremcphphl 23779 A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Theoremcphnlm 23780 A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Theoremcphngp 23781 A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)

Theoremcphlmod 23782 A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)

Theoremcphlvec 23783 A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Theoremcphnvc 23784 A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Theoremcphsubrglem 23785 Lemma for cphsubrg 23788. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       (𝜑 → (𝐹 = (ℂflds 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Theoremcphreccllem 23786 Lemma for cphreccl 23789. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       ((𝜑𝑋𝐾𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)

Theoremcphsca 23787 A subcomplex pre-Hilbert space is a vector space over a subfield of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))

Theoremcphsubrg 23788 The scalar field of a subcomplex pre-Hilbert space is a subring of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))

Theoremcphreccl 23789 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 ∧ 𝐴𝐾𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾)

Theoremcphdivcl 23790 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 ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcphcjcl 23791 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 ∧ 𝐴𝐾) → (∗‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl 23792 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 ∧ (𝐴𝐾𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)

Theoremcphabscl 23793 The scalar field of a subcomplex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (abs‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl2 23794 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl3 23795 If the scalar field of a subcomplex pre-Hilbert space contains the imaginary unit i, then it is closed under square roots (i.e., it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾𝐴𝐾) → (√‘𝐴) ∈ 𝐾)

Theoremcphqss 23796 The scalar field of a subcomplex pre-Hilbert space contains the rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾)

Theoremcphclm 23797 A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)

Theoremcphnmvs 23798 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((abs‘𝑋) · (𝑁𝑌)))

Theoremcphipcl 23799 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ ℂ)

Theoremcphnmfval 23800* The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

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