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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclmgrp 23601 A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
 
Theoremclmabl 23602 A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
 
Theoremclmring 23603 The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
 
Theoremclmfgrp 23604 The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
 
Theoremclm0 23605 The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 0 = (0g𝐹))
 
Theoremclm1 23606 The identity of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 1 = (1r𝐹))
 
Theoremclmadd 23607 The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → + = (+g𝐹))
 
Theoremclmmul 23608 The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → · = (.r𝐹))
 
Theoremclmcj 23609 The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → ∗ = (*𝑟𝐹))
 
Theoremisclmi 23610 Reverse direction of isclm 23597. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝐹 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
 
Theoremclmzss 23611 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)
 
Theoremclmsscn 23612 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)
 
Theoremclmsub 23613 Subtraction in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾𝐵𝐾) → (𝐴𝐵) = (𝐴(-g𝐹)𝐵))
 
Theoremclmneg 23614 Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → -𝐴 = ((invg𝐹)‘𝐴))
 
Theoremclmneg1 23615 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)
 
Theoremclmabs 23616 Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
 
Theoremclmacl 23617 Closure of ring addition for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremclmmcl 23618 Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremclmsubcl 23619 Closure of ring subtraction for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋𝑌) ∈ 𝐾)
 
Theoremlmhmclm 23620 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))
 
Theoremclmvscl 23621 Closure of scalar product for a subcomplex module. Analogue of lmodvscl 19582. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑄𝐾𝑋𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
 
Theoremclmvsass 23622 Associative law for scalar product. Analogue of lmodvsass 19590. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
Theoremclmvscom 23623 Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))
 
Theoremclmvsdir 23624 Distributive law for scalar product (right-distributivity). (lmodvsdir 19589 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
 
Theoremclmvsdi 23625 Distributive law for scalar product (left-distributivity). (lmodvsdi 19588 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝐴𝐾𝑋𝑉𝑌𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
 
Theoremclmvs1 23626 Scalar product with ring unit. (lmodvs1 19593 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)
 
Theoremclmvs2 23627 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 23628 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 19598 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )
 
Theoremclmopfne 23629 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 23630* 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 23631* 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 23632 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 19608 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))
 
Theoremclmvsneg 23633 Multiplication of a vector by a negated scalar. (lmodvsneg 19609 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 23634 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 23635 Scalar multiplication distributive law for subtraction. (lmodsubdir 19623 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 23636 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 23637 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 23638 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 23639 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 23640 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 23641 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 23642 Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 19620. (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 23643 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 23644 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 23645 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 23646 The scalar product of a subcomplex module matches the scalar product of the derived -module, which implies, together with zlmbas 20595 and zlmplusg 20596, 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 23647* Lemma for nmoleub2a 23650 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 23648* Lemma for nmoleub2a 23650 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 23649* Lemma for nmoleub2a 23650 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 23650* 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 23651* 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 23652* 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 23653 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 23654 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 23655 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 23657 and characterized in iscvs 23660. These include rational vector spaces (qcvs 23680), real vector spaces (recvs 23679) and complex vector spaces (cncvs 23678).

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 23645) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 23681), because the ring ZZ is not a division ring (see zringndrg 20567).

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 23656 Syntax for the class of subcomplex vector spaces.
class ℂVec
 
Definitiondf-cvs 23657 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 23658 A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ LVec)
 
Theoremcvsclm 23659 A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ ℂMod)
 
Theoremiscvs 23660 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 23661* 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 23662* 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 23663* 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 23664 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 23665 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 23666 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 23667 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 23668 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 23669 Lemma 1 for cnlmod 23673. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Base‘𝑊) = ℂ
 
Theoremcnlmodlem2 23670 Lemma 2 for cnlmod 23673. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (+g𝑊) = +
 
Theoremcnlmodlem3 23671 Lemma 3 for cnlmod 23673. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       (Scalar‘𝑊) = ℂfld
 
Theoremcnlmod4 23672 Lemma 4 for cnlmod 23673. (Contributed by AV, 20-Sep-2021.)
𝑊 = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), + ⟩} ∪ {⟨(Scalar‘ndx), ℂfld⟩, ⟨( ·𝑠 ‘ndx), · ⟩})       ( ·𝑠𝑊) = ·
 
Theoremcnlmod 23673 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 23674 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 23675 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 23676 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 23677 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 23678 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 23679 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 23680 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 23681 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 23682. 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 23682* 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 23683* 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 23684* 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 23685 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 23686 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 23687 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 23688 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 23689 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 23690 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 23691 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 23692 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 23693) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ (NrmVec ∩ ℂVec)
 
Theoremcnnm 23693 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 23694 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 23695 The metric induced on the complex numbers. cnmet 23309 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 20485 and also cnmet 23309. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
𝑇 = (ℂfld toNrmGrp abs)       (dist‘𝑇) = (abs ∘ − )
 
Theoremcnncvsaddassdemo 23696 Derive the associative law for complex number addition addass 10613 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 23697 Derive the associative law for complex number multiplication mulass 10614 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 23698 Derive the absolute value of a negative complex number absneg 14627 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 23699 Extend class notation with the class of subcomplex pre-Hilbert spaces.
class ℂPreHil
 
Syntaxctcph 23700 Function to put a norm on a pre-Hilbert space.
class toℂPreHil
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