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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem0vfval 26601 Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       (𝑈𝑉𝑍 = (GId‘𝐺))

Theoremnmcvfval 26602 Value of the norm function in a normed complex vector space. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
𝑁 = (normCV𝑈)       𝑁 = (2nd𝑈)

Theoremnvop2 26603 A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑊 = (1st𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Theoremnvvop 26604 The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑊 = (1st𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑊 = ⟨𝐺, 𝑆⟩)

Theoremisnvlem 26605* Lemma for isnv 26607. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremnvex 26606 The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)
(⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V))

Theoremisnv 26607* The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ↔ (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))

Theoremisnvi 26608* Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝐺, 𝑆⟩ ∈ CVecOLD    &   𝑁:𝑋⟶ℝ    &   ((𝑥𝑋 ∧ (𝑁𝑥) = 0) → 𝑥 = 𝑍)    &   ((𝑦 ∈ ℂ ∧ 𝑥𝑋) → (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)))    &   ((𝑥𝑋𝑦𝑋) → (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))    &   𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁       𝑈 ∈ NrmCVec

Theoremnvi 26609* The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (⟨𝐺, 𝑆⟩ ∈ CVecOLD𝑁:𝑋⟶ℝ ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 → 𝑥 = 𝑍) ∧ ∀𝑦 ∈ ℂ (𝑁‘(𝑦𝑆𝑥)) = ((abs‘𝑦) · (𝑁𝑥)) ∧ ∀𝑦𝑋 (𝑁‘(𝑥𝐺𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))

Theoremnvvc 26610 The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑊 = (1st𝑈)       (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD)

Theoremnvablo 26611 The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Theoremnvgrp 26612 The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)

Theoremnvgf 26613 Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋)

Theoremnvsf 26614 Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

Theoremnvgcl 26615 Closure law for the vector addition (group) operation of a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)

Theoremnvcom 26616 The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))

Theoremnvass 26617 The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Theoremnvadd12 26618 Commutative/associative law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐺(𝐵𝐺𝐶)) = (𝐵𝐺(𝐴𝐺𝐶)))

Theoremnvadd32 26619 Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Theoremnvrcan 26620 Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Theoremnvlcan 26621 Left cancellation law for vector addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Theoremnvadd4 26622 Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷)))

Theoremnvscl 26623 Closure law for the scalar product operation of a normed complex vector space. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝐴𝑆𝐵) ∈ 𝑋)

Theoremnvsid 26624 Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Theoremnvsass 26625 Associative law for the scalar product of a normed complex vector space. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Theoremnvscom 26626 Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → (𝐴𝑆(𝐵𝑆𝐶)) = (𝐵𝑆(𝐴𝑆𝐶)))

Theoremnvdi 26627 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶)))

Theoremnvdir 26628 Distributive law for the scalar product of a complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶)))

Theoremnv2 26629 A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴))

Theoremvsfval 26630 Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       𝑀 = ( /𝑔𝐺)

Theoremnvzcl 26631 Closure law for the zero vector of a normed complex vector space. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)       (𝑈 ∈ NrmCVec → 𝑍𝑋)

Theoremnv0rid 26632 The zero vector is a right identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺𝑍) = 𝐴)

Theoremnv0lid 26633 The zero vector is a left identity element. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝐺𝐴) = 𝐴)

Theoremnv0 26634 Zero times a vector is the zero vector. (Contributed by NM, 27-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (0𝑆𝐴) = 𝑍)

Theoremnvsz 26635 Anything times the zero vector is the zero vector. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Theoremnvinv 26636 Minus 1 times a vector is the underlying group's inverse element. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = (inv‘𝐺)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (-1𝑆𝐴) = (𝑀𝐴))

Theoremnvinvfval 26637 Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (𝑆(2nd ↾ ({-1} × V)))       (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Theoremnvm 26638 Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = ( /𝑔𝐺)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))

Theoremnvmval 26639 Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵)))

Theoremnvmval2 26640 Value of vector subtraction on a normed complex vector space. (Contributed by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴))

Theoremnvmfval 26641* Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑀 = ( −𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝑀 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥𝐺(-1𝑆𝑦))))

Theoremnvzs 26642 Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑍𝑀𝐴) = (-1𝑆𝐴))

Theoremnvmf 26643 Mapping for the vector subtraction operation. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋)

Theoremnvmcl 26644 Closure law for the vector subtraction operation of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀𝐵) ∈ 𝑋)

Theoremnvnnncan1 26645 Cancellation law for vector subtraction. (nnncan1 10068 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵))

Theoremnvnnncan2 26646 Cancellation law for vector subtraction. (nnncan2 10069 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐶)𝑀(𝐵𝑀𝐶)) = (𝐴𝑀𝐵))

Theoremnvmdi 26647 Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑋𝐶𝑋)) → (𝐴𝑆(𝐵𝑀𝐶)) = ((𝐴𝑆𝐵)𝑀(𝐴𝑆𝐶)))

Theoremnvnegneg 26648 Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴)

Theoremnvmul0or 26649 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → ((𝐴𝑆𝐵) = 𝑍 ↔ (𝐴 = 0 ∨ 𝐵 = 𝑍)))

Theoremnvrinv 26650 A vector minus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍)

Theoremnvlinv 26651 Minus a vector plus itself. (Contributed by NM, 4-Dec-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍)

Theoremnvsubadd 26652 Relationship between vector subtraction and addition. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵) = 𝐶 ↔ (𝐵𝐺𝐶) = 𝐴))

Theoremnvpncan2 26653 Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝑀𝐴) = 𝐵)

Theoremnvpncan 26654 Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴)

Theoremnvaddsubass 26655 Associative-type law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = (𝐴𝐺(𝐵𝑀𝐶)))

Theoremnvaddsub 26656 Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))

Theoremnvnpcan 26657 Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑀𝐵)𝐺𝐵) = 𝐴)

Theoremnvaddsub4 26658 Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷)))

Theoremnvsubsub23 26659 Swap subtrahend and result of vector subtraction. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝑀𝐵) = 𝐶 ↔ (𝐴𝑀𝐶) = 𝐵))

Theoremnvnncan 26660 Cancellation law for a normed complex vector space. (Contributed by NM, 17-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝑀(𝐴𝑀𝐵)) = 𝐵)

Theoremnvmeq0 26661 The difference between two vectors is zero iff they are equal. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝑀𝐵) = 𝑍𝐴 = 𝐵))

Theoremnvmid 26662 A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑍 = (0vec𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝑀𝐴) = 𝑍)

Theoremnvf 26663 Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ)

Theoremnvcl 26664 The norm of a normed complex vector space is a real number. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) ∈ ℝ)

Theoremnvcli 26665 The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)    &   𝑈 ∈ NrmCVec    &   𝐴𝑋       (𝑁𝐴) ∈ ℝ

Theoremnvdm 26666 Two ways to express the set of vectors in a normed complex vector space. (Contributed by NM, 31-Jan-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (𝑋 = dom 𝑁𝑋 = ran 𝐺))

Theoremnvs 26667 Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁𝐵)))

Theoremnvsge0 26668 The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵𝑋) → (𝑁‘(𝐴𝑆𝐵)) = (𝐴 · (𝑁𝐵)))

Theoremnvm1 26669 The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁𝐴))

Theoremnvdif 26670 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺(-1𝑆𝐵))) = (𝑁‘(𝐵𝐺(-1𝑆𝐴))))

Theoremnvpi 26671 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺(i𝑆𝐵))) = (𝑁‘(𝐵𝐺(-i𝑆𝐴))))

Theoremnvsub 26672 The norm of the difference between two vectors. (Contributed by NM, 14-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝑀𝐵)) = (𝑁‘(𝐵𝑀𝐴)))

Theoremnvz0 26673 The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → (𝑁𝑍) = 0)

Theoremnvz 26674 The norm of a vector is zero iff the vector is zero. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → ((𝑁𝐴) = 0 ↔ 𝐴 = 𝑍))

Theoremnvtri 26675 Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))

Theoremnvmtri 26676 Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))

Theoremnvmtri2 26677 Triangle inequality for the norm of a vector difference. (Contributed by NM, 24-Feb-2008.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝑁‘(𝐴𝑀𝐶)) ≤ ((𝑁‘(𝐴𝑀𝐵)) + (𝑁‘(𝐵𝑀𝐶))))

Theoremnvabs 26678 Norm difference property of a normed complex vector space. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (abs‘((𝑁𝐴) − (𝑁𝐵))) ≤ (𝑁‘(𝐴𝐺(-1𝑆𝐵))))

Theoremnvge0 26679 The norm of a normed complex vector space is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → 0 ≤ (𝑁𝐴))

Theoremnvgt0 26680 A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝐴𝑍 ↔ 0 < (𝑁𝐴)))

Theoremnv1 26681 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐴𝑍) → (𝑁‘((1 / (𝑁𝐴))𝑆𝐴)) = 1)

Theoremnvop 26682 A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)       (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Theoremnvoprne 26683 The vector addition and scalar product operations are not identical. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
(⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec → 𝐺𝑆)

18.3.2  Examples of normed complex vector spaces

Theoremcnnv 26684 The set of complex numbers is a normed complex vector space. The vector operation is +, the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       𝑈 ∈ NrmCVec

Theoremcnnvg 26685 The vector addition (group) operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        + = ( +𝑣𝑈)

Theoremcnnvba 26686 The base set of the normed complex vector space of complex numbers. (Contributed by NM, 7-Nov-2007.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       ℂ = (BaseSet‘𝑈)

Theoremcnnvdemo 26687 Derive the associative law for complex number addition addass 9778 to demonstrate the use of cnnv 26684, cnnvg 26685, and cnnvba 26686. (Contributed by NM, 12-Jan-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremcnnvs 26688 The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        · = ( ·𝑠OLD𝑈)

Theoremcnnvnm 26689 The norm operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩       abs = (normCV𝑈)

Theoremcnnvm 26690 The vector subtraction operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
𝑈 = ⟨⟨ + , · ⟩, abs⟩        − = ( −𝑣𝑈)

Theoremelimnv 26691 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑈 ∈ NrmCVec       if(𝐴𝑋, 𝐴, 𝑍) ∈ 𝑋

Theoremelimnvu 26692 Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.)
if(𝑈 ∈ NrmCVec, 𝑈, ⟨⟨ + , · ⟩, abs⟩) ∈ NrmCVec

18.3.3  Induced metric of a normed complex vector space

Theoremimsval 26693 Value of the induced metric of a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 = (𝑁𝑀))

Theoremimsdval 26694 Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑀 = ( −𝑣𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵)))

Theoremimsdval2 26695 Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))

Theoremnvnd 26696 The norm of a normed complex vector space expressed in terms of the distance function of its induced metric. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)       ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐷𝑍))

Theoremimsdf 26697 Mapping for the induced metric distance function of a normed complex vector space. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷:(𝑋 × 𝑋)⟶ℝ)

Theoremimsmetlem 26698 Lemma for imsmet 26699. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐺 = ( +𝑣𝑈)    &   𝑀 = (inv‘𝐺)    &   𝑆 = ( ·𝑠OLD𝑈)    &   𝑍 = (0vec𝑈)    &   𝑁 = (normCV𝑈)    &   𝐷 = (IndMet‘𝑈)    &   𝑈 ∈ NrmCVec       𝐷 ∈ (Met‘𝑋)

Theoremimsmet 26699 The induced metric of a normed complex vector space is a metric space. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋))

Theoremimsxmet 26700 The induced metric of a normed complex vector space is an extended metric space. (Contributed by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)
𝑋 = (BaseSet‘𝑈)    &   𝐷 = (IndMet‘𝑈)       (𝑈 ∈ NrmCVec → 𝐷 ∈ (∞Met‘𝑋))

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