HomeTheorem List (p. 308 of 501)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | nvscl 30701 | 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 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝐴𝑆𝐵) ∈ 𝑋) | ||
| Theorem | nvsid 30702 | 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 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) | ||
| Theorem | nvsass 30703 | 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 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶))) | ||
| Theorem | nvscom 30704 | 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 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝑆𝐶)) = (𝐵𝑆(𝐴𝑆𝐶))) | ||
| Theorem | nvdi 30705 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝐺𝐶)) = ((𝐴𝑆𝐵)𝐺(𝐴𝑆𝐶))) | ||
| Theorem | nvdir 30706 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐵)𝑆𝐶) = ((𝐴𝑆𝐶)𝐺(𝐵𝑆𝐶))) | ||
| Theorem | nv2 30707 | A vector plus itself is two times the vector. (Contributed by NM, 9-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝐴) = (2𝑆𝐴)) | ||
| Theorem | vsfval 30708 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ 𝑀 = ( /𝑔 ‘𝐺) | ||
| Theorem | nvzcl 30709 | 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 = (0vec‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) | ||
| Theorem | nv0rid 30710 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴) | ||
| Theorem | nv0lid 30711 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝐺𝐴) = 𝐴) | ||
| Theorem | nv0 30712 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) | ||
| Theorem | nvsz 30713 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍) | ||
| Theorem | nvinv 30714 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑀 = (inv‘𝐺) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆𝐴) = (𝑀‘𝐴)) | ||
| Theorem | nvinvfval 30715 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V))) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺)) | ||
| Theorem | nvm 30716 | Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = ( /𝑔 ‘𝐺) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵)) | ||
| Theorem | nvmval 30717 | Value of vector subtraction on a normed complex vector space. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝐺(-1𝑆𝐵))) | ||
| Theorem | nvmval2 30718 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = ((-1𝑆𝐵)𝐺𝐴)) | ||
| Theorem | nvmfval 30719* | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥𝐺(-1𝑆𝑦)))) | ||
| Theorem | nvmf 30720 | 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 = ( −𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑀:(𝑋 × 𝑋)⟶𝑋) | ||
| Theorem | nvmcl 30721 | 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 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) ∈ 𝑋) | ||
| Theorem | nvnnncan1 30722 | Cancellation law for vector subtraction. (nnncan1 11417 analog.) (Contributed by NM, 7-Mar-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝑀𝐵)𝑀(𝐴𝑀𝐶)) = (𝐶𝑀𝐵)) | ||
| Theorem | nvmdi 30723 | Distributive law for scalar product over subtraction. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑆(𝐵𝑀𝐶)) = ((𝐴𝑆𝐵)𝑀(𝐴𝑆𝐶))) | ||
| Theorem | nvnegneg 30724 | Double negative of a vector. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (-1𝑆(-1𝑆𝐴)) = 𝐴) | ||
| Theorem | nvmul0or 30725 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑆𝐵) = 𝑍 ↔ (𝐴 = 0 ∨ 𝐵 = 𝑍))) | ||
| Theorem | nvrinv 30726 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(-1𝑆𝐴)) = 𝑍) | ||
| Theorem | nvlinv 30727 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((-1𝑆𝐴)𝐺𝐴) = 𝑍) | ||
| Theorem | nvpncan2 30728 | Cancellation law for vector subtraction. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐴) = 𝐵) | ||
| Theorem | nvpncan 30729 | Cancellation law for vector subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐺𝐵)𝑀𝐵) = 𝐴) | ||
| Theorem | nvaddsub 30730 | Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵)) | ||
| Theorem | nvnpcan 30731 | Cancellation law for a normed complex vector space. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵)𝐺𝐵) = 𝐴) | ||
| Theorem | nvaddsub4 30732 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀(𝐶𝐺𝐷)) = ((𝐴𝑀𝐶)𝐺(𝐵𝑀𝐷))) | ||
| Theorem | nvmeq0 30733 | 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 = ( −𝑣 ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑀𝐵) = 𝑍 ↔ 𝐴 = 𝐵)) | ||
| Theorem | nvmid 30734 | A vector minus itself is the zero vector. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑀𝐴) = 𝑍) | ||
| Theorem | nvf 30735 | Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁:𝑋⟶ℝ) | ||
| Theorem | nvcl 30736 | 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 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) | ||
| Theorem | nvcli 30737 | 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 = (normCV‘𝑈) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝐴 ∈ 𝑋 ⇒ ⊢ (𝑁‘𝐴) ∈ ℝ | ||
| Theorem | nvs 30738 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) | ||
| Theorem | nvsge0 30739 | The norm of a scalar product with a nonnegative real. (Contributed by NM, 1-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑆𝐵)) = (𝐴 · (𝑁‘𝐵))) | ||
| Theorem | nvm1 30740 | The norm of the negative of a vector. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘(-1𝑆𝐴)) = (𝑁‘𝐴)) | ||
| Theorem | nvdif 30741 | The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(-1𝑆𝐵))) = (𝑁‘(𝐵𝐺(-1𝑆𝐴)))) | ||
| Theorem | nvpi 30742 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺(i𝑆𝐵))) = (𝑁‘(𝐵𝐺(-i𝑆𝐴)))) | ||
| Theorem | nvz0 30743 | The norm of a zero vector is zero. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.) |
| ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑁‘𝑍) = 0) | ||
| Theorem | nvz 30744 | 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 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 𝑍)) | ||
| Theorem | nvtri 30745 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝐺𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
| Theorem | nvmtri 30746 | Triangle inequality for the norm of a vector difference. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴𝑀𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
| Theorem | nvabs 30747 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘((𝑁‘𝐴) − (𝑁‘𝐵))) ≤ (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) | ||
| Theorem | nvge0 30748 | 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.) (Proof shortened by AV, 10-Jul-2022.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) | ||
| Theorem | nvgt0 30749 | A nonzero norm is positive. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 𝑍 ↔ 0 < (𝑁‘𝐴))) | ||
| Theorem | nv1 30750 | 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 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍) → (𝑁‘((1 / (𝑁‘𝐴))𝑆𝐴)) = 1) | ||
| Theorem | nvop 30751 | A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩) | ||
| Theorem | cnnv 30752 | 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 | ||
| Theorem | cnnvg 30753 | 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⟩ ⇒ ⊢ + = ( +𝑣 ‘𝑈) | ||
| Theorem | cnnvba 30754 | 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) | ||
| Theorem | cnnvs 30755 | 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⟩ ⇒ ⊢ · = ( ·𝑠OLD ‘𝑈) | ||
| Theorem | cnnvnm 30756 | 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 = (normCV‘𝑈) | ||
| Theorem | cnnvm 30757 | 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⟩ ⇒ ⊢ − = ( −𝑣 ‘𝑈) | ||
| Theorem | elimnv 30758 | 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 = (0vec‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 | ||
| Theorem | elimnvu 30759 | 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 | ||
| Theorem | imsval 30760 | 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 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐷 = (𝑁 ∘ 𝑀)) | ||
| Theorem | imsdval 30761 | 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 = ( −𝑣 ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝑀𝐵))) | ||
| Theorem | imsdval2 30762 | 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 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴𝐺(-1𝑆𝐵)))) | ||
| Theorem | nvnd 30763 | 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 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐷𝑍)) | ||
| Theorem | imsdf 30764 | 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) | ||
| Theorem | imsmetlem 30765 | Lemma for imsmet 30766. (Contributed by NM, 29-Nov-2006.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑀 = (inv‘𝐺) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐷 = (IndMet‘𝑈) & ⊢ 𝑈 ∈ NrmCVec ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
| Theorem | imsmet 30766 | 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)) | ||
| Theorem | imsxmet 30767 | 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)) | ||
| Theorem | cnims 30768 | The metric induced on the complex numbers. cnmet 24715 proves that it is a metric. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by NM, 15-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩ & ⊢ 𝐷 = (abs ∘ − ) ⇒ ⊢ 𝐷 = (IndMet‘𝑈) | ||
| Theorem | vacn 30769 | Vector addition is jointly continuous in both arguments. (Contributed by Jeff Hankins, 16-Jun-2009.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | ||
| Theorem | nmcvcn 30770 | The norm of a normed complex vector space is a continuous function. (Contributed by NM, 16-May-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2014.) (New usage is discouraged.) |
| ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (topGen‘ran (,)) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | nmcnc 30771 | The norm of a normed complex vector space is a continuous function to ℂ. (For ℝ, see nmcvcn 30770.) (Contributed by NM, 12-Aug-2007.) (New usage is discouraged.) |
| ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑁 ∈ (𝐽 Cn 𝐾)) | ||
| Theorem | smcnlem 30772* | Lemma for smcn 30773. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐾 = (TopOpen‘ℂfld) & ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑈 ∈ NrmCVec & ⊢ 𝑇 = (1 / (1 + ((((𝑁‘𝑦) + (abs‘𝑥)) + 1) / 𝑟))) ⇒ ⊢ 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽) | ||
| Theorem | smcn 30773 | Scalar multiplication is jointly continuous in both arguments. (Contributed by NM, 16-Jun-2009.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑆 ∈ ((𝐾 ×t 𝐽) Cn 𝐽)) | ||
| Theorem | vmcn 30774 | Vector subtraction is jointly continuous in both arguments. (Contributed by Mario Carneiro, 6-May-2014.) (New usage is discouraged.) |
| ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑀 ∈ ((𝐽 ×t 𝐽) Cn 𝐽)) | ||
| Syntax | cdip 30775 | Extend class notation with the class inner product functions. |
| class ·𝑖OLD | ||
| Definition | df-dip 30776* | Define a function that maps a normed complex vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st ‘𝑤), the scalar product is (2nd ‘𝑤), and the norm is 𝑛. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.) |
| ⊢ ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4))) | ||
| Theorem | dipfval 30777* | The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))) | ||
| Theorem | ipval 30778* | Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, the norm is 𝑁, and the set of vectors is 𝑋. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4)) | ||
| Theorem | ipval2lem2 30779 | Lemma for ipval3 30784. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℝ) | ||
| Theorem | ipval2lem3 30780 | Lemma for ipval3 30784. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘(𝐴𝐺𝐵))↑2) ∈ ℝ) | ||
| Theorem | ipval2lem4 30781 | Lemma for ipval3 30784. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ ℂ) → ((𝑁‘(𝐴𝐺(𝐶𝑆𝐵)))↑2) ∈ ℂ) | ||
| Theorem | ipval2 30782 | Expansion of the inner product value ipval 30778. (Contributed by NM, 31-Jan-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2)))) / 4)) | ||
| Theorem | 4ipval2 30783 | Four times the inner product value ipval3 30784, useful for simplifying certain proofs. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴𝑃𝐵)) = ((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝐺(-i𝑆𝐵)))↑2))))) | ||
| Theorem | ipval3 30784 | Expansion of the inner product value ipval 30778. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝑀 = ( −𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (((((𝑁‘(𝐴𝐺𝐵))↑2) − ((𝑁‘(𝐴𝑀𝐵))↑2)) + (i · (((𝑁‘(𝐴𝐺(i𝑆𝐵)))↑2) − ((𝑁‘(𝐴𝑀(i𝑆𝐵)))↑2)))) / 4)) | ||
| Theorem | ipidsq 30785 | The inner product of a vector with itself is the square of the vector's norm. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2)) | ||
| Theorem | ipnm 30786 | Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (√‘(𝐴𝑃𝐴))) | ||
| Theorem | dipcl 30787 | An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) ∈ ℂ) | ||
| Theorem | ipf 30788 | Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃:(𝑋 × 𝑋)⟶ℂ) | ||
| Theorem | dipcj 30789 | The complex conjugate of an inner product reverses its arguments. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∗‘(𝐴𝑃𝐵)) = (𝐵𝑃𝐴)) | ||
| Theorem | ipipcj 30790 | An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) · (𝐵𝑃𝐴)) = ((abs‘(𝐴𝑃𝐵))↑2)) | ||
| Theorem | diporthcom 30791 | Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0)) | ||
| Theorem | dip0r 30792 | Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝑍) = 0) | ||
| Theorem | dip0l 30793 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝑍𝑃𝐴) = 0) | ||
| Theorem | ipz 30794 | The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑍 = (0vec‘𝑈) & ⊢ 𝑃 = (·𝑖OLD‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → ((𝐴𝑃𝐴) = 0 ↔ 𝐴 = 𝑍)) | ||
| Theorem | dipcn 30795 | Inner product is jointly continuous in both arguments. (Contributed by NM, 21-Aug-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.) |
| ⊢ 𝑃 = (·𝑖OLD‘𝑈) & ⊢ 𝐶 = (IndMet‘𝑈) & ⊢ 𝐽 = (MetOpen‘𝐶) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑃 ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) | ||
| Syntax | css 30796 | Extend class notation with the class of all subspaces of normed complex vector spaces. |
| class SubSp | ||
| Definition | df-ssp 30797* | Define the class of all subspaces of normed complex vector spaces. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ SubSp = (𝑢 ∈ NrmCVec ↦ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ ( +𝑣 ‘𝑢) ∧ ( ·𝑠OLD ‘𝑤) ⊆ ( ·𝑠OLD ‘𝑢) ∧ (normCV‘𝑤) ⊆ (normCV‘𝑢))}) | ||
| Theorem | sspval 30798* | The set of all subspaces of a normed complex vector space. (Contributed by NM, 26-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}) | ||
| Theorem | isssp 30799 | The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
| ⊢ 𝐺 = ( +𝑣 ‘𝑈) & ⊢ 𝐹 = ( +𝑣 ‘𝑊) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) & ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) & ⊢ 𝑁 = (normCV‘𝑈) & ⊢ 𝑀 = (normCV‘𝑊) & ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) | ||
| Theorem | sspid 30800 | A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
| ⊢ 𝐻 = (SubSp‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) | ||
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