![]() |
Metamath
Proof Explorer Theorem List (p. 285 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | nvgf 28401 | Mapping for the vector addition operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋) | ||
Theorem | nvsf 28402 | Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) ⇒ ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋) | ||
Theorem | nvgcl 28403 | 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 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) | ||
Theorem | nvcom 28404 | The vector addition (group) operation is commutative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) | ||
Theorem | nvass 28405 | The vector addition (group) operation is associative. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶))) | ||
Theorem | nvadd32 28406 | Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵)) | ||
Theorem | nvrcan 28407 | Right cancellation law for vector addition. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | nvadd4 28408 | Rearrangement of 4 terms in a vector sum. (Contributed by NM, 8-Feb-2008.) (New usage is discouraged.) |
⊢ 𝑋 = (BaseSet‘𝑈) & ⊢ 𝐺 = ( +𝑣 ‘𝑈) ⇒ ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺(𝐶𝐺𝐷)) = ((𝐴𝐺𝐶)𝐺(𝐵𝐺𝐷))) | ||
Theorem | nvscl 28409 | 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 28410 | 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 28411 | 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 28412 | 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 28413 | 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 28414 | 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 28415 | 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 28416 | 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 28417 | 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 28418 | 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 28419 | 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 28420 | 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 28421 | 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 28422 | 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 28423 | 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 28424 | 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 28425 | 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 28426 | 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 28427* | 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 28428 | 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 28429 | 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 28430 | Cancellation law for vector subtraction. (nnncan1 10911 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 28431 | 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 28432 | 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 28433 | 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 28434 | 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 28435 | 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 28436 | 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 28437 | 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 28438 | 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 28439 | 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 28440 | 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 28441 | 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 28442 | 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 28443 | 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 28444 | 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 28445 | 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 28446 | 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 28447 | 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 28448 | 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 28449 | 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 28450 | 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 28451 | 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 28452 | 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 28453 | 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 28454 | 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 28455 | 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 28456 | 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 28457 | 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 28458 | 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 28459 | 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 28460 | 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 28461 | 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 28462 | 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 28463 | 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 28464 | 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 28465 | 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 28466 | 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 28467 | 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 28468 | 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 28469 | 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 28470 | 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 28471 | 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 28472 | 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 28473 | Lemma for imsmet 28474. (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 28474 | 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 28475 | 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 28476 | The metric induced on the complex numbers. cnmet 23377 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 28477 | 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 28478 | 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 28479 | The norm of a normed complex vector space is a continuous function to ℂ. (For ℝ, see nmcvcn 28478.) (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 28480* | Lemma for smcn 28481. (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 28481 | 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 28482 | 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 28483 | Extend class notation with the class inner product functions. |
class ·𝑖OLD | ||
Definition | df-dip 28484* | 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 28485* | 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 28486* | 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 28487 | Lemma for ipval3 28492. (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 28488 | Lemma for ipval3 28492. (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 28489 | Lemma for ipval3 28492. (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 28490 | Expansion of the inner product value ipval 28486. (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 28491 | Four times the inner product value ipval3 28492, 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 28492 | Expansion of the inner product value ipval 28486. (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 28493 | 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 28494 | 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 28495 | 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 28496 | 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 28497 | 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 28498 | 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 28499 | 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 28500 | 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) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |