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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | clmvsdi 24401 | Distributive law for scalar product (left-distributivity). (lmodvsdi 20292 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌))) | ||
Theorem | clmvs1 24402 | Scalar product with ring unity. (lmodvs1 20297 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋) | ||
Theorem | clmvs2 24403 | A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴)) | ||
Theorem | clm0vs 24404 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 20302 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (0 · 𝑋) = 0 ) | ||
Theorem | clmopfne 24405 | 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.) |
⊢ · = ( ·sf ‘𝑊) & ⊢ + = (+𝑓‘𝑊) ⇒ ⊢ (𝑊 ∈ ℂMod → + ≠ · ) | ||
Theorem | isclmp 24406* | The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))) | ||
Theorem | isclmi0 24407* | Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑆 = (ℂfld ↾s 𝐾) & ⊢ 𝑊 ∈ Grp & ⊢ 𝐾 ∈ (SubRing‘ℂfld) & ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))) ⇒ ⊢ 𝑊 ∈ ℂMod | ||
Theorem | clmvneg1 24408 | Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20312 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (invg‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋)) | ||
Theorem | clmvsneg 24409 | Multiplication of a vector by a negated scalar. (lmodvsneg 20313 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (invg‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂMod) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋)) | ||
Theorem | clmmulg 24410 | The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ∙ = (.g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉) → (𝐴 ∙ 𝐵) = (𝐴 · 𝐵)) | ||
Theorem | clmsubdir 24411 | Scalar multiplication distributive law for subtraction. (lmodsubdir 20327 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ − = (-g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂMod) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝑋) = ((𝐴 · 𝑋) − (𝐵 · 𝑋))) | ||
Theorem | clmpm1dir 24412 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐾 = (Base‘(Scalar‘𝑊)) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶)))) | ||
Theorem | clmnegneg 24413 | Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴) | ||
Theorem | clmnegsubdi2 24414 | Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴))) | ||
Theorem | clmsub4 24415 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷)))) | ||
Theorem | clmvsrinv 24416 | A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 ) | ||
Theorem | clmvslinv 24417 | Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴) + 𝐴) = 0 ) | ||
Theorem | clmvsubval 24418 | Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 20324. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵))) | ||
Theorem | clmvsubval2 24419 | Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴)) | ||
Theorem | clmvz 24420 | Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴)) | ||
Theorem | zlmclm 24421 | The ℤ-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) | ||
Theorem | clmzlmvsca 24422 | The scalar product of a subcomplex module matches the scalar product of the derived ℤ-module, which implies, together with zlmbas 20866 and zlmplusg 20868, that any module over ℤ is structure-equivalent to the canonical ℤ-module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.) |
⊢ 𝑊 = (ℤMod‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋)) → (𝐴( ·𝑠 ‘𝐺)𝐵) = (𝐴( ·𝑠 ‘𝑊)𝐵)) | ||
Theorem | nmoleub2lem 24423* | Lemma for nmoleub2a 24426 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴) & ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦))) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝜓 → (𝐿‘𝑥) ≤ 𝑅)) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
Theorem | nmoleub2lem3 24424* | Lemma for nmoleub2a 24426 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ℚ ⊆ 𝐾) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑆)) & ⊢ (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴))) & ⊢ (𝜑 → ¬ (𝑀‘(𝐹‘𝐵)) ≤ (𝐴 · (𝐿‘𝐵))) ⇒ ⊢ ¬ 𝜑 | ||
Theorem | nmoleub2lem2 24425* | Lemma for nmoleub2a 24426 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ℚ ⊆ 𝐾) & ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅)) & ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅)) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
Theorem | nmoleub2a 24426* | 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ℚ ⊆ 𝐾) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) ≤ 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
Theorem | nmoleub2b 24427* | 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → ℚ ⊆ 𝐾) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
Theorem | nmoleub3 24428* | 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) & ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod)) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) & ⊢ (𝜑 → 𝐴 ∈ ℝ*) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 0 ≤ 𝐴) & ⊢ (𝜑 → ℝ ⊆ 𝐾) ⇒ ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴))) | ||
Theorem | nmhmcn 24429 | 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)))) | ||
Theorem | cmodscexp 24430 | 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) | ||
Theorem | cmodscmulexp 24431 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋) | ||
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 24433 and characterized in iscvs 24436. These include rational vector spaces (qcvs 24457), real vector spaces (recvs 24455) and complex vector spaces (cncvs 24454). 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 24421) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 24458), because the ring ZZ is not a division ring (see zringndrg 20836). 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. | ||
Syntax | ccvs 24432 | Syntax for the class of subcomplex vector spaces. |
class ℂVec | ||
Definition | df-cvs 24433 | 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) | ||
Theorem | cvslvec 24434 | A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
⊢ (𝜑 → 𝑊 ∈ ℂVec) ⇒ ⊢ (𝜑 → 𝑊 ∈ LVec) | ||
Theorem | cvsclm 24435 | A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.) |
⊢ (𝜑 → 𝑊 ∈ ℂVec) ⇒ ⊢ (𝜑 → 𝑊 ∈ ℂMod) | ||
Theorem | iscvs 24436 | 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)) | ||
Theorem | iscvsp 24437* | The predicate "is a subcomplex vector space". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) ⇒ ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))) | ||
Theorem | iscvsi 24438* | Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑊 ∈ Grp & ⊢ 𝑆 = (ℂfld ↾s 𝐾) & ⊢ 𝑆 ∈ DivRing & ⊢ 𝐾 ∈ (SubRing‘ℂfld) & ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥))) & ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))) ⇒ ⊢ 𝑊 ∈ ℂVec | ||
Theorem | cvsi 24439* | 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) & ⊢ + = (+g‘𝑊) & ⊢ 𝑆 = (Base‘(Scalar‘𝑊)) & ⊢ ∙ = ( ·sf ‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ (𝑊 ∈ ℂVec → (𝑊 ∈ Abel ∧ (𝑆 ⊆ ℂ ∧ ∙ :(𝑆 × 𝑋)⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑆 (∀𝑧 ∈ 𝑋 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝑆 (((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥)) ∧ ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥))))))) | ||
Theorem | cvsunit 24440 | 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)) | ||
Theorem | cvsdiv 24441 | 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 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r‘𝐹)𝐵)) | ||
Theorem | cvsdivcl 24442 | 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 ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾) | ||
Theorem | cvsmuleqdivd 24443 | An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌)) ⇒ ⊢ (𝜑 → 𝑋 = ((𝐵 / 𝐴) · 𝑌)) | ||
Theorem | cvsdiveqd 24444 | An equality involving ratios in a subcomplex vector space. (Contributed by Thierry Arnoux, 22-May-2019.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ≠ 0) & ⊢ (𝜑 → 𝐵 ≠ 0) & ⊢ (𝜑 → 𝑋 = ((𝐴 / 𝐵) · 𝑌)) ⇒ ⊢ (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌) | ||
Theorem | cnlmodlem1 24445 | Lemma 1 for cnlmod 24449. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (Base‘𝑊) = ℂ | ||
Theorem | cnlmodlem2 24446 | Lemma 2 for cnlmod 24449. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (+g‘𝑊) = + | ||
Theorem | cnlmodlem3 24447 | Lemma 3 for cnlmod 24449. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ (Scalar‘𝑊) = ℂfld | ||
Theorem | cnlmod4 24448 | Lemma 4 for cnlmod 24449. (Contributed by AV, 20-Sep-2021.) |
⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) ⇒ ⊢ ( ·𝑠 ‘𝑊) = · | ||
Theorem | cnlmod 24449 | 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 | ||
Theorem | cnstrcvs 24450 | 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 | ||
Theorem | cnrbas 24451 | 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 | ||
Theorem | cnrlmod 24452 | 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 | ||
Theorem | cnrlvec 24453 | 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 | ||
Theorem | cncvs 24454 | 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 | ||
Theorem | recvs 24455 | 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.) (Proof shortened by SN, 23-Nov-2024.) |
⊢ 𝑅 = (ringLMod‘ℝfld) ⇒ ⊢ 𝑅 ∈ ℂVec | ||
Theorem | recvsOLD 24456 | Obsolete version of recvs 24455 as of 23-Nov-2024. (Contributed by AV, 22-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑅 = (ringLMod‘ℝfld) ⇒ ⊢ 𝑅 ∈ ℂVec | ||
Theorem | qcvs 24457 | 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‘(ℂfld ↾s ℚ)) ⇒ ⊢ 𝑄 ∈ ℂVec | ||
Theorem | zclmncvs 24458 | 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) | ||
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 24459. 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. | ||
Theorem | isncvsngp 24459* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))) | ||
Theorem | isncvsngpd 24460* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂVec) & ⊢ (𝜑 → 𝑊 ∈ NrmGrp) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑘 ∈ 𝐾)) → (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) ⇒ ⊢ (𝜑 → 𝑊 ∈ (NrmVec ∩ ℂVec)) | ||
Theorem | ncvsi 24461* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))) | ||
Theorem | ncvsprp 24462 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = ((abs‘𝐴) · (𝑁‘𝐵))) | ||
Theorem | ncvsge0 24463 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ (𝐾 ∩ ℝ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 · 𝐵)) = (𝐴 · (𝑁‘𝐵))) | ||
Theorem | ncvsm1 24464 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉) → (𝑁‘(-1 · 𝐴)) = (𝑁‘𝐴)) | ||
Theorem | ncvsdif 24465 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴)))) | ||
Theorem | ncvspi 24466 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴)))) | ||
Theorem | ncvs1 24467 | 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 = (0g‘𝐺) & ⊢ · = ( ·𝑠 ‘𝐺) & ⊢ 𝐹 = (Scalar‘𝐺) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) ∧ (1 / (𝑁‘𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁‘𝐴)) · 𝐴)) = 1) | ||
Theorem | cnrnvc 24468 | 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 | ||
Theorem | cnncvs 24469 | 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 24470) . (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.) |
⊢ 𝐶 = (ringLMod‘ℂfld) ⇒ ⊢ 𝐶 ∈ (NrmVec ∩ ℂVec) | ||
Theorem | cnnm 24470 | 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 | ||
Theorem | ncvspds 24471 | 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) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵)))) | ||
Theorem | cnindmet 24472 | The metric induced on the complex numbers. cnmet 24081 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 20753 and also cnmet 24081. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.) |
⊢ 𝑇 = (ℂfld toNrmGrp abs) ⇒ ⊢ (dist‘𝑇) = (abs ∘ − ) | ||
Theorem | cnncvsaddassdemo 24473 | Derive the associative law for complex number addition addass 11096 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))) | ||
Theorem | cnncvsmulassdemo 24474 | Derive the associative law for complex number multiplication mulass 11097 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))) | ||
Theorem | cnncvsabsnegdemo 24475 | Derive the absolute value of a negative complex number absneg 15116 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)) | ||
Syntax | ccph 24476 | Extend class notation with the class of subcomplex pre-Hilbert spaces. |
class ℂPreHil | ||
Syntax | ctcph 24477 | Function to put a norm on a pre-Hilbert space. |
class toℂPreHil | ||
Definition | df-cph 24478* | Define the class of subcomplex pre-Hilbert spaces. By restricting the scalar field to a subfield of ℂfld closed under square roots of nonnegative reals, we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))} | ||
Definition | df-tcph 24479* | Define a function to augment a pre-Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed subcomplex pre-Hilbert space (see tcphcph 24547). (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))) | ||
Theorem | iscph 24480* | A subcomplex pre-Hilbert space is exactly a pre-Hilbert space over a subfield of the field of complex numbers closed under square roots of nonnegative reals equipped with a norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) | ||
Theorem | cphphl 24481 | A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | ||
Theorem | cphnlm 24482 | A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) | ||
Theorem | cphngp 24483 | A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | ||
Theorem | cphlmod 24484 | A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | ||
Theorem | cphlvec 24485 | A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec) | ||
Theorem | cphnvc 24486 | A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec) | ||
Theorem | cphsubrglem 24487 | Lemma for cphsubrg 24490. (Contributed by Mario Carneiro, 9-Oct-2015.) |
⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴)) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ (𝜑 → (𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld))) | ||
Theorem | cphreccllem 24488 | Lemma for cphreccl 24491. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐴)) & ⊢ (𝜑 → 𝐹 ∈ DivRing) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾 ∧ 𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾) | ||
Theorem | cphsca 24489 | A subcomplex pre-Hilbert space is a vector space over a subfield of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂPreHil → 𝐹 = (ℂfld ↾s 𝐾)) | ||
Theorem | cphsubrg 24490 | The scalar field of a subcomplex pre-Hilbert space is a subring of ℂfld. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld)) | ||
Theorem | cphreccl 24491 | The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾) | ||
Theorem | cphdivcl 24492 | The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾) | ||
Theorem | cphcjcl 24493 | The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (∗‘𝐴) ∈ 𝐾) | ||
Theorem | cphsqrtcl 24494 | The scalar field of a subcomplex pre-Hilbert space is closed under square roots of nonnegative reals. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾) | ||
Theorem | cphabscl 24495 | The scalar field of a subcomplex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾) → (abs‘𝐴) ∈ 𝐾) | ||
Theorem | cphsqrtcl2 24496 | The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾) | ||
Theorem | cphsqrtcl3 24497 | If the scalar field of a subcomplex pre-Hilbert space contains the imaginary unit i, then it is closed under square roots (i.e., it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾 ∧ 𝐴 ∈ 𝐾) → (√‘𝐴) ∈ 𝐾) | ||
Theorem | cphqss 24498 | The scalar field of a subcomplex pre-Hilbert space contains the rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾) | ||
Theorem | cphclm 24499 | A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | ||
Theorem | cphnmvs 24500 | Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((abs‘𝑋) · (𝑁‘𝑌))) |
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