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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | slmdvsass 29101 | Associative law for scalar product. (ax-hvmulass 27248 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ × = (.r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋))) | ||
Theorem | slmd0cl 29102 | The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾) | ||
Theorem | slmd1cl 29103 | The ring unit in a semiring left module belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ (𝑊 ∈ SLMod → 1 ∈ 𝐾) | ||
Theorem | slmdvs1 29104 | Scalar product with ring unit. (ax-hvmulid 27247 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 1 = (1r‘𝐹) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋) | ||
Theorem | slmd0vcl 29105 | The zero vector is a vector. (ax-hv0cl 27244 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝑉) | ||
Theorem | slmd0vlid 29106 | Left identity law for the zero vector. (hvaddid2 27264 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋) | ||
Theorem | slmd0vrid 29107 | Right identity law for the zero vector. (ax-hvaddid 27245 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋) | ||
Theorem | slmd0vs 29108 | Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27251 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 ) | ||
Theorem | slmdvs0 29109 | Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 27265 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 ) | ||
Theorem | gsumle 29110 | A finite sum in an ordered monoid is monotonic. This proof would be much easier in an ordered group, where an inverse element would be available. (Contributed by Thierry Arnoux, 13-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ ≤ = (le‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ oMnd) & ⊢ (𝜑 → 𝑀 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐹 ∘𝑟 ≤ 𝐺) ⇒ ⊢ (𝜑 → (𝑀 Σg 𝐹) ≤ (𝑀 Σg 𝐺)) | ||
Theorem | gsummpt2co 29111* | Split a finite sum into a sum of a collection of sums over disjoint subsets. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐸 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ 𝐸) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐷) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ 𝐸 ↦ (𝑊 Σg (𝑥 ∈ (◡𝐹 “ {𝑦}) ↦ 𝐶))))) | ||
Theorem | gsummpt2d 29112* | Express a finite sum over a two-dimensional range as a double sum. See also gsum2d 18194. (Contributed by Thierry Arnoux, 27-Apr-2020.) |
⊢ Ⅎ𝑧𝐶 & ⊢ Ⅎ𝑦𝜑 & ⊢ 𝐵 = (Base‘𝑊) & ⊢ (𝑥 = 〈𝑦, 𝑧〉 → 𝐶 = 𝐷) & ⊢ (𝜑 → Rel 𝐴) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ CMnd) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑊 Σg (𝑦 ∈ dom 𝐴 ↦ (𝑊 Σg (𝑧 ∈ (𝐴 “ {𝑦}) ↦ 𝐷))))) | ||
Theorem | gsumvsca1 29113* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑃 ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))) | ||
Theorem | gsumvsca2 29114* | Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐺 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝑊 ∈ SLMod) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄)) | ||
Theorem | gsummptres 29115* | Extend a finite group sum by padding outside with zeroes. Proof generated using OpenAI's proof assistant. (Contributed by Thierry Arnoux, 11-Jul-2020.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ CMnd) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ 𝐷)) → 𝐶 = 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑥 ∈ (𝐴 ∩ 𝐷) ↦ 𝐶))) | ||
Theorem | xrge0tsmsd 29116* | Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.) |
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) & ⊢ (𝜑 → 𝑆 = sup(ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))), ℝ*, < )) ⇒ ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝑆}) | ||
Theorem | xrge0tsmsbi 29117 | Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.) |
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) | ||
Theorem | xrge0tsmseq 29118 | Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) & ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) ⇒ ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹)) | ||
Theorem | rngurd 29119* | Deduce the unit of a ring from its properties. (Contributed by Thierry Arnoux, 6-Sep-2016.) |
⊢ (𝜑 → 𝐵 = (Base‘𝑅)) & ⊢ (𝜑 → · = (.r‘𝑅)) & ⊢ (𝜑 → 1 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) = 𝑥) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 · 1 ) = 𝑥) ⇒ ⊢ (𝜑 → 1 = (1r‘𝑅)) | ||
Theorem | ress1r 29120 | 1r is unaffected by restriction. This is a bit more generic than subrg1 18613. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑆 = (𝑅 ↾s 𝐴) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 1 ∈ 𝐴 ∧ 𝐴 ⊆ 𝐵) → 1 = (1r‘𝑆)) | ||
Theorem | dvrdir 29121 | Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ / = (/r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 + 𝑌) / 𝑍) = ((𝑋 / 𝑍) + (𝑌 / 𝑍))) | ||
Theorem | rdivmuldivd 29122 | Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑊 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑋 / 𝑌) · (𝑍 / 𝑊)) = ((𝑋 · 𝑍) / (𝑌 · 𝑊))) | ||
Theorem | ringinvval 29123* | The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ∗ = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝑁 = (invr‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) | ||
Theorem | dvrcan5 29124 | Cancellation law for common factor in ratio. (divcan5 10606 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) & ⊢ / = (/r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑍) / (𝑌 · 𝑍)) = (𝑋 / 𝑌)) | ||
Theorem | subrgchr 29125 | If 𝐴 is a subring of 𝑅, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.) |
⊢ (𝐴 ∈ (SubRing‘𝑅) → (chr‘(𝑅 ↾s 𝐴)) = (chr‘𝑅)) | ||
Syntax | corng 29126 | Extend class notation with the class of all ordered rings. |
class oRing | ||
Syntax | cofld 29127 | Extend class notation with the class of all ordered fields. |
class oField | ||
Definition | df-orng 29128* | Define class of all ordered rings. An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ oRing = {𝑟 ∈ (Ring ∩ oGrp) ∣ [(Base‘𝑟) / 𝑣][(0g‘𝑟) / 𝑧][(.r‘𝑟) / 𝑡][(le‘𝑟) / 𝑙]∀𝑎 ∈ 𝑣 ∀𝑏 ∈ 𝑣 ((𝑧𝑙𝑎 ∧ 𝑧𝑙𝑏) → 𝑧𝑙(𝑎𝑡𝑏))} | ||
Definition | df-ofld 29129 | Define class of all ordered fields. An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.) |
⊢ oField = (Field ∩ oRing) | ||
Theorem | isorng 29130* | An ordered ring is a ring with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 18-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ ≤ = (le‘𝑅) ⇒ ⊢ (𝑅 ∈ oRing ↔ (𝑅 ∈ Ring ∧ 𝑅 ∈ oGrp ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (( 0 ≤ 𝑎 ∧ 0 ≤ 𝑏) → 0 ≤ (𝑎 · 𝑏)))) | ||
Theorem | orngring 29131 | An ordered ring is a ring. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | ||
Theorem | orngogrp 29132 | An ordered ring is an ordered group. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | ||
Theorem | isofld 29133 | An ordered field is a field with a total ordering compatible with its operations. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ (𝐹 ∈ oField ↔ (𝐹 ∈ Field ∧ 𝐹 ∈ oRing)) | ||
Theorem | orngmul 29134 | In an ordered ring, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ≤ = (le‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ oRing ∧ (𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) ∧ (𝑌 ∈ 𝐵 ∧ 0 ≤ 𝑌)) → 0 ≤ (𝑋 · 𝑌)) | ||
Theorem | orngsqr 29135 | In an ordered ring, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ ≤ = (le‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ oRing ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝑋 · 𝑋)) | ||
Theorem | ornglmulle 29136 | In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ ≤ = (le‘𝑅) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 0 ≤ 𝑍) ⇒ ⊢ (𝜑 → (𝑍 · 𝑋) ≤ (𝑍 · 𝑌)) | ||
Theorem | orngrmulle 29137 | In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ ≤ = (le‘𝑅) & ⊢ (𝜑 → 𝑋 ≤ 𝑌) & ⊢ (𝜑 → 0 ≤ 𝑍) ⇒ ⊢ (𝜑 → (𝑋 · 𝑍) ≤ (𝑌 · 𝑍)) | ||
Theorem | ornglmullt 29138 | In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → 0 < 𝑍) ⇒ ⊢ (𝜑 → (𝑍 · 𝑋) < (𝑍 · 𝑌)) | ||
Theorem | orngrmullt 29139 | In an ordered ring, multiplication with a positive does not change strict comparison. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 < 𝑌) & ⊢ (𝜑 → 0 < 𝑍) ⇒ ⊢ (𝜑 → (𝑋 · 𝑍) < (𝑌 · 𝑍)) | ||
Theorem | orngmullt 29140 | In an ordered ring, the strict ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ < = (lt‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ oRing) & ⊢ (𝜑 → 𝑅 ∈ DivRing) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 0 < 𝑋) & ⊢ (𝜑 → 0 < 𝑌) ⇒ ⊢ (𝜑 → 0 < (𝑋 · 𝑌)) | ||
Theorem | ofldfld 29141 | An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
⊢ (𝐹 ∈ oField → 𝐹 ∈ Field) | ||
Theorem | ofldtos 29142 | An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.) |
⊢ (𝐹 ∈ oField → 𝐹 ∈ Toset) | ||
Theorem | orng0le1 29143 | In an ordered ring, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ 0 = (0g‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ ≤ = (le‘𝐹) ⇒ ⊢ (𝐹 ∈ oRing → 0 ≤ 1 ) | ||
Theorem | ofldlt1 29144 | In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ 0 = (0g‘𝐹) & ⊢ 1 = (1r‘𝐹) & ⊢ < = (lt‘𝐹) ⇒ ⊢ (𝐹 ∈ oField → 0 < 1 ) | ||
Theorem | ofldchr 29145 | The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.) (Proof shortened by AV, 6-Oct-2020.) |
⊢ (𝐹 ∈ oField → (chr‘𝐹) = 0) | ||
Theorem | suborng 29146 | Every subring of an ordered ring is also an ordered ring. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ((𝑅 ∈ oRing ∧ (𝑅 ↾s 𝐴) ∈ Ring) → (𝑅 ↾s 𝐴) ∈ oRing) | ||
Theorem | subofld 29147 | Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ((𝐹 ∈ oField ∧ (𝐹 ↾s 𝐴) ∈ Field) → (𝐹 ↾s 𝐴) ∈ oField) | ||
Theorem | isarchiofld 29148* | Axiom of Archimedes : a characterization of the Archimedean property for ordered fields. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐻 = (ℤRHom‘𝑊) & ⊢ < = (lt‘𝑊) ⇒ ⊢ (𝑊 ∈ oField → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∃𝑛 ∈ ℕ 𝑥 < (𝐻‘𝑛))) | ||
Theorem | rhmdvdsr 29149 | A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ 𝑋 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ / = (∥r‘𝑆) ⇒ ⊢ (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐴 ∥ 𝐵) → (𝐹‘𝐴) / (𝐹‘𝐵)) | ||
Theorem | rhmopp 29150 | A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.) |
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆))) | ||
Theorem | elrhmunit 29151 | Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆)) | ||
Theorem | rhmdvd 29152 | A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.) |
⊢ 𝑈 = (Unit‘𝑆) & ⊢ 𝑋 = (Base‘𝑅) & ⊢ / = (/r‘𝑆) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ∧ ((𝐹‘𝐵) ∈ 𝑈 ∧ (𝐹‘𝐶) ∈ 𝑈)) → ((𝐹‘𝐴) / (𝐹‘𝐵)) = ((𝐹‘(𝐴 · 𝐶)) / (𝐹‘(𝐵 · 𝐶)))) | ||
Theorem | rhmunitinv 29153 | Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴))) | ||
Theorem | kerunit 29154 | If a unit element lies in the kernel of a ring homomorphism, then 0 = 1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.) |
⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑆) & ⊢ 1 = (1r‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ (𝑈 ∩ (◡𝐹 “ { 0 })) ≠ ∅) → 1 = 0 ) | ||
Syntax | cresv 29155 | Extend class notation with the scalar restriction operation. |
class ↾v | ||
Definition | df-resv 29156* | Define an operator to restrict the scalar field component of an extended structure. (Contributed by Thierry Arnoux, 5-Sep-2018.) |
⊢ ↾v = (𝑤 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑤)) ⊆ 𝑥, 𝑤, (𝑤 sSet 〈(Scalar‘ndx), ((Scalar‘𝑤) ↾s 𝑥)〉))) | ||
Theorem | reldmresv 29157 | The scalar restriction is a proper operator, so it can be used with ovprc1 6582. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ Rel dom ↾v | ||
Theorem | resvval 29158 | Value of structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = if(𝐵 ⊆ 𝐴, 𝑊, (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉))) | ||
Theorem | resvid2 29159 | General behavior of trivial restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = 𝑊) | ||
Theorem | resvval2 29160 | Value of nontrivial structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ 𝑋 ∧ 𝐴 ∈ 𝑌) → 𝑅 = (𝑊 sSet 〈(Scalar‘ndx), (𝐹 ↾s 𝐴)〉)) | ||
Theorem | resvsca 29161 | Base set of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹 ↾s 𝐴) = (Scalar‘𝑅)) | ||
Theorem | resvlem 29162 | Other elements of a structure restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝑅 = (𝑊 ↾v 𝐴) & ⊢ 𝐶 = (𝐸‘𝑊) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 ≠ 5 ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐶 = (𝐸‘𝑅)) | ||
Theorem | resvbas 29163 | Base is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐵 = (Base‘𝐻)) | ||
Theorem | resvplusg 29164 | +g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → + = (+g‘𝐻)) | ||
Theorem | resvvsca 29165 | ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = ( ·𝑠 ‘𝐻)) | ||
Theorem | resvmulr 29166 | ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ · = (.r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻)) | ||
Theorem | resv0g 29167 | 0g is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 0 = (0g‘𝐻)) | ||
Theorem | resv1r 29168 | 1r is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) & ⊢ 1 = (1r‘𝐺) ⇒ ⊢ (𝐴 ∈ 𝑉 → 1 = (1r‘𝐻)) | ||
Theorem | resvcmn 29169 | Scalar restriction preserves commutative monoids. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐻 = (𝐺 ↾v 𝐴) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐺 ∈ CMnd ↔ 𝐻 ∈ CMnd)) | ||
Theorem | gzcrng 29170 | The gaussian integers form a commutative ring. (Contributed by Thierry Arnoux, 18-Mar-2018.) |
⊢ (ℂfld ↾s ℤ[i]) ∈ CRing | ||
Theorem | reofld 29171 | The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.) |
⊢ ℝfld ∈ oField | ||
Theorem | nn0omnd 29172 | The nonnegative integers form an ordered monoid. (Contributed by Thierry Arnoux, 23-Mar-2018.) |
⊢ (ℂfld ↾s ℕ0) ∈ oMnd | ||
Theorem | rearchi 29173 | The field of the real numbers is Archimedean. See also arch 11166. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
⊢ ℝfld ∈ Archi | ||
Theorem | nn0archi 29174 | The monoid of the nonnegative integers is Archimedean. (Contributed by Thierry Arnoux, 16-Sep-2018.) |
⊢ (ℂfld ↾s ℕ0) ∈ Archi | ||
Theorem | xrge0slmod 29175 | The extended nonnegative real numbers form a semiring left module. One could also have used subringAlg to get the same structure. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) & ⊢ 𝑊 = (𝐺 ↾v (0[,)+∞)) ⇒ ⊢ 𝑊 ∈ SLMod | ||
Theorem | symgfcoeu 29176* | Uniqueness property of permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ∈ 𝐺 ∧ 𝑄 ∈ 𝐺) → ∃!𝑝 ∈ 𝐺 𝑄 = (𝑃 ∘ 𝑝)) | ||
Theorem | psgndmfi 29177 | For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝑆 = (pmSgn‘𝐷) & ⊢ 𝐺 = (Base‘(SymGrp‘𝐷)) ⇒ ⊢ (𝐷 ∈ Fin → 𝑆 Fn 𝐺) | ||
Theorem | psgnid 29178 | Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) | ||
Theorem | pmtrprfv2 29179 | In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ ((𝐷 ∈ 𝑉 ∧ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷 ∧ 𝑋 ≠ 𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋) | ||
Theorem | pmtrto1cl 29180 | Useful lemma for the following theorems. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑇 = (pmTrsp‘𝐷) ⇒ ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑇‘{𝐾, (𝐾 + 1)}) ∈ ran 𝑇) | ||
Theorem | psgnfzto1stlem 29181* | Lemma for psgnfzto1st 29186. Our permutation of rank (𝑛 + 1) can be written as a permutation of rank 𝑛 composed with a transposition. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) ⇒ ⊢ ((𝐾 ∈ ℕ ∧ (𝐾 + 1) ∈ 𝐷) → (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, (𝐾 + 1), if(𝑖 ≤ (𝐾 + 1), (𝑖 − 1), 𝑖))) = (((pmTrsp‘𝐷)‘{𝐾, (𝐾 + 1)}) ∘ (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐾, if(𝑖 ≤ 𝐾, (𝑖 − 1), 𝑖))))) | ||
Theorem | fzto1stfv1 29182* | Value of our permutation 𝑃 at 1. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑃‘1) = 𝐼) | ||
Theorem | fzto1st1 29183* | Special case where the permutation defined in psgnfzto1st 29186 is the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) ⇒ ⊢ (𝐼 = 1 → 𝑃 = ( I ↾ 𝐷)) | ||
Theorem | fzto1st 29184* | The function moving one element to the first position (and shifting all elements before it) is a permutation. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → 𝑃 ∈ 𝐵) | ||
Theorem | fzto1stinvn 29185* | Value of the inverse of our permutation 𝑃 at 𝐼 (Contributed by Thierry Arnoux, 23-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) ⇒ ⊢ (𝐼 ∈ 𝐷 → (◡𝑃‘𝐼) = 1) | ||
Theorem | psgnfzto1st 29186* | The permutation sign for moving one element to the first position. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
⊢ 𝐷 = (1...𝑁) & ⊢ 𝑃 = (𝑖 ∈ 𝐷 ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝐺 = (SymGrp‘𝐷) & ⊢ 𝐵 = (Base‘𝐺) & ⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐼 ∈ 𝐷 → (𝑆‘𝑃) = (-1↑(𝐼 + 1))) | ||
Syntax | csmat 29187 | Syntax for a function generating submatrixes. |
class subMat1 | ||
Definition | df-smat 29188* | Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 20202. (Contributed by Thierry Arnoux, 18-Aug-2020.) |
⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ 〈if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))〉)))) | ||
Theorem | smatfval 29189* | Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ 〈if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))〉))) | ||
Theorem | smatrcl 29190 | Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) ⇒ ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑𝑚 ((1...(𝑀 − 1)) × (1...(𝑁 − 1))))) | ||
Theorem | smatlem 29191 | Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) & ⊢ (𝜑 → 𝐼 ∈ ℕ) & ⊢ (𝜑 → 𝐽 ∈ ℕ) & ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋) & ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌) ⇒ ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌)) | ||
Theorem | smattl 29192 | Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) & ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) & ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) ⇒ ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽)) | ||
Theorem | smattr 29193 | Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) & ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) & ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿)) ⇒ ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽)) | ||
Theorem | smatbl 29194 | Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) & ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾)) & ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) ⇒ ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1))) | ||
Theorem | smatbr 29195 | Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑀)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑𝑚 ((1...𝑀) × (1...𝑁)))) & ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀)) & ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁)) ⇒ ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1))) | ||
Theorem | smatcl 29196 | Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indexes in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅)) & ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐿 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐶) | ||
Theorem | matmpt2 29197* | Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ (𝑀 ∈ 𝐵 → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗))) | ||
Theorem | 1smat1 29198 | The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20208. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
⊢ 1 = (1r‘((1...𝑁) Mat 𝑅)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅))) | ||
Theorem | submat1n 29199 | One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁)) | ||
Theorem | submatres 29200 | Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1))))) |
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