HomeTheorem List (p. 307 of 443)
GIF version.
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List
| Home |
Metamath
Proof Explorer Theorem List (p. 307 of 443) | < 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: | Metamath Proof Explorer
(1-28464) |
Hilbert Space Explorer
(28465-29989) |
Users' Mathboxes
(29990-44213) |
| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | drngdimgt0 30601 | The dimension of a vector space that is also a division ring is greater than zero. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹)) | ||
| Theorem | lmhmlvec2 30602 | A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec) | ||
| Theorem | kerlmhm 30603 | The kernel of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ 0 = (0g‘𝑈) & ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 })) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec) | ||
| Theorem | imlmhm 30604 | The image of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.) |
| ⊢ 𝐼 = (𝑈 ↾s ran 𝐹) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec) | ||
| Theorem | lindsunlem 30605 | Lemma for lindsun 30606. (Contributed by Thierry Arnoux, 9-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 }) & ⊢ 𝑂 = (0g‘(Scalar‘𝑊)) & ⊢ 𝐹 = (Base‘(Scalar‘𝑊)) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) & ⊢ (𝜑 → 𝐾 ∈ (𝐹 ∖ {𝑂})) & ⊢ (𝜑 → (𝐾( ·𝑠 ‘𝑊)𝐶) ∈ (𝑁‘((𝑈 ∪ 𝑉) ∖ {𝐶}))) ⇒ ⊢ (𝜑 → ⊥) | ||
| Theorem | lindsun 30606 | Condition for the union of two independent sets to be an independent set. (Contributed by Thierry Arnoux, 9-May-2023.) |
| ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → 𝑉 ∈ (LIndS‘𝑊)) & ⊢ (𝜑 → ((𝑁‘𝑈) ∩ (𝑁‘𝑉)) = { 0 }) ⇒ ⊢ (𝜑 → (𝑈 ∪ 𝑉) ∈ (LIndS‘𝑊)) | ||
| Theorem | lbsdiflsp0 30607 | The linear spans of two disjunct independent sets only have a trivial intersection. This can be seen as the opposite direction of lindsun 30606. (Contributed by Thierry Arnoux, 17-May-2023.) |
| ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐵 ∈ 𝐽 ∧ 𝑉 ⊆ 𝐵) → ((𝑁‘(𝐵 ∖ 𝑉)) ∩ (𝑁‘𝑉)) = { 0 }) | ||
| Theorem | dimkerim 30608 | Given a linear map 𝐹 between vector spaces 𝑉 and 𝑈, the dimension of the vector space 𝑉 is the sum of the dimension of 𝐹 's kernel and the dimension of 𝐹's image. Second part of theorem 5.3 in [Lang] p. 141 This can also be described as the Rank-nullity theorem, (dim‘𝐼) being the rank of 𝐹 (the dimension of its image), and (dim‘𝐾) its nullity (the dimension of its kernel). (Contributed by Thierry Arnoux, 17-May-2023.) |
| ⊢ 0 = (0g‘𝑈) & ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 })) & ⊢ 𝐼 = (𝑈 ↾s ran 𝐹) ⇒ ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (dim‘𝑉) = ((dim‘𝐾) +𝑒 (dim‘𝐼))) | ||
| Theorem | qusdimsum 30609 | Let 𝑊 be a vector space, and let 𝑋 be a subspace. Then the dimension of 𝑊 is the sum of the dimension of 𝑋 and the dimension of the quotient space of 𝑋. First part of theorem 5.3 in [Lang] p. 141 (Contributed by Thierry Arnoux, 20-May-2023.) |
| ⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑌 = (𝑊 /s (𝑊 ~QG 𝑈)) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑊) = ((dim‘𝑋) +𝑒 (dim‘𝑌))) | ||
| Theorem | fedgmullem1 30610* | Lemma for fedgmul 30612 (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) & ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)) & ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) & ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵)) & ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐴)) & ⊢ (𝜑 → 𝐿:𝑌⟶(Base‘(Scalar‘𝐵))) & ⊢ (𝜑 → 𝐿 finSupp (0g‘(Scalar‘𝐵))) & ⊢ (𝜑 → 𝑍 = (𝐵 Σg (𝑗 ∈ 𝑌 ↦ ((𝐿‘𝑗)( ·𝑠 ‘𝐵)𝑗)))) & ⊢ (𝜑 → 𝐺:𝑌⟶((Base‘(Scalar‘𝐶)) ↑𝑚 𝑋)) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) finSupp (0g‘(Scalar‘𝐶))) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐿‘𝑗) = (𝐶 Σg (𝑖 ∈ 𝑋 ↦ (((𝐺‘𝑗)‘𝑖)( ·𝑠 ‘𝐶)𝑖)))) ⇒ ⊢ (𝜑 → (𝐻 finSupp (0g‘(Scalar‘𝐴)) ∧ 𝑍 = (𝐴 Σg (𝐻 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)))) | ||
| Theorem | fedgmullem2 30611* | Lemma for fedgmul 30612 (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) & ⊢ 𝐷 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ (𝑖(.r‘𝐸)𝑗)) & ⊢ 𝐻 = (𝑗 ∈ 𝑌, 𝑖 ∈ 𝑋 ↦ ((𝐺‘𝑗)‘𝑖)) & ⊢ (𝜑 → 𝑋 ∈ (LBasis‘𝐶)) & ⊢ (𝜑 → 𝑌 ∈ (LBasis‘𝐵)) & ⊢ (𝜑 → 𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))) & ⊢ (𝜑 → (𝐴 Σg (𝑊 ∘𝑓 ( ·𝑠 ‘𝐴)𝐷)) = (0g‘𝐴)) ⇒ ⊢ (𝜑 → 𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))})) | ||
| Theorem | fedgmul 30612 | The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, we have [𝐸:𝐾] = [𝐸:𝐹][𝐹:𝐾]. Proposition 1.2 of [Lang], p. 224. Here (dim‘𝐴) is the degree of the extension 𝐸 of 𝐾, (dim‘𝐵) is the degree of the extension 𝐸 of 𝐹, and (dim‘𝐶) is the degree of the extension 𝐹 of 𝐾. This proof is valid for infinite dimensions, and is actually valid for division ring extensions, not just field extensions. (Contributed by Thierry Arnoux, 25-Jul-2023.) |
| ⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑉) & ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) & ⊢ 𝐶 = ((subringAlg ‘𝐹)‘𝑉) & ⊢ 𝐹 = (𝐸 ↾s 𝑈) & ⊢ 𝐾 = (𝐸 ↾s 𝑉) & ⊢ (𝜑 → 𝐸 ∈ DivRing) & ⊢ (𝜑 → 𝐹 ∈ DivRing) & ⊢ (𝜑 → 𝐾 ∈ DivRing) & ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) & ⊢ (𝜑 → 𝑉 ∈ (SubRing‘𝐹)) ⇒ ⊢ (𝜑 → (dim‘𝐴) = ((dim‘𝐵) ·e (dim‘𝐶))) | ||
| Syntax | cfldext 30613 | Syntax for the field extension relation. |
| class /FldExt | ||
| Syntax | cfinext 30614 | Syntax for the finite field extension relation. |
| class /FinExt | ||
| Syntax | calgext 30615 | Syntax for the algebraic field extension relation. |
| class /AlgExt | ||
| Syntax | cextdg 30616 | Syntax for the field extension degree operation. |
| class [:] | ||
| Definition | df-fldext 30617* | Definition of the field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /FldExt = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 ↾s (Base‘𝑓)) ∧ (Base‘𝑓) ∈ (SubRing‘𝑒)))} | ||
| Definition | df-extdg 30618* | Definition of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓)))) | ||
| Definition | df-finext 30619* | Definition of the finite field extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /FinExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)} | ||
| Definition | df-algext 30620* | Definition of the algebraic extension relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ /AlgExt = {⟨𝑒, 𝑓⟩ ∣ (𝑒/FldExt𝑓 ∧ ∀𝑥 ∈ (Base‘𝑒)∃𝑝 ∈ (Poly1‘𝑓)(((eval1‘𝑓)‘𝑝)‘𝑥) = (0g‘𝑒))} | ||
| Theorem | relfldext 30621 | The field extension is a relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ Rel /FldExt | ||
| Theorem | brfldext 30622 | The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)))) | ||
| Theorem | ccfldextrr 30623 | The field of the complex numbers is an extension of the field of the real numbers. (Contributed by Thierry Arnoux, 20-Jul-2023.) |
| ⊢ ℂfld/FldExtℝfld | ||
| Theorem | fldextfld1 30624 | A field extension is only defined if the extension is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | ||
| Theorem | fldextfld2 30625 | A field extension is only defined if the subfield is a field. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | ||
| Theorem | fldextsubrg 30626 | Field extension implies a subring relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ 𝑈 = (Base‘𝐹) ⇒ ⊢ (𝐸/FldExt𝐹 → 𝑈 ∈ (SubRing‘𝐸)) | ||
| Theorem | fldextress 30627 | Field extension implies a structure restriction relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 𝐹 = (𝐸 ↾s (Base‘𝐹))) | ||
| Theorem | brfinext 30628 | The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0)) | ||
| Theorem | extdgval 30629 | Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) | ||
| Theorem | fldextsralvec 30630 | The subring algebra associated with a field extension is a vector space. (Contributed by Thierry Arnoux, 3-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec) | ||
| Theorem | extdgcl 30631 | Closure of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) ∈ ℕ0*) | ||
| Theorem | extdggt0 30632 | Degrees of field extension are greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ (𝐸/FldExt𝐹 → 0 < (𝐸[:]𝐹)) | ||
| Theorem | fldexttr 30633 | Field extension is a transitive relation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → 𝐸/FldExt𝐾) | ||
| Theorem | fldextid 30634 | The field extension relation is reflexive. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ (𝐹 ∈ Field → 𝐹/FldExt𝐹) | ||
| Theorem | extdgid 30635 | A trivial field extension has degree one. (Contributed by Thierry Arnoux, 4-Aug-2023.) |
| ⊢ (𝐸 ∈ Field → (𝐸[:]𝐸) = 1) | ||
| Theorem | extdgmul 30636 | The multiplicativity formula for degrees of field extensions. Given 𝐸 a field extension of 𝐹, itself a field extension of 𝐾, the degree of the extension 𝐸/FldExt𝐾 is the product of the degrees of the extensions 𝐸/FldExt𝐹 and 𝐹/FldExt𝐾. Proposition 1.2 of [Lang], p. 224. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸[:]𝐾) = ((𝐸[:]𝐹) ·e (𝐹[:]𝐾))) | ||
| Theorem | finexttrb 30637 | The extension 𝐸 of 𝐾 is finite if and only if 𝐸 is finite over 𝐹 and 𝐹 is finite over 𝐾. Corollary 1.3 of [Lang] , p. 225. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ 𝐹/FldExt𝐾) → (𝐸/FinExt𝐾 ↔ (𝐸/FinExt𝐹 ∧ 𝐹/FinExt𝐾))) | ||
| Theorem | extdg1id 30638 | If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.) |
| ⊢ ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹) | ||
| Theorem | extdg1b 30639 | The degree of the extension 𝐸/FldExt𝐹 is 1 iff 𝐸 and 𝐹 are the same structure. (Contributed by Thierry Arnoux, 6-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → ((𝐸[:]𝐹) = 1 ↔ 𝐸 = 𝐹)) | ||
| Theorem | fldextchr 30640 | The characteristic of a subfield is the same as the characteristic of the larger field. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ (𝐸/FldExt𝐹 → (chr‘𝐹) = (chr‘𝐸)) | ||
| Theorem | ccfldsrarelvec 30641 | The subring algebra of the complex numbers over the real numbers is a left vector space. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ ((subringAlg ‘ℂfld)‘ℝ) ∈ LVec | ||
| Theorem | ccfldextdgrr 30642 | The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.) |
| ⊢ (ℂfld[:]ℝfld) = 2 | ||
| Theorem | symgfcoeu 30643* | 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 30644 | 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 30645 | Permutation sign of the identity. (Contributed by Thierry Arnoux, 21-Aug-2020.) |
| ⊢ 𝑆 = (pmSgn‘𝐷) ⇒ ⊢ (𝐷 ∈ Fin → (𝑆‘( I ↾ 𝐷)) = 1) | ||
| Theorem | pmtrprfv2 30646 | 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 30647 | 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 30648* | Lemma for psgnfzto1st 30653. 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 30649* | 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 30650* | Special case where the permutation defined in psgnfzto1st 30653 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 30651* | 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 30652* | 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 30653* | 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))) | ||
| Theorem | pmtridf1o 30654 | Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) | ||
| Theorem | pmtridfv1 30655 | Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌) | ||
| Theorem | pmtridfv2 30656 | Value at Y of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.) |
| ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋) | ||
| Syntax | csmat 30657 | Syntax for a function generating submatrices. |
| class subMat1 | ||
| Definition | df-smat 30658* | 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 20880. (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 30659* | 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 30660 | 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 30661 | 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 30662 | 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 30663 | 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 30664 | 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 30665 | 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 30666 | Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices 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 | matmpo 30667* | 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 30668 | The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20886. (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 30669 | 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 30670 | 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))))) | ||
| Theorem | submateqlem1 30671 | Lemma for submateq 30673. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝐾 ≤ 𝑀) ⇒ ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾}))) | ||
| Theorem | submateqlem2 30672 | Lemma for submateq 30673. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) & ⊢ (𝜑 → 𝑀 < 𝐾) ⇒ ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) | ||
| Theorem | submateq 30673* | Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐸 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗)) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽)) | ||
| Theorem | submatminr1 30674 | If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
| ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽)) | ||
| Syntax | clmat 30675 | Extend class notation with the literal matrix conversion function. |
| class litMat | ||
| Definition | df-lmat 30676* | Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1)))) | ||
| Theorem | lmatval 30677* | Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1)))) | ||
| Theorem | lmatfval 30678* | Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1))) | ||
| Theorem | lmatfvlem 30679* | Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ 𝐾 ∈ ℕ0 & ⊢ 𝐿 ∈ ℕ0 & ⊢ 𝐼 ≤ 𝑁 & ⊢ 𝐽 ≤ 𝑁 & ⊢ (𝐾 + 1) = 𝐼 & ⊢ (𝐿 + 1) = 𝐽 & ⊢ (𝑊‘𝐾) = 𝑋 & ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌) ⇒ ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌) | ||
| Theorem | lmatcl 30680* | Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘𝑊) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉) & ⊢ (𝜑 → (♯‘𝑊) = 𝑁) & ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝑂 = ((1...𝑁) Mat 𝑅) & ⊢ 𝑃 = (Base‘𝑂) & ⊢ (𝜑 → 𝑅 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝑀 ∈ 𝑃) | ||
| Theorem | lmat22lem 30681* | Lemma for lmat22e11 30682 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2) | ||
| Theorem | lmat22e11 30682 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (1𝑀1) = 𝐴) | ||
| Theorem | lmat22e12 30683 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (1𝑀2) = 𝐵) | ||
| Theorem | lmat22e21 30684 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (2𝑀1) = 𝐶) | ||
| Theorem | lmat22e22 30685 | Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → (2𝑀2) = 𝐷) | ||
| Theorem | lmat22det 30686 | The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.) |
| ⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ 𝑉 = (Base‘𝑅) & ⊢ 𝐽 = ((1...2) maDet 𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵))) | ||
| Theorem | mdetpmtr1 30687* | The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗)) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸))) | ||
| Theorem | mdetpmtr2 30688* | The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗))) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸))) | ||
| Theorem | mdetpmtr12 30689* | The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐺 = (Base‘(SymGrp‘𝑁)) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀(𝑄‘𝑗))) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ∈ 𝐺) & ⊢ (𝜑 → 𝑄 ∈ 𝐺) ⇒ ⊢ (𝜑 → (𝐷‘𝑀) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝐸))) | ||
| Theorem | mdetlap1 30690* | A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.) |
| ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐷 = (𝑁 maDet 𝑅) & ⊢ 𝐾 = (𝑁 maAdju 𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼))))) | ||
| Theorem | madjusmdetlem1 30691* | Lemma for madjusmdet 30695. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁))) & ⊢ 𝑆 = (pmSgn‘(1...𝑁)) & ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗))) & ⊢ (𝜑 → 𝑃 ∈ 𝐺) & ⊢ (𝜑 → 𝑄 ∈ 𝐺) & ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼) & ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽) & ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | madjusmdetlem2 30692* | Lemma for madjusmdet 30695. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋)) | ||
| Theorem | madjusmdetlem3 30693* | Lemma for madjusmdet 30695. (Contributed by Thierry Arnoux, 27-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) & ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗))) & ⊢ (𝜑 → 𝑈 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁)) | ||
| Theorem | madjusmdetlem4 30694* | Lemma for madjusmdet 30695. (Contributed by Thierry Arnoux, 22-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) & ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖))) & ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖))) & ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗))) & ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗))) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | madjusmdet 30695 | Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽)))) | ||
| Theorem | mdetlap 30696* | Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.) |
| ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝐴 = ((1...𝑁) Mat 𝑅) & ⊢ 𝐷 = ((1...𝑁) maDet 𝑅) & ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑍 = (ℤRHom‘𝑅) & ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝐼 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝐽 ∈ (1...𝑁)) & ⊢ (𝜑 → 𝑀 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗))))))) | ||
| Theorem | fvproj 30697* | Value of a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.) |
| ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩) | ||
| Theorem | fimaproj 30698* | Image of a cartesian product for a function on pairs, given two projections 𝐹 and 𝐺. (Contributed by Thierry Arnoux, 30-Dec-2019.) |
| ⊢ 𝐻 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐵) & ⊢ (𝜑 → 𝑋 ⊆ 𝐴) & ⊢ (𝜑 → 𝑌 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌))) | ||
| Theorem | txomap 30699* | Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.) |
| ⊢ (𝜑 → 𝐹:𝑋⟶𝑍) & ⊢ (𝜑 → 𝐺:𝑌⟶𝑇) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍)) & ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀) & ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾)) & ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ⇒ ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀)) | ||
| Theorem | qtopt1 30700* | If every equivalence class is closed, then the quotient space is T1 . (Contributed by Thierry Arnoux, 5-Jan-2020.) |
| ⊢ 𝑋 = ∪ 𝐽 & ⊢ (𝜑 → 𝐽 ∈ Fre) & ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹 “ {𝑥}) ∈ (Clsd‘𝐽)) ⇒ ⊢ (𝜑 → (𝐽 qTop 𝐹) ∈ Fre) | ||
| < Previous Next > |
| Copyright terms: Public domain | < Previous Next > |