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Type | Label | Description |
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Statement | ||
Theorem | lsslvec 21001 | A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec) | ||
Theorem | lmhmlvec 21002 | The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec)) | ||
Theorem | lvecvs0or 21003 | If a scalar product is zero, one of its factors must be zero. (hvmul0or 30855 analog.) (Contributed by NM, 2-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) = 0 ↔ (𝐴 = 𝑂 ∨ 𝑋 = 0 ))) | ||
Theorem | lvecvsn0 21004 | A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑂 = (0g‘𝐹) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) ≠ 0 ↔ (𝐴 ≠ 𝑂 ∧ 𝑋 ≠ 0 ))) | ||
Theorem | lssvs0or 21005 | If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) ∈ 𝑈 ↔ (𝐴 = 0 ∨ 𝑋 ∈ 𝑈))) | ||
Theorem | lvecvscan 21006 | Cancellation law for scalar multiplication. (hvmulcan 30902 analog.) (Contributed by NM, 2-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | lvecvscan2 21007 | Cancellation law for scalar multiplication. (hvmulcan2 30903 analog.) (Contributed by NM, 2-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ≠ 0 ) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵)) | ||
Theorem | lvecinv 21008 | Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) & ⊢ 𝐼 = (invr‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐴 ∈ (𝐾 ∖ { 0 })) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼‘𝐴) · 𝑋))) | ||
Theorem | lspsnvs 21009 | A nonzero scalar product does not change the span of a singleton. (spansncol 31398 analog.) (Contributed by NM, 23-Apr-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ (𝑅 ∈ 𝐾 ∧ 𝑅 ≠ 0 ) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅 · 𝑋)}) = (𝑁‘{𝑋})) | ||
Theorem | lspsneleq 21010 | Membership relation that implies equality of spans. (spansneleq 31400 analog.) (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋})) | ||
Theorem | lspsncmp 21011 | Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌}))) | ||
Theorem | lspsnne1 21012 | Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) | ||
Theorem | lspsnne2 21013 | Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | ||
Theorem | lspsnnecom 21014 | Swap two vectors with different spans. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋})) | ||
Theorem | lspabs2 21015 | Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | ||
Theorem | lspabs3 21016 | Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)})) | ||
Theorem | lspsneq 21017* | Equal spans of singletons must have proportional vectors. See lspsnss2 20896 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑘 · 𝑌))) | ||
Theorem | lspsneu 21018* | Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ 𝑂 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃!𝑘 ∈ (𝐾 ∖ {𝑂})𝑋 = (𝑘 · 𝑌))) | ||
Theorem | lspsnel4 21019 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 31403 analog.) (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈)) | ||
Theorem | lspdisj 21020 | The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }) | ||
Theorem | lspdisjb 21021 | A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })) | ||
Theorem | lspdisj2 21022 | Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 }) | ||
Theorem | lspfixed 21023* | Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)})) | ||
Theorem | lspexch 21024 | Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 21025 versus lspexchn2 21026); look for lspexch 21024 and prcom 4741 in same proof. TODO: would a hypothesis of ¬ 𝑋 ∈ (𝑁‘{𝑍}) instead of (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) be better overall? This would be shorter and also satisfy the 𝑋 ≠ 0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the ≠ pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) | ||
Theorem | lspexchn1 21025 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21024 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})) | ||
Theorem | lspexchn2 21026 | Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 21024 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})) | ||
Theorem | lspindpi 21027 | Partial independence property. (Contributed by NM, 23-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) | ||
Theorem | lspindp1 21028 | Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌}))) | ||
Theorem | lspindp2l 21029 | Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))) | ||
Theorem | lspindp2 21030 | Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋}))) | ||
Theorem | lspindp3 21031 | Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑋 + 𝑌)})) | ||
Theorem | lspindp4 21032 | (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑋 + 𝑌)})) | ||
Theorem | lvecindp 21033 | Compute the 𝑋 coefficient in a sum with an independent vector 𝑋 (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions 𝑌 and 𝑍 (second conjunct). Typically, 𝑈 is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 19-Jul-2022.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → ((𝐴 · 𝑋) + 𝑌) = ((𝐵 · 𝑋) + 𝑍)) ⇒ ⊢ (𝜑 → (𝐴 = 𝐵 ∧ 𝑌 = 𝑍)) | ||
Theorem | lvecindp2 21034 | Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) & ⊢ (𝜑 → 𝐷 ∈ 𝐾) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌))) ⇒ ⊢ (𝜑 → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | lspsnsubn0 21035 | Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 0 ) | ||
Theorem | lsmcv 21036 | Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 31482 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 = (𝑇 ⊕ (𝑁‘{𝑋}))) | ||
Theorem | lspsolvlem 21037* | Lemma for lspsolv 21038. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑄 = {𝑧 ∈ 𝑉 ∣ ∃𝑟 ∈ 𝐵 (𝑧 + (𝑟 · 𝑌)) ∈ (𝑁‘𝐴)} & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌}))) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ 𝐵 (𝑋 + (𝑟 · 𝑌)) ∈ (𝑁‘𝐴)) | ||
Theorem | lspsolv 21038 | If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ (𝐴 ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁‘𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋}))) | ||
Theorem | lssacsex 21039* | In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 20858 by lspsolv 21038. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐴 = (LSubSp‘𝑊) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝑋 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝐴 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))) | ||
Theorem | lspsnat 21040 | There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 31411 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑉) ∧ 𝑈 ⊆ (𝑁‘{𝑋})) → (𝑈 = (𝑁‘{𝑋}) ∨ 𝑈 = { 0 })) | ||
Theorem | lspsncv0 21041* | The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.) (Revised by AV, 13-Jul-2022.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ¬ ∃𝑦 ∈ 𝑆 ({ 0 } ⊊ 𝑦 ∧ 𝑦 ⊊ (𝑁‘{𝑋}))) | ||
Theorem | lsppratlem1 21042 | Lemma for lspprat 21048. Let 𝑥 ∈ (𝑈 ∖ {0}) (if there is no such 𝑥 then 𝑈 is the zero subspace), and let 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) (assuming the conclusion is false). The goal is to write 𝑋, 𝑌 in terms of 𝑥, 𝑦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 21038 (hence the name), which we use extensively below. In this lemma, we show that since 𝑥 ∈ (𝑁‘{𝑋, 𝑌}), either 𝑥 ∈ (𝑁‘{𝑌}) or 𝑋 ∈ (𝑁‘{𝑥, 𝑌}). (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) & ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) | ||
Theorem | lsppratlem2 21043 | Lemma for lspprat 21048. Show that if 𝑋 and 𝑌 are both in (𝑁‘{𝑥, 𝑦}) (which will be our goal for each of the two cases above), then (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈, contradicting the hypothesis for 𝑈. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) & ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) & ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) | ||
Theorem | lsppratlem3 21044 | Lemma for lspprat 21048. In the first case of lsppratlem1 21042, since 𝑥 ∉ (𝑁‘∅), also 𝑌 ∈ (𝑁‘{𝑥}), and since 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑥}) and 𝑦 ∉ (𝑁‘{𝑥}), we have 𝑋 ∈ (𝑁‘{𝑥, 𝑦}) as desired. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) & ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) & ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) | ||
Theorem | lsppratlem4 21045 | Lemma for lspprat 21048. In the second case of lsppratlem1 21042, 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑌}) and 𝑦 ∉ (𝑁‘{𝑥}) implies 𝑌 ∈ (𝑁‘{𝑥, 𝑦}) and thus 𝑋 ∈ (𝑁‘{𝑥, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑦}) as well. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) & ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) & ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) | ||
Theorem | lsppratlem5 21046 | Lemma for lspprat 21048. Combine the two cases and show a contradiction to 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}) under the assumptions on 𝑥 and 𝑦. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) & ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) & ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) | ||
Theorem | lsppratlem6 21047 | Lemma for lspprat 21048. Negating the assumption on 𝑦, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥}))) | ||
Theorem | lspprat 21048* | A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if 𝑧 is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑈 = (𝑁‘{𝑧})) | ||
Theorem | islbs2 21049* | An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥}))))) | ||
Theorem | islbs3 21050* | An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑠(𝑠 ⊊ 𝐵 → (𝑁‘𝑠) ⊊ 𝑉)))) | ||
Theorem | lbsacsbs 21051 | Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 21049. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐴 = (LSubSp‘𝑊) & ⊢ 𝑁 = (mrCls‘𝐴) & ⊢ 𝑋 = (Base‘𝑊) & ⊢ 𝐼 = (mrInd‘𝐴) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → (𝑆 ∈ 𝐽 ↔ (𝑆 ∈ 𝐼 ∧ (𝑁‘𝑆) = 𝑋))) | ||
Theorem | lvecdim 21052 | The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 21039 and lbsacsbs 21051 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 18558. (Contributed by David Moews, 1-May-2017.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑇 ∈ 𝐽) → 𝑆 ≈ 𝑇) | ||
Theorem | lbsextlem1 21053* | Lemma for lbsext 21058. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐶 ⊆ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) & ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} ⇒ ⊢ (𝜑 → 𝑆 ≠ ∅) | ||
Theorem | lbsextlem2 21054* | Lemma for lbsext 21058. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of ∪ 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐶 ⊆ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) & ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} & ⊢ 𝑃 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝐴) & ⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⇒ ⊢ (𝜑 → (𝑇 ∈ 𝑃 ∧ (∪ 𝐴 ∖ {𝑥}) ⊆ 𝑇)) | ||
Theorem | lbsextlem3 21055* | Lemma for lbsext 21058. A chain in 𝑆 has an upper bound in 𝑆. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐶 ⊆ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) & ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} & ⊢ 𝑃 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝑆) & ⊢ (𝜑 → 𝐴 ≠ ∅) & ⊢ (𝜑 → [⊊] Or 𝐴) & ⊢ 𝑇 = ∪ 𝑢 ∈ 𝐴 (𝑁‘(𝑢 ∖ {𝑥})) ⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑆) | ||
Theorem | lbsextlem4 21056* | Lemma for lbsext 21058. lbsextlem3 21055 satisfies the conditions for the application of Zorn's lemma zorn 10538 (thus invoking AC), and so there is a maximal linearly independent set extending 𝐶. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝐶 ⊆ 𝑉) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) & ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} & ⊢ (𝜑 → 𝒫 𝑉 ∈ dom card) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) | ||
Theorem | lbsextg 21057* | For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (((𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card) ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) | ||
Theorem | lbsext 21058* | For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LVec ∧ 𝐶 ⊆ 𝑉 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠 ∈ 𝐽 𝐶 ⊆ 𝑠) | ||
Theorem | lbsexg 21059 | Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((CHOICE ∧ 𝑊 ∈ LVec) → 𝐽 ≠ ∅) | ||
Theorem | lbsex 21060 | Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → 𝐽 ≠ ∅) | ||
Theorem | lvecprop2d 21061* | If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 21062 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ 𝐹 = (Scalar‘𝐾) & ⊢ 𝐺 = (Scalar‘𝐿) & ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) & ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec)) | ||
Theorem | lvecpropd 21062* | If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) & ⊢ 𝑃 = (Base‘𝐹) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec)) | ||
Syntax | csra 21063 | Extend class notation with the subring algebra generator. |
class subringAlg | ||
Syntax | crglmod 21064 | Extend class notation with the left module induced by a ring over itself. |
class ringLMod | ||
Definition | df-sra 21065* | Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet 〈(Scalar‘ndx), (𝑤 ↾s 𝑠)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑤)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑤)〉))) | ||
Definition | df-rgmod 21066 | Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤))) | ||
Theorem | sraval 21067 | Lemma for srabase 21070 through sravsca 21078. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ ((𝑊 ∈ 𝑉 ∧ 𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet 〈(Scalar‘ndx), (𝑊 ↾s 𝑆)〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | ||
Theorem | sralem 21068 | Lemma for srabase 21070 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (Scalar‘ndx) ≠ (𝐸‘ndx) & ⊢ ( ·𝑠 ‘ndx) ≠ (𝐸‘ndx) & ⊢ (·𝑖‘ndx) ≠ (𝐸‘ndx) ⇒ ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴)) | ||
Theorem | sralemOLD 21069 | Obsolete version of sralem 21068 as of 29-Oct-2024. Lemma for srabase 21070 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ (𝑁 < 5 ∨ 8 < 𝑁) ⇒ ⊢ (𝜑 → (𝐸‘𝑊) = (𝐸‘𝐴)) | ||
Theorem | srabase 21070 | Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴)) | ||
Theorem | srabaseOLD 21071 | Obsolete proof of srabase 21070 as of 29-Oct-2024. Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴)) | ||
Theorem | sraaddg 21072 | Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴)) | ||
Theorem | sraaddgOLD 21073 | Obsolete proof of sraaddg 21072 as of 29-Oct-2024. Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴)) | ||
Theorem | sramulr 21074 | Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴)) | ||
Theorem | sramulrOLD 21075 | Obsolete proof of sramulr 21074 as of 29-Oct-2024. Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = (.r‘𝐴)) | ||
Theorem | srasca 21076 | The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) | ||
Theorem | srascaOLD 21077 | Obsolete proof of srasca 21076 as of 12-Nov-2024. The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 12-Nov-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (𝑊 ↾s 𝑆) = (Scalar‘𝐴)) | ||
Theorem | sravsca 21078 | The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof shortened by AV, 12-Nov-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) | ||
Theorem | sravscaOLD 21079 | Obsolete proof of sravsca 21078 as of 12-Nov-2024. The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = ( ·𝑠 ‘𝐴)) | ||
Theorem | sraip 21080 | The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (.r‘𝑊) = (·𝑖‘𝐴)) | ||
Theorem | sratset 21081 | Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴)) | ||
Theorem | sratsetOLD 21082 | Obsolete proof of sratset 21081 as of 29-Oct-2024. Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴)) | ||
Theorem | sratopn 21083 | Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴)) | ||
Theorem | srads 21084 | Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) | ||
Theorem | sradsOLD 21085 | Obsolete proof of srads 21084 as of 29-Oct-2024. Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) | ||
Theorem | sraring 21086 | Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.) |
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring) | ||
Theorem | sralmod 21087 | The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆) ⇒ ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod) | ||
Theorem | sralmod0 21088 | The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.) |
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) & ⊢ (𝜑 → 0 = (0g‘𝑊)) & ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) ⇒ ⊢ (𝜑 → 0 = (0g‘𝐴)) | ||
Theorem | issubrgd 21089* | Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.) |
⊢ (𝜑 → 𝑆 = (𝐼 ↾s 𝐷)) & ⊢ (𝜑 → 0 = (0g‘𝐼)) & ⊢ (𝜑 → + = (+g‘𝐼)) & ⊢ (𝜑 → 𝐷 ⊆ (Base‘𝐼)) & ⊢ (𝜑 → 0 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 + 𝑦) ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((invg‘𝐼)‘𝑥) ∈ 𝐷) & ⊢ (𝜑 → 1 = (1r‘𝐼)) & ⊢ (𝜑 → · = (.r‘𝐼)) & ⊢ (𝜑 → 1 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷) → (𝑥 · 𝑦) ∈ 𝐷) & ⊢ (𝜑 → 𝐼 ∈ Ring) ⇒ ⊢ (𝜑 → 𝐷 ∈ (SubRing‘𝐼)) | ||
Theorem | rlmfn 21090 | ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ ringLMod Fn V | ||
Theorem | rlmval 21091 | Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊)) | ||
Theorem | rlmval2 21092 | Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.) |
⊢ (𝑊 ∈ 𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet 〈(Scalar‘ndx), 𝑊〉) sSet 〈( ·𝑠 ‘ndx), (.r‘𝑊)〉) sSet 〈(·𝑖‘ndx), (.r‘𝑊)〉)) | ||
Theorem | rlmbas 21093 | Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | ||
Theorem | rlmplusg 21094 | Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (+g‘𝑅) = (+g‘(ringLMod‘𝑅)) | ||
Theorem | rlm0 21095 | Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | ||
Theorem | rlmsub 21096 | Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅)) | ||
Theorem | rlmmulr 21097 | Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ (.r‘𝑅) = (.r‘(ringLMod‘𝑅)) | ||
Theorem | rlmsca 21098 | Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
⊢ (𝑅 ∈ 𝑋 → 𝑅 = (Scalar‘(ringLMod‘𝑅))) | ||
Theorem | rlmsca2 21099 | Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.) |
⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | ||
Theorem | rlmvsca 21100 | Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) |
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