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Type | Label | Description |
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Statement | ||
Theorem | lspidm 21001 | The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈)) | ||
Theorem | lspun 21002 | The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑇 ∪ 𝑈)) = (𝑁‘((𝑁‘𝑇) ∪ (𝑁‘𝑈)))) | ||
Theorem | lspssp 21003 | If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) | ||
Theorem | mrclsp 21004 | Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
⊢ 𝑈 = (LSubSp‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝐹 = (mrCls‘𝑈) ⇒ ⊢ (𝑊 ∈ LMod → 𝐾 = 𝐹) | ||
Theorem | lspsnss 21005 | The span of the singleton of a subspace member is included in the subspace. (spansnss 31599 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) | ||
Theorem | ellspsn3 21006 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 31600 analog.) (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑈) | ||
Theorem | lspprss 21007 | The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) | ||
Theorem | lspsnid 21008 | A vector belongs to the span of its singleton. (spansnid 31591 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) | ||
Theorem | ellspsn6 21009 | Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) | ||
Theorem | ellspsn5b 21010 | Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) | ||
Theorem | ellspsn5 21011 | Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) | ||
Theorem | lspprid1 21012 | A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspprid2 21013 | A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspprvacl 21014 | The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lssats2 21015* | A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) | ||
Theorem | ellspsni 21016 | A scalar product with a vector belongs to the span of its singleton. (spansnmul 31592 analog.) (Contributed by NM, 2-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) | ||
Theorem | lspsn 21017* | Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) | ||
Theorem | ellspsn 21018* | Member of span of the singleton of a vector. (elspansn 31594 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋))) | ||
Theorem | lspsnvsi 21019 | Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅 · 𝑋)}) ⊆ (𝑁‘{𝑋})) | ||
Theorem | lspsnss2 21020* | Comparable spans of singletons must have proportional vectors. See lspsneq 21141 for equal span version. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ 𝐾 𝑋 = (𝑘 · 𝑌))) | ||
Theorem | lspsnneg 21021 | Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑀 = (invg‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑀‘𝑋)}) = (𝑁‘{𝑋})) | ||
Theorem | lspsnsub 21022 | Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)})) | ||
Theorem | lspsn0 21023 | Span of the singleton of the zero vector. (spansn0 31569 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) | ||
Theorem | lsp0 21024 | Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) | ||
Theorem | lspuni0 21025 | Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → ∪ (𝑁‘∅) = 0 ) | ||
Theorem | lspun0 21026 | The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) | ||
Theorem | lspsneq0 21027 | Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) | ||
Theorem | lspsneq0b 21028 | Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 = 0 ↔ 𝑌 = 0 )) | ||
Theorem | lmodindp1 21029 | Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) | ||
Theorem | lsslsp 21030 | Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) Terms in the equation were swapped as proposed by NM on 15-Mar-2015. (Revised by AV, 18-Apr-2025.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑀 = (LSpan‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑋) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑁‘𝐺) = (𝑀‘𝐺)) | ||
Theorem | lsslspOLD 21031 | Obsolete version of lsslsp 21030 as of 25-Apr-2025. Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap 𝑀‘𝐺 and 𝑁‘𝐺 since we are computing a property of 𝑁‘𝐺? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015. (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑀 = (LSpan‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑋) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) = (𝑁‘𝐺)) | ||
Theorem | lss0v 21032 | The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑍 = (0g‘𝑋) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) | ||
Theorem | lsspropd 21033* | If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) ⇒ ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) | ||
Theorem | lsppropd 21034* | If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) & ⊢ (𝜑 → 𝐾 ∈ 𝑋) & ⊢ (𝜑 → 𝐿 ∈ 𝑌) ⇒ ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) | ||
Syntax | clmhm 21035 | Extend class notation with the generator of left module hom-sets. |
class LMHom | ||
Syntax | clmim 21036 | The class of left module isomorphism sets. |
class LMIso | ||
Syntax | clmic 21037 | The class of the left module isomorphism relation. |
class ≃𝑚 | ||
Definition | df-lmhm 21038* | A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))}) | ||
Definition | df-lmim 21039* | An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)}) | ||
Definition | df-lmic 21040 | Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ ≃𝑚 = (◡ LMIso “ (V ∖ 1o)) | ||
Theorem | reldmlmhm 21041 | Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ Rel dom LMHom | ||
Theorem | lmimfn 21042 | Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ LMIso Fn (LMod × LMod) | ||
Theorem | islmhm 21043* | Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐸 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) | ||
Theorem | islmhm3 21044* | Property of a module homomorphism, similar to ismhm 18810. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐸 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) | ||
Theorem | lmhmlem 21045 | Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾))) | ||
Theorem | lmhmsca 21046 | A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾) | ||
Theorem | lmghm 21047 | A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | ||
Theorem | lmhmlmod2 21048 | A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod) | ||
Theorem | lmhmlmod1 21049 | A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) | ||
Theorem | lmhmf 21050 | A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵⟶𝐶) | ||
Theorem | lmhmlin 21051 | A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐸 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹‘𝑌))) | ||
Theorem | lmodvsinv 21052 | Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (invg‘𝑊) & ⊢ 𝑀 = (invg‘𝐹) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑅) · 𝑋) = (𝑁‘(𝑅 · 𝑋))) | ||
Theorem | lmodvsinv2 21053 | Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (invg‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐵) → (𝑅 · (𝑁‘𝑋)) = (𝑁‘(𝑅 · 𝑋))) | ||
Theorem | islmhm2 21054* | A one-equation proof of linearity of a left module homomorphism, similar to df-lss 20947. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) & ⊢ 𝐸 = (Base‘𝐾) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐸 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹‘𝑦)) ⨣ (𝐹‘𝑧))))) | ||
Theorem | islmhmd 21055* | Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) & ⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐽 = (Scalar‘𝑇) & ⊢ 𝑁 = (Base‘𝐾) & ⊢ (𝜑 → 𝑆 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐽 = 𝐾) & ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇)) | ||
Theorem | 0lmhm 21056 | The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 0 = (0g‘𝑁) & ⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝑆 = (Scalar‘𝑀) & ⊢ 𝑇 = (Scalar‘𝑁) ⇒ ⊢ ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁)) | ||
Theorem | idlmhm 21057 | The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀)) | ||
Theorem | invlmhm 21058 | The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ 𝐼 = (invg‘𝑀) ⇒ ⊢ (𝑀 ∈ LMod → 𝐼 ∈ (𝑀 LMHom 𝑀)) | ||
Theorem | lmhmco 21059 | The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.) |
⊢ ((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂)) | ||
Theorem | lmhmplusg 21060 | The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.) |
⊢ + = (+g‘𝑁) ⇒ ⊢ ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁)) | ||
Theorem | lmhmvsca 21061 | The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.) |
⊢ 𝑉 = (Base‘𝑀) & ⊢ · = ( ·𝑠 ‘𝑁) & ⊢ 𝐽 = (Scalar‘𝑁) & ⊢ 𝐾 = (Base‘𝐽) ⇒ ⊢ ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘f · 𝐹) ∈ (𝑀 LMHom 𝑁)) | ||
Theorem | lmhmf1o 21062 | A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ 𝑋 = (Base‘𝑆) & ⊢ 𝑌 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆))) | ||
Theorem | lmhmima 21063 | The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑋 = (LSubSp‘𝑆) & ⊢ 𝑌 = (LSubSp‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑋) → (𝐹 “ 𝑈) ∈ 𝑌) | ||
Theorem | lmhmpreima 21064 | The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑋 = (LSubSp‘𝑆) & ⊢ 𝑌 = (LSubSp‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋) | ||
Theorem | lmhmlsp 21065 | Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑆) & ⊢ 𝐾 = (LSpan‘𝑆) & ⊢ 𝐿 = (LSpan‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ⊆ 𝑉) → (𝐹 “ (𝐾‘𝑈)) = (𝐿‘(𝐹 “ 𝑈))) | ||
Theorem | lmhmrnlss 21066 | The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇)) | ||
Theorem | lmhmkerlss 21067 | The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝐾 = (◡𝐹 “ { 0 }) & ⊢ 0 = (0g‘𝑇) & ⊢ 𝑈 = (LSubSp‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾 ∈ 𝑈) | ||
Theorem | reslmhm 21068 | Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
⊢ 𝑈 = (LSubSp‘𝑆) & ⊢ 𝑅 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝑈) → (𝐹 ↾ 𝑋) ∈ (𝑅 LMHom 𝑇)) | ||
Theorem | reslmhm2 21069 | Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) & ⊢ 𝐿 = (LSubSp‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇)) | ||
Theorem | reslmhm2b 21070 | Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) & ⊢ 𝐿 = (LSubSp‘𝑇) ⇒ ⊢ ((𝑇 ∈ LMod ∧ 𝑋 ∈ 𝐿 ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈))) | ||
Theorem | lmhmeql 21071 | The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑈 = (LSubSp‘𝑆) ⇒ ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ 𝑈) | ||
Theorem | lspextmo 21072* | A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐾 = (LSpan‘𝑆) ⇒ ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹) | ||
Theorem | pwsdiaglmhm 21073* | Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌)) | ||
Theorem | pwssplit0 21074* | Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s 𝑈) & ⊢ 𝑍 = (𝑊 ↑s 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑊 ∈ 𝑇 ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵⟶𝐶) | ||
Theorem | pwssplit1 21075* | Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s 𝑈) & ⊢ 𝑍 = (𝑊 ↑s 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑊 ∈ Mnd ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹:𝐵–onto→𝐶) | ||
Theorem | pwssplit2 21076* | Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s 𝑈) & ⊢ 𝑍 = (𝑊 ↑s 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑊 ∈ Grp ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 GrpHom 𝑍)) | ||
Theorem | pwssplit3 21077* | Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝑌 = (𝑊 ↑s 𝑈) & ⊢ 𝑍 = (𝑊 ↑s 𝑉) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍)) | ||
Theorem | islmim 21078 | An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶)) | ||
Theorem | lmimf1o 21079 | An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵–1-1-onto→𝐶) | ||
Theorem | lmimlmhm 21080 | An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆)) | ||
Theorem | lmimgim 21081 | An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆)) | ||
Theorem | islmim2 21082 | An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 LMHom 𝑅))) | ||
Theorem | lmimcnv 21083 | The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ (𝐹 ∈ (𝑆 LMIso 𝑇) → ◡𝐹 ∈ (𝑇 LMIso 𝑆)) | ||
Theorem | brlmic 21084 | The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | ||
Theorem | brlmici 21085 | Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅 ≃𝑚 𝑆) | ||
Theorem | lmiclcl 21086 | Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) | ||
Theorem | lmicrcl 21087 | Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod) | ||
Theorem | lmicsym 21088 | Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) | ||
Theorem | lmhmpropd 21089* | Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐽)) & ⊢ (𝜑 → 𝐺 = (Scalar‘𝐾)) & ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿)) & ⊢ (𝜑 → 𝐺 = (Scalar‘𝑀)) & ⊢ 𝑃 = (Base‘𝐹) & ⊢ 𝑄 = (Base‘𝐺) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐽)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑄 ∧ 𝑦 ∈ 𝐶)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀)) | ||
Syntax | clbs 21090 | Extend class notation with the set of bases for a vector space. |
class LBasis | ||
Definition | df-lbs 21091* | Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
⊢ LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛‘𝑏) = (Base‘𝑤) ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑦( ·𝑠 ‘𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))}) | ||
Theorem | islbs 21092* | The predicate "𝐵 is a basis for the left module or vector space 𝑊". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝐵 ∈ 𝐽 ↔ (𝐵 ⊆ 𝑉 ∧ (𝑁‘𝐵) = 𝑉 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥}))))) | ||
Theorem | lbsss 21093 | A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ (𝐵 ∈ 𝐽 → 𝐵 ⊆ 𝑉) | ||
Theorem | lbsel 21094 | An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) ⇒ ⊢ ((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → 𝐸 ∈ 𝑉) | ||
Theorem | lbssp 21095 | The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝐵 ∈ 𝐽 → (𝑁‘𝐵) = 𝑉) | ||
Theorem | lbsind 21096 | A basis is linearly independent; that is, every element has a span which trivially intersects the span of the remainder of the basis. (Contributed by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (((𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐵 ∖ {𝐸}))) | ||
Theorem | lbsind2 21097 | A basis is linearly independent; that is, every element is not in the span of the remainder of the basis. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 12-Jan-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 1 = (1r‘𝐹) & ⊢ 0 = (0g‘𝐹) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐸 ∈ 𝐵) → ¬ 𝐸 ∈ (𝑁‘(𝐵 ∖ {𝐸}))) | ||
Theorem | lbspss 21098 | No proper subset of a basis spans the space. (Contributed by Mario Carneiro, 25-Jun-2014.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 1 = (1r‘𝐹) & ⊢ 0 = (0g‘𝐹) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 1 ≠ 0 ) ∧ 𝐵 ∈ 𝐽 ∧ 𝐶 ⊊ 𝐵) → (𝑁‘𝐶) ≠ 𝑉) | ||
Theorem | lsmcl 21099 | The sum of two subspaces is a subspace. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆) | ||
Theorem | lsmspsn 21100* | Member of subspace sum of spans of singletons. (Contributed by NM, 8-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑈 ∈ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ↔ ∃𝑗 ∈ 𝐾 ∃𝑘 ∈ 𝐾 𝑈 = ((𝑗 · 𝑋) + (𝑘 · 𝑌)))) |
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