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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lspprid2 19701 | 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 19702 | 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 19703* | 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 | lspsneli 19704 | A scalar product with a vector belongs to the span of its singleton. (spansnmul 29269 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 19705* | 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 | lspsnel 19706* | Member of span of the singleton of a vector. (elspansn 29271 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 19707 | 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 19708* | Comparable spans of singletons must have proportional vectors. See lspsneq 19825 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 19709 | 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 19710 | 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 19711 | Span of the singleton of the zero vector. (spansn0 29246 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 19712 | Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) | ||
Theorem | lspuni0 19713 | 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 19714 | 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 19715 | 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 19716 | 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 19717 | 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 19718 | 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. |
⊢ 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 19719 | 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 19720* | 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 19721* | 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.) |
⊢ (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 ∈ V) & ⊢ (𝜑 → 𝐿 ∈ V) ⇒ ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) | ||
Syntax | clmhm 19722 | Extend class notation with the generator of left module hom-sets. |
class LMHom | ||
Syntax | clmim 19723 | The class of left module isomorphism sets. |
class LMIso | ||
Syntax | clmic 19724 | The class of the left module isomorphism relation. |
class ≃𝑚 | ||
Definition | df-lmhm 19725* | 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 19726* | 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 19727 | 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 19728 | Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ Rel dom LMHom | ||
Theorem | lmimfn 19729 | Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ LMIso Fn (LMod × LMod) | ||
Theorem | islmhm 19730* | 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 19731* | Property of a module homomorphism, similar to ismhm 17948. (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 19732 | 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 19733 | 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 19734 | 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 19735 | 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 19736 | 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 19737 | 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 19738 | 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 19739 | 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 19740 | 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 19741* | A one-equation proof of linearity of a left module homomorphism, similar to df-lss 19635. (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 19742* | 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 19743 | 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 19744 | 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 19745 | 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 19746 | 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 19747 | 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 19748 | 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 19749 | 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 19750 | 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 19751 | 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 19752 | 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 19753 | 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 19754 | 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 19755 | 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 19756 | 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 19757 | 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 19758 | 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 19759* | 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 19760* | 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 19761* | 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 19762* | 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 19763* | 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 19764* | 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 19765 | 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 19766 | 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 19767 | 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 19768 | 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 19769 | 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 19770 | 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 19771 | The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | ||
Theorem | brlmici 19772 | Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅 ≃𝑚 𝑆) | ||
Theorem | lmiclcl 19773 | Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) | ||
Theorem | lmicrcl 19774 | Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod) | ||
Theorem | lmicsym 19775 | Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) | ||
Theorem | lmhmpropd 19776* | 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 19777 | Extend class notation with the set of bases for a vector space. |
class LBasis | ||
Definition | df-lbs 19778* | 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 19779* | 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 19780 | 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 19781 | 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 19782 | 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 19783 | 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 19784 | 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 19785 | 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 19786 | 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 19787* | 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)))) | ||
Theorem | lsmelval2 19788* | Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧}))))) | ||
Theorem | lsmsp 19789 | Subspace sum in terms of span. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) = (𝑁‘(𝑇 ∪ 𝑈))) | ||
Theorem | lsmsp2 19790 | Subspace sum of spans of subsets is the span of their union. (spanuni 29249 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → ((𝑁‘𝑇) ⊕ (𝑁‘𝑈)) = (𝑁‘(𝑇 ∪ 𝑈))) | ||
Theorem | lsmssspx 19791 | Subspace sum (in its extended domain) is a subset of the span of the union of its arguments. (Contributed by NM, 6-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ (𝜑 → 𝑇 ⊆ 𝑉) & ⊢ (𝜑 → 𝑈 ⊆ 𝑉) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → (𝑇 ⊕ 𝑈) ⊆ (𝑁‘(𝑇 ∪ 𝑈))) | ||
Theorem | lsmpr 19792 | The span of a pair of vectors equals the sum of the spans of their singletons. (Contributed by NM, 13-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) | ||
Theorem | lsppreli 19793 | A vector expressed as a sum belongs to the span of its components. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝐵 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lsmelpr 19794 | Two ways to say that a vector belongs to the span of a pair of vectors. (Contributed by NM, 14-Jan-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})))) | ||
Theorem | lsppr0 19795 | The span of a vector paired with zero equals the span of the singleton of the vector. (Contributed by NM, 29-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 0 }) = (𝑁‘{𝑋})) | ||
Theorem | lsppr 19796* | Span of a pair of vectors. (Contributed by NM, 22-Aug-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑣 = ((𝑘 · 𝑋) + (𝑙 · 𝑌))}) | ||
Theorem | lspprel 19797* | Member of the span of a pair of vectors. (Contributed by NM, 10-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑍 ∈ (𝑁‘{𝑋, 𝑌}) ↔ ∃𝑘 ∈ 𝐾 ∃𝑙 ∈ 𝐾 𝑍 = ((𝑘 · 𝑋) + (𝑙 · 𝑌)))) | ||
Theorem | lspprabs 19798 | Absorption of vector sum into span of pair. (Contributed by NM, 27-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspvadd 19799 | The span of a vector sum is included in the span of its arguments. (Contributed by NM, 22-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspsntri 19800 | Triangle-type inequality for span of a singleton. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) |
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