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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | lmhmlem 19801 | 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 19802 | 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 19803 | 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 19804 | 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 19805 | 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 19806 | 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 19807 | 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 19808 | 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 19809 | 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 19810* | A one-equation proof of linearity of a left module homomorphism, similar to df-lss 19704. (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 19811* | 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 19812 | 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 19813 | 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 19814 | 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 19815 | 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 19816 | 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 19817 | 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 19818 | 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 19819 | 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 19820 | 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 19821 | 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 19822 | 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 19823 | 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 19824 | 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 19825 | 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 19826 | 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 19827 | 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 19828* | 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 19829* | 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 19830* | 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 19831* | 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 19832* | 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 19833* | 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 19834 | 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 19835 | 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 19836 | 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 19837 | 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 19838 | 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 19839 | 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 19840 | The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | ||
Theorem | brlmici 19841 | Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅 ≃𝑚 𝑆) | ||
Theorem | lmiclcl 19842 | Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) | ||
Theorem | lmicrcl 19843 | Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ∈ LMod) | ||
Theorem | lmicsym 19844 | Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ (𝑅 ≃𝑚 𝑆 → 𝑆 ≃𝑚 𝑅) | ||
Theorem | lmhmpropd 19845* | 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 19846 | Extend class notation with the set of bases for a vector space. |
class LBasis | ||
Definition | df-lbs 19847* | 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 19848* | 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 19849 | 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 19850 | 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 19851 | 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 19852 | 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 19853 | 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 19854 | 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 19855 | 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 19856* | 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 19857* | 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 19858 | 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 19859 | Subspace sum of spans of subsets is the span of their union. (spanuni 29321 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 19860 | 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 19861 | 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 19862 | 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 19863 | 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 19864 | 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 19865* | 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 19866* | 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 19867 | 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 19868 | 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 19869 | 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}))) | ||
Theorem | lspsntrim 19870 | Triangle-type inequality for span of a singleton of vector difference. (Contributed by NM, 25-Apr-2014.) (Revised by Mario Carneiro, 21-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ⊆ ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌}))) | ||
Theorem | lbspropd 19871* | If two structures have the same components (properties), they have the same set of bases. (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)) & ⊢ 𝐹 = (Scalar‘𝐾) & ⊢ 𝐺 = (Scalar‘𝐿) & ⊢ (𝜑 → 𝑃 = (Base‘𝐹)) & ⊢ (𝜑 → 𝑃 = (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦)) & ⊢ (𝜑 → 𝐾 ∈ V) & ⊢ (𝜑 → 𝐿 ∈ V) ⇒ ⊢ (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿)) | ||
Theorem | pj1lmhm 19872 | The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐿 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑃 = (proj1‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝐿) & ⊢ (𝜑 → 𝑈 ∈ 𝐿) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) ⇒ ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom 𝑊)) | ||
Theorem | pj1lmhm2 19873 | The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐿 = (LSubSp‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑃 = (proj1‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ∈ 𝐿) & ⊢ (𝜑 → 𝑈 ∈ 𝐿) & ⊢ (𝜑 → (𝑇 ∩ 𝑈) = { 0 }) ⇒ ⊢ (𝜑 → (𝑇𝑃𝑈) ∈ ((𝑊 ↾s (𝑇 ⊕ 𝑈)) LMHom (𝑊 ↾s 𝑇))) | ||
Syntax | clvec 19874 | Extend class notation with class of all left vector spaces. |
class LVec | ||
Definition | df-lvec 19875 | Define the class of all left vector spaces. A left vector space over a division ring is an Abelian group (vectors) together with a division ring (scalars) and a left scalar product connecting them. Some authors call this a "left module over a division ring", reserving "vector space" for those where the division ring is commutative, i.e., is a field. (Contributed by NM, 11-Nov-2013.) |
⊢ LVec = {𝑓 ∈ LMod ∣ (Scalar‘𝑓) ∈ DivRing} | ||
Theorem | islvec 19876 | The predicate "is a left vector space". (Contributed by NM, 11-Nov-2013.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) | ||
Theorem | lvecdrng 19877 | The set of scalars of a left vector space is a division ring. (Contributed by NM, 17-Apr-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) | ||
Theorem | lveclmod 19878 | A left vector space is a left module. (Contributed by NM, 9-Dec-2013.) |
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | ||
Theorem | lsslvec 19879 | 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 | lvecvs0or 19880 | If a scalar product is zero, one of its factors must be zero. (hvmul0or 28802 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 19881 | 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 19882 | 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 19883 | Cancellation law for scalar multiplication. (hvmulcan 28849 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 19884 | Cancellation law for scalar multiplication. (hvmulcan2 28850 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 19885 | 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 19886 | A nonzero scalar product does not change the span of a singleton. (spansncol 29345 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 19887 | Membership relation that implies equality of spans. (spansneleq 29347 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 19888 | 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 19889 | 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 19890 | 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 19891 | 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 19892 | 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 19893 | 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 19894* | Equal spans of singletons must have proportional vectors. See lspsnss2 19777 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 19895* | 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 19896 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 29350 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 19897 | 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 19898 | 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 19899 | 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 19900* | 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)})) |
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