P. 206 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frlmbasfsupp 20501	Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 finSupp 0 )
Theorem	frlmbasmap 20502	Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝑁 ↑𝑚 𝐼))
Theorem	frlmbasf 20503	Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝑁)
Theorem	frlmlvec 20504	The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LVec)
Theorem	frlmfibas 20505	The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑁 ↑𝑚 𝐼) = (Base‘𝐹))
Theorem	elfrlmbasn0 20506	If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ≠ ∅) → (𝑋 ∈ 𝐵 → 𝑋 ≠ ∅))
Theorem	frlmplusgval 20507	Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘𝑓 + 𝐺))
Theorem	frlmsubgval 20508	Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑀 = (-g‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘𝑓 − 𝐺))
Theorem	frlmvscafval 20509	Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ ∙ = ( ·𝑠 ‘𝑌)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋))
Theorem	frlmvplusgvalc 20510	Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    ⇒   ⊢ (𝜑 → ((𝑋 ✚ 𝑌)‘𝐽) = ((𝑋‘𝐽) + (𝑌‘𝐽)))
Theorem	frlmvscaval 20511	Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ ∙ = ( ·𝑠 ‘𝑌)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))
Theorem	frlmplusgvalb 20512*	Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    ⇒   ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))
Theorem	frlmvscavalb 20513*	Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
Theorem	frlmvplusgscavalb 20514*	Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
Theorem	frlmgsum 20515*	Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
Theorem	frlmsplit2 20516*	Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝑈)    &   ⊢ 𝑍 = (𝑅 freeLMod 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
Theorem	frlmsslss 20517*	A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (LSubSp‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ 𝐽) = (𝐽 × { 0 })}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)
Theorem	frlmsslss2 20518*	A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (LSubSp‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)
Theorem	frlmbas3 20519	An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑉 = (Base‘𝐹)    ⇒   ⊢ (((𝑅 ∈ 𝑊 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵)
Theorem	mpt2frlmd 20520*	Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.)
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   ⊢ 𝑉 = (Base‘𝐹)    &   ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))    ⇒   ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴))
Theorem	frlmip 20521*	The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑𝑚 𝐼), 𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌))
Theorem	frlmipval 20522	The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    ⇒   ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹 ∘𝑓 · 𝐺)))
Theorem	frlmphllem 20523*	Lemma for frlmphl 20524. (Contributed by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    &   ⊢ 𝑂 = (0g‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 )
Theorem	frlmphl 20524*	Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    &   ⊢ 𝑂 = (0g‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑌 ∈ PreHil)
11.1.3  Standard basis (unit vectors) According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces.
Syntax	cuvc 20525	Class of basic unit vectors for an explicit free module.
class unitVec
Definition	df-uvc 20526*	((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
Theorem	uvcfval 20527*	Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Theorem	uvcval 20528*	Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Theorem	uvcvval 20529	Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Theorem	uvcvvcl 20530	A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 })
Theorem	uvcvvcl2 20531	A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵)
Theorem	uvcvv1 20532	The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )
Theorem	uvcvv0 20533	The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )
Theorem	uvcff 20534	Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵)
Theorem	uvcf1 20535	In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵)
Theorem	uvcresum 20536	Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘𝑓 · 𝑈)))
Theorem	frlmssuvc1 20537*	A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
Theorem	frlmssuvc2 20538*	A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ (𝐼 ∖ 𝐽))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐾 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ¬ (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)
Theorem	frlmsslsp 20539*	A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐾 = (LSpan‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶)
Theorem	frlmlbs 20540	The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐽 = (LBasis‘𝐹)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽)
Theorem	frlmup1 20541*	Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇))
Theorem	frlmup2 20542*	The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌))
Theorem	frlmup3 20543*	The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    &   ⊢ 𝐾 = (LSpan‘𝑇)    ⇒   ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))
Theorem	frlmup4 20544*	Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ 𝑅 = (Scalar‘𝑇)    &   ⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚 ∘ 𝑈) = 𝐴)
Theorem	ellspd 20545*	The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝑁 = (LSpan‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ V)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹)))))
Theorem	elfilspd 20546*	Simplified version of ellspd 20545 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑁 = (LSpan‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ · = ( ·𝑠 ‘𝑀)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘𝑓 · 𝐹))))
11.1.4  Independent sets and families According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over A) if whenever we have a linear combination ∑x∈Saxx which is equal to 0, then ax = 0 for all x ∈ S", and according to the Definition in [Lang] p. 130: "a familiy {xi}i∈I of elements of M is said to be linearly independent (over A) if whenever we have a linear combination ∑i∈Iaixi = 0, then ai = 0 for all i ∈ I." These definitions correspond to the definitions df-linds 20550 and df-lindf 20549 respectively, where it is claimed that a nonzero summand can be extracted (∑i∈{I\{j}}aixi = -ajxj) and be represented as a linear combination of the remaining elements of the family.
Syntax	clindf 20547	The class relationship of independent families in a module.
class LIndF
Syntax	clinds 20548	The class generator of independent sets in a module.
class LIndS
Definition	df-lindf 20549*	An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set. This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power. We can almost define family independence as a family of unequal elements with independent range, as islindf3 20569, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring. This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 20581) and only one representation for each element of the range (islindf5 20582). (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
Definition	df-linds 20550*	An independent set is a set which is independent as a family. See also islinds3 20577 and islinds4 20578. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
Theorem	rellindf 20551	The independent-family predicate is a proper relation and can be used with brrelex1i 5406. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ Rel LIndF
Theorem	islinds 20552	Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
Theorem	linds1 20553	An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵)
Theorem	linds2 20554	An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ (𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)
Theorem	islindf 20555*	Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
Theorem	islinds2 20556*	Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))
Theorem	islindf2 20557*	Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝑁 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
Theorem	lindff 20558	Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵)
Theorem	lindfind 20559	A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind 20560	A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    &   ⊢ 0 = (0g‘𝐿)    &   ⊢ 𝐾 = (Base‘𝐿)    ⇒   ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
Theorem	lindfind2 20561	In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
Theorem	lindsind2 20562	In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))
Theorem	lindff1 20563	A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹–1-1→𝐵)
Theorem	lindfrn 20564	The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))
Theorem	f1lindf 20565	Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊)
Theorem	lindfres 20566	Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊)
Theorem	lindsss 20567	Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊))
Theorem	f1linds 20568	A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊)
Theorem	islindf3 20569	In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐿 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊))))
Theorem	lindfmm 20570	Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺 ∘ 𝐹) LIndF 𝑇))
Theorem	lindsmm 20571	Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ⊆ 𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺 “ 𝐹) ∈ (LIndS‘𝑇)))
Theorem	lindsmm2 20572	The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝐶 = (Base‘𝑇)    ⇒   ⊢ ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵–1-1→𝐶 ∧ 𝐹 ∈ (LIndS‘𝑆)) → (𝐺 “ 𝐹) ∈ (LIndS‘𝑇))
Theorem	lsslindf 20573	Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ ran 𝐹 ⊆ 𝑆) → (𝐹 LIndF 𝑋 ↔ 𝐹 LIndF 𝑊))
Theorem	lsslinds 20574	Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝑈 = (LSubSp‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s 𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝑈 ∧ 𝐹 ⊆ 𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊)))
Theorem	islbs4 20575	A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) ‘𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    ⇒   ⊢ (𝑋 ∈ 𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾‘𝑋) = 𝐵))
Theorem	lbslinds 20576	A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ 𝐽 ⊆ (LIndS‘𝑊)
Theorem	islinds3 20577	A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ 𝑋 = (𝑊 ↾s (𝐾‘𝑌))    &   ⊢ 𝐽 = (LBasis‘𝑋)    ⇒   ⊢ (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌 ∈ 𝐽))
Theorem	islinds4 20578*	A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏 ∈ 𝐽 𝑌 ⊆ 𝑏))
11.1.5  Characterization of free modules
Theorem	lmimlbs 20579	The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑆)    &   ⊢ 𝐾 = (LBasis‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵 ∈ 𝐽) → (𝐹 “ 𝐵) ∈ 𝐾)
Theorem	lmiclbs 20580	Having a basis is an isomorphism invariant. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑆)    &   ⊢ 𝐾 = (LBasis‘𝑇)    ⇒   ⊢ (𝑆 ≃𝑚 𝑇 → (𝐽 ≠ ∅ → 𝐾 ≠ ∅))
Theorem	islindf4 20581*	A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (0g‘𝑅)    &   ⊢ 𝐿 = (Base‘(𝑅 freeLMod 𝐼))    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐿 ((𝑊 Σg (𝑥 ∘𝑓 · 𝐹)) = 0 → 𝑥 = (𝐼 × {𝑌}))))
Theorem	islindf5 20582*	A family is independent iff the linear combinations homomorphism is injective. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐶)    ⇒   ⊢ (𝜑 → (𝐴 LIndF 𝑇 ↔ 𝐸:𝐵–1-1→𝐶))
Theorem	indlcim 20583*	An independent, spanning family extends to an isomorphism from a free module. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐶 = (Base‘𝑇)    &   ⊢ · = ( ·𝑠 ‘𝑇)    &   ⊢ 𝑁 = (LSpan‘𝑇)    &   ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘𝑓 · 𝐴)))    &   ⊢ (𝜑 → 𝑇 ∈ LMod)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇))    &   ⊢ (𝜑 → 𝐴:𝐼–onto→𝐽)    &   ⊢ (𝜑 → 𝐴 LIndF 𝑇)    &   ⊢ (𝜑 → (𝑁‘𝐽) = 𝐶)    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMIso 𝑇))
Theorem	lbslcic 20584	A module with a basis is isomorphic to a free module with the same cardinality. (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐽 ∧ 𝐼 ≈ 𝐵) → 𝑊 ≃𝑚 (𝐹 freeLMod 𝐼))
Theorem	lmisfree 20585*	A module has a basis iff it is isomorphic to a free module. In settings where isomorphic objects are not distinguished, it is common to define "free module" as any module with a basis; thus for instance lbsex 19562 might be described as "every vector space is free". (Contributed by Stefan O'Rear, 26-Feb-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → (𝐽 ≠ ∅ ↔ ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘)))
Theorem	lvecisfrlm 20586*	Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → ∃𝑘 𝑊 ≃𝑚 (𝐹 freeLMod 𝑘))
Theorem	lmimco 20587	The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 LMIso 𝑇))
Theorem	lmictra 20588	Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ≃𝑚 𝑆 ∧ 𝑆 ≃𝑚 𝑇) → 𝑅 ≃𝑚 𝑇)
Theorem	uvcf1o 20589	In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1-onto→ran 𝑈)
Theorem	uvcendim 20590	In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝐼 ≈ ran 𝑈)
Theorem	frlmisfrlm 20591	A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌 ∧ 𝐼 ≈ 𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽))
Theorem	frlmiscvec 20592	Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.)
⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅})))
11.2  Matrices According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 20490) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 20507) and scalar multiplication (see frlmvscafval 20509) for free modules. Actually, there isn't a definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 20594. By this, a statement like "Then the set of m x n matrices in R is a module (i.e. an R-module)" as in [Lang] p. 504 follows immediately from frlmlmod 20492. However, for square matrices there is the definition df-mat 20618, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication. A "usual" matrix (aij), (i= 1,..., m and j = 1,... n) would be represented as element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))), and a square matrix (aij), (i= 1,..., n and j = 1,... n) would be represented as element of (the base set of) ((1...𝑛) Mat 𝑅). Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which is excluded in the definition of many authors, e.g. in [Lang] p. 503. It is shown in mat0dimbas0 20677 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). The determinant is also defined for such an empty matrix, see mdet0pr 20803.
11.2.1  The matrix multiplication This section is about the multiplication of m x n matrices.
Syntax	cmmul 20593	Syntax for the matrix multiplication operator.
class maMul
Definition	df-mamu 20594*	The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
⊢ maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ ⦋(1st ‘(1st ‘𝑜)) / 𝑚⦌⦋(2nd ‘(1st ‘𝑜)) / 𝑛⦌⦋(2nd ‘𝑜) / 𝑝⦌(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖 ∈ 𝑚, 𝑘 ∈ 𝑝 ↦ (𝑟 Σg (𝑗 ∈ 𝑛 ↦ ((𝑖𝑥𝑗)(.r‘𝑟)(𝑗𝑦𝑘)))))))
Theorem	mamufval 20595*	Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)) ↦ (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
Theorem	mamuval 20596*	Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → (𝑋𝐹𝑌) = (𝑖 ∈ 𝑀, 𝑘 ∈ 𝑃 ↦ (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
Theorem	mamufv 20597*	A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
Theorem	mamudm 20598	The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))
Theorem	mamufacex 20599	Every solution of the equation 𝐴∗𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   ⊢ 𝐵 = (Base‘𝐸)    &   ⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   ⊢ 𝐶 = (Base‘𝐹)    &   ⊢ × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐺 = (𝑅 freeLMod (𝑀 × 𝑃))    &   ⊢ 𝐷 = (Base‘𝐺)    ⇒   ⊢ (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅 ∈ 𝑉 ∧ 𝑌 ∈ 𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌 → 𝑍 ∈ 𝐶))
Theorem	mamures 20600	Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   ⊢ 𝐺 = (𝑅 maMul ⟨𝐼, 𝑁, 𝑃⟩)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 ∈ Fin)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑃 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ⊆ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑𝑚 (𝑀 × 𝑁)))    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑𝑚 (𝑁 × 𝑃)))    ⇒   ⊢ (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌))