P. 343 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34201-34300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	selvply1rhmlem3 34201*	Lemma for selvply1rhm 34204. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ (ℕ0 ↑m 1o))    ⇒   ⊢ (𝜑 → ((𝐻‘𝐹)‘𝑁) = ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)‘{⟨𝑋, (𝑁‘∅)⟩}))
Theorem	selvply1rhmlem4 34202*	Lemma for selvply1rhm 34204: The mapping 𝐻 is linear. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘(𝐹(+g‘𝑃)𝐺)) = ((𝐻‘𝐹)(+g‘𝑄)(𝐻‘𝐺)))
Theorem	selvply1rhmlem5 34203*	Lemma for selvply1rhm 34204. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝑀 = (𝑞 ∈ (Base‘({𝑋} mPoly 𝑈)) ↦ (𝑠 ∈ (ℕ0 ↑m 1o) ↦ (𝑞‘{⟨𝑋, (𝑠‘∅)⟩})))    ⇒   ⊢ (𝜑 → (𝐻‘𝐹) = (𝑀‘(((𝐼 selectVars 𝑅)‘{𝑋})‘𝐹)))
Theorem	selvply1rhm 34204*	Build a ring homomorphism 𝐻 between the multivariate polynomials 𝑃 with variables in 𝐼 and the univariate polynomials 𝑄 in a single variable 𝑋 element of 𝐼. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (𝑃 RingHom 𝑄))
Theorem	selvply1rhm0 34205*	The ring homomorphism 𝐻 built in selvply1rhm 34204 is injective. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑈 = ((𝐼 ∖ {𝑋}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((((𝐼 selectVars 𝑅)‘{𝑋})‘𝑓)‘{⟨𝑋, (𝑛‘∅)⟩})))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 0 = (0g‘𝑄)    &   ⊢ 𝑍 = (0g‘𝑃)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐻‘𝐹) = 0 )    ⇒   ⊢ (𝜑 → 𝐹 = 𝑍)
Theorem	mplidomlem 34206*	Lemma for mplidom 34207. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ 𝐻 = (𝑓 ∈ 𝐶 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (((((𝑗 ∪ {𝑥}) selectVars 𝑅)‘{𝑥})‘𝑓)‘{⟨𝑥, (𝑛‘∅)⟩})))    &   ⊢ 𝐶 = (Base‘𝑆)    &   ⊢ 𝑆 = ((𝑗 ∪ {𝑥}) mPoly 𝑅)    &   ⊢ 𝑈 = (((𝑗 ∪ {𝑥}) ∖ {𝑥}) mPoly 𝑅)    &   ⊢ 𝑄 = (Poly1‘𝑈)    ⇒   ⊢ (𝜑 → 𝑃 ∈ IDomn)
Theorem	mplidom 34207	The multivariate polynomials over an integral domain form an integral domain. See ply1idom 26449. (Contributed by Thierry Arnoux, 4-May-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑃 ∈ IDomn)
Syntax	cextv 34208	Extend class notation with the "variable extension" function.
class extendVars
Definition	df-extv 34209*	Define the "variable extension" function. The function ((𝐼extendVars𝑅)‘𝐴) converts polynomials with variables indexed by (𝐼 ∖ {𝐴}) into polynomials indexed by 𝐼, and therefore maps elements of ((𝐼 ∖ {𝐴}) mPoly 𝑅) onto (𝐼 mPoly 𝑅). (Contributed by Thierry Arnoux, 20-Jan-2026.)
⊢ extendVars = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑎 ∈ 𝑖 ↦ (𝑓 ∈ (Base‘((𝑖 ∖ {𝑎}) mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝑖 ∖ {𝑎}))), (0g‘𝑟))))))
Theorem	extvval 34210*	Value of the "variable extension" function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑎})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   ⊢ (𝜑 → (𝐼extendVars𝑅) = (𝑎 ∈ 𝐼 ↦ (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝑎) = 0, (𝑓‘(𝑥 ↾ (𝐼 ∖ {𝑎}))), 0 )))))
Theorem	extvfval 34211*	The "variable extension" function evaluated for adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴) = (𝑓 ∈ 𝑀 ↦ (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝑓‘(𝑥 ↾ 𝐽)), 0 ))))
Theorem	extvfv 34212*	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) = (𝑥 ∈ 𝐷 ↦ if((𝑥‘𝐴) = 0, (𝐹‘(𝑥 ↾ 𝐽)), 0 )))
Theorem	extvfvv 34213*	The "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) = if((𝑋‘𝐴) = 0, (𝐹‘(𝑋 ↾ 𝐽)), 0 ))
Theorem	extvfvvcl 34214*	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((((𝐼extendVars𝑅)‘𝐴)‘𝐹)‘𝑋) ∈ 𝐵)
Theorem	extvfvcl 34215*	Closure for the "variable extension" function evaluated for converting a given polynomial 𝐹 by adding a variable with index 𝐴. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝐴)‘𝐹) ∈ 𝑁)
Theorem	extvfvalf 34216*	The "variable extension" function maps polynomials with variables indexed in 𝐽 to polynomials with variables indexed in 𝐼. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝐴})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐼)    &   ⊢ 𝑁 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼extendVars𝑅)‘𝐴):𝑀⟶𝑁)
Theorem	mvrvalind 34217*	Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋})    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))
Theorem	mplmulmvr 34218*	Multiply a polynomial 𝐹 with a variable 𝑋 (i.e. with a monic monomial). (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑋 = ((𝐼 mVar 𝑅)‘𝑌)    &   ⊢ 𝑀 = (Base‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑌})    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    ⇒   ⊢ (𝜑 → (𝑋 · 𝐹) = (𝑏 ∈ 𝐷 ↦ if((𝑏‘𝑌) = 0, 0 , (𝐹‘(𝑏 ∘f − 𝐴)))))
Theorem	evlscaval 34219	Polynomial evaluation for scalars. See evlsscaval 22441. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐿:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐴‘𝑋))‘𝐿) = 𝑋)
Theorem	evlvarval 34220	Polynomial evaluation builder for a variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑆)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑆)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑍)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ 𝑉 = (𝐼 mVar 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑉‘𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉‘𝑋))‘𝐴) = (𝐴‘𝑋)))
Theorem	evlextv 34221	Evaluating a variable-extended polynomial is the same as evaluating the polynomial in the original set of variables (in both cases, the additionial variable is ignored). (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (𝐼extendVars𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))
Theorem	mplvrpmlem 34222*	Lemma for mplvrpmga 34224 and others. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    &   ⊢ (𝜑 → 𝑋 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})    ⇒   ⊢ (𝜑 → (𝑋 ∘ 𝐷) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
Theorem	mplvrpmfgalem 34223*	Permuting variables in a multivariate polynomial conserves finite support. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑃)    ⇒   ⊢ (𝜑 → (𝑄𝐴𝐹) finSupp 0 )
Theorem	mplvrpmga 34224*	The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))
Theorem	mplvrpmmhm 34225*	The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑊 MndHom 𝑊))
Theorem	mplvrpmrhm 34226*	The action of permuting variables in a multivariate polynomial is a ring homomorphism. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑓 ∈ 𝑀 ↦ (𝐷𝐴𝑓))    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑊 RingHom 𝑊))
Theorem	psrgsum 34227*	Finite commutative sums of power series are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
Theorem	psrmon 34228*	A monomial is a power series. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) ∈ 𝐵)
Theorem	psrmonmul 34229*	The product of two power series monomials adds the exponent vectors together. For example, the product of (𝑥↑2)(𝑦↑2) with (𝑦↑1)(𝑧↑3) is (𝑥↑2)(𝑦↑3)(𝑧↑3), where the exponent vectors ⟨2, 2, 0⟩ and ⟨0, 1, 3⟩ are added to give ⟨2, 3, 3⟩. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 1 , 0 )) · (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 1 , 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), 1 , 0 )))
Theorem	psrmonmul2 34230*	The product of two power series monomials adds the exponent vectors together. Here, the function 𝐺 is a monomial builder, which maps a bag of variables with the monic monomial with only those variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐷)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → ((𝐺‘𝑋) · (𝐺‘𝑌)) = (𝐺‘(𝑋 ∘f + 𝑌)))
Theorem	psrmonprod 34231*	Finite product of bags of variables in a power series. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
Theorem	mplgsum 34232*	Finite commutative sums of polynomials are taken componentwise. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝑃 Σg 𝐹) = (𝑦 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘)‘𝑦)))))
Theorem	mplmonprod 34233*	Finite product of monomials. Here the function 𝐺 maps a bag of variables to the corresponding monomial. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐷)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑃)    &   ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 = 𝑦, 1 , 0 )))    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝐺 ∘ 𝐹)) = (𝐺‘(𝑖 ∈ 𝐼 ↦ (ℂfld Σg (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)‘𝑖))))))
21.4.10.50  The ring of symmetric polynomials
Syntax	csply 34234	Extend class notation with the symmetric polynomials.
class SymPoly
Syntax	cesply 34235	Extend class notation with the elementary symmetric polynomials.
class eSymPoly
Definition	df-sply 34236*	Define symmetric polynomials. See splyval 34238 for a more readable expression. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ SymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((Base‘(𝑖 mPoly 𝑟))FixPts(𝑑 ∈ (Base‘(SymGrp‘𝑖)), 𝑓 ∈ (Base‘(𝑖 mPoly 𝑟)) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))))
Definition	df-esply 34237*	Define elementary symmetric polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ eSymPoly = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑟) ∘ ((𝟭‘{ℎ ∈ (ℕ0 ↑m 𝑖) ∣ ℎ finSupp 0})‘((𝟭‘𝑖) “ {𝑐 ∈ 𝒫 𝑖 ∣ (♯‘𝑐) = 𝑘})))))
Theorem	splyval 34238*	The symmetric polynomials for a given index 𝐼 of variables and base ring 𝑅. These are the fixed points of the action 𝐴 which permutes variables. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐼SymPoly𝑅) = (𝑀FixPts𝐴))
Theorem	splysubrg 34239*	The symmetric polynomials form a subring of the ring of polynomials. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐼SymPoly𝑅) ∈ (SubRing‘(𝐼 mPoly 𝑅)))
Theorem	issply 34240*	Conditions for being a symmetric polynomial. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝑆 = (SymGrp‘𝐼)    &   ⊢ 𝑃 = (Base‘𝑆)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑀)    &   ⊢ (((𝜑 ∧ 𝑝 ∈ 𝑃) ∧ 𝑥 ∈ 𝐷) → (𝐹‘(𝑥 ∘ 𝑝)) = (𝐹‘𝑥))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝐼SymPoly𝑅))
Theorem	esplyval 34241*	The elementary polynomials for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐼eSymPoly𝑅) = (𝑘 ∈ ℕ0 ↦ ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝑘})))))
Theorem	esplyfval 34242*	The 𝐾-th elementary polynomial for a given index 𝐼 of variables and base ring 𝑅. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = ((ℤRHom‘𝑅) ∘ ((𝟭‘𝐷)‘((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}))))
Theorem	esplyfval0 34243	The 0-th elementary symmetric polynomial is the constant 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝑈 = (1r‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘0) = 𝑈)
Theorem	esplyfval2 34244*	When 𝐾 is out-of-bounds, the 𝐾-th elementary symmetric polynomial is zero. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (ℕ0 ∖ (0...(♯‘𝐼))))    &   ⊢ 𝑍 = (0g‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = 𝑍)
Theorem	esplylem 34245*	Lemma for esplyfv 34249 and others. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    ⇒   ⊢ (𝜑 → ((𝟭‘𝐼) “ {𝑐 ∈ 𝒫 𝐼 ∣ (♯‘𝑐) = 𝐾}) ⊆ 𝐷)
Theorem	esplympl 34246*	Elementary symmetric polynomials are polynomials. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ 𝑀)
Theorem	esplymhp 34247*	The 𝐾-th elementary symmetric polynomial is homogeneous of degree 𝐾. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑅)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐻‘𝐾))
Theorem	esplyfv1 34248*	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹 where variables are not raised to a power. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → ran 𝐹 ⊆ {0, 1})    ⇒   ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((♯‘(𝐹 supp 0)) = 𝐾, 1 , 0 ))
Theorem	esplyfv 34249*	Coefficient for the 𝐾-th elementary symmetric polynomial and a bag of variables 𝐹: the coefficient is 1 for the bags of exactly 𝐾 variables, having exponent at most 1. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐷)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → (((𝐼eSymPoly𝑅)‘𝐾)‘𝐹) = if((ran 𝐹 ⊆ {0, 1} ∧ (♯‘(𝐹 supp 0)) = 𝐾), 1 , 0 ))
Theorem	esplysply 34250*	The 𝐾-th elementary symmetric polynomial is symmetric. (Contributed by Thierry Arnoux, 18-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐼)))    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) ∈ (𝐼SymPoly𝑅))
Theorem	esplyfval3 34251*	Alternate expression for the value of the 𝐾-th elementary symmetric polynomial. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ0)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (𝑓 ∈ 𝐷 ↦ if((ran 𝑓 ⊆ {0, 1} ∧ (♯‘(𝑓 supp 0)) = 𝐾), 1 , 0 )))
Theorem	esplyfval1 34252	The first elementary symmetric polynomial is the sum of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐸‘1) = (𝑊 Σg 𝑉))
Theorem	esplyfvaln 34253	The last elementary symmetric polynomial is the product of all variables. (Contributed by Thierry Arnoux, 16-Mar-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ 𝑁 = (♯‘𝐼)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    ⇒   ⊢ (𝜑 → (𝐸‘𝑁) = (𝑀 Σg 𝑉))
Theorem	esplyind 34254*	A recursive formula for the elementary symmetric polynomials. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑉 = (𝐼 mVar 𝑅)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = (.r‘𝑊)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}    &   ⊢ 𝐺 = ((𝐼extendVars𝑅)‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝐸 = (𝐽eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐾 ∈ (1...(♯‘𝐼)))    &   ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}    ⇒   ⊢ (𝜑 → ((𝐼eSymPoly𝑅)‘𝐾) = (((𝑉‘𝑌) · (𝐺‘(𝐸‘(𝐾 − 1)))) + (𝐺‘(𝐸‘𝐾))))
Theorem	esplyindfv 34255*	A recursive formula for the elementary symmetric polynomials, evaluated at a given set of points 𝑍. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ 𝐸 = (𝐽eSymPoly𝑅)    &   ⊢ (𝜑 → 𝐾 ∈ (0...(♯‘𝐽)))    &   ⊢ 𝐶 = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}    &   ⊢ 𝐹 = ((𝐼eSymPoly𝑅)‘(𝐾 + 1))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝑍) = (((𝑍‘𝑌) · ((𝑂‘(𝐸‘𝐾))‘(𝑍 ↾ 𝐽))) + ((𝑂‘(𝐸‘(𝐾 + 1)))‘(𝑍 ↾ 𝐽))))
Theorem	esplyfvn 34256	Express the last elementary symmetric polynomial, evaluated at a given set of points 𝑍, in terms of the last elementary symmetric polynomial with one less variable. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝑂 = (𝐽 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝐹 = (𝐽eSymPoly𝑅)    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ 𝐾 = (♯‘𝐽)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → ((𝑄‘(𝐸‘𝐻))‘𝑍) = ((𝑍‘𝑌) · ((𝑂‘(𝐹‘𝐾))‘(𝑍 ↾ 𝐽))))
Theorem	vietadeg1 34257*	The degree of a product of 𝐻 of linear polynomials of the form 𝑋 − 𝑍 is 𝐻. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ 𝐷 = (deg1‘𝑅)    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = 𝐻)
Theorem	vietalem 34258*	Lemma for vieta 34259: induction step. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ (𝜑 → 𝐾 ∈ (0...𝐻))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐼)    &   ⊢ 𝐽 = (𝐼 ∖ {𝑌})    &   ⊢ (𝜑 → ∀𝑧 ∈ (𝐵 ↑m 𝐽)∀𝑘 ∈ (0...(♯‘𝐽))((coe1‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘(𝑧‘𝑛))))))‘((♯‘𝐽) − 𝑘)) = ((𝑘 ↑ (𝑁‘ 1 )) · (((𝐽 eval 𝑅)‘((𝐽eSymPoly𝑅)‘𝑘))‘𝑧)))    &   ⊢ (𝜑 → ((deg1‘𝑅)‘(𝑀 Σg (𝑛 ∈ 𝐽 ↦ (𝑋 − (𝐴‘((𝑍 ↾ 𝐽)‘𝑛)))))) = (♯‘𝐽))    ⇒   ⊢ (𝜑 → ((coe1‘𝐹)‘𝐾) = (((𝐻 − 𝐾) ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘(𝐻 − 𝐾)))‘𝑍)))
Theorem	vieta 34259*	Vieta's Formulas: Coefficients of a monic polynomial 𝐹 expressed as a product of linear polynomials of the form 𝑋 − 𝑍 can be expressed in terms of elementary symmetric polynomials. The formulas appear in Chapter 6 of [Lang], p. 190. Theorem vieta1 26641 is a special case for the complex numbers, for the case 𝐾 = 1. (Contributed by Thierry Arnoux, 15-Feb-2026.)
⊢ 𝑊 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑀 = (mulGrp‘𝑊)    &   ⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐸 = (𝐼eSymPoly𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑋 = (var1‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐻 = (♯‘𝐼)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑍:𝐼⟶𝐵)    &   ⊢ 𝐹 = (𝑀 Σg (𝑛 ∈ 𝐼 ↦ (𝑋 − (𝐴‘(𝑍‘𝑛)))))    &   ⊢ (𝜑 → 𝐾 ∈ (0...𝐻))    &   ⊢ 𝐶 = (coe1‘𝐹)    ⇒   ⊢ (𝜑 → (𝐶‘(𝐻 − 𝐾)) = ((𝐾 ↑ (𝑁‘ 1 )) · ((𝑄‘(𝐸‘𝐾))‘𝑍)))
21.4.10.51  The subring algebra
Theorem	sra1r 34260	The unity element of a subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 1 = (1r‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → 1 = (1r‘𝐴))
Theorem	sradrng 34261	Condition for a subring algebra to be a division ring. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ DivRing)
Theorem	sraidom 34262	Condition for a subring algebra to be an integral domain. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ IDomn)    &   ⊢ (𝜑 → 𝑉 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ IDomn)
Theorem	srasubrg 34263	A subring of the original structure is also a subring of the constructed subring algebra. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑊))    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐴))
Theorem	sralvec 34264	Given a sub division ring 𝐹 of a division ring 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    ⇒   ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
Theorem	srafldlvec 34265	Given a subfield 𝐹 of a field 𝐸, 𝐸 may be considered as a vector space over 𝐹, which becomes the field of scalars. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐴 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    ⇒   ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
Theorem	resssra 34266	The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝑆 = (𝑅 ↾s 𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((subringAlg ‘𝑆)‘𝐶) = (((subringAlg ‘𝑅)‘𝐶) ↾s 𝐵))
Theorem	lsssra 34267	A subring is a subspace of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
⊢ 𝑊 = ((subringAlg ‘𝑅)‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑅)    &   ⊢ 𝑆 = (𝑅 ↾s 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐶 ∈ (SubRing‘𝑆))    ⇒   ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑊))
Theorem	srapwov 34268	The "power" operation on a subring algebra. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊))    ⇒   ⊢ (𝜑 → (.g‘(mulGrp‘𝑊)) = (.g‘(mulGrp‘𝐴)))
21.4.10.52  Division Ring Extensions
Theorem	drgext0g 34269	The additive neutral element of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (0g‘𝐸) = (0g‘𝐵))
Theorem	drgextvsca 34270	The scalar multiplication operation of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘𝐵))
Theorem	drgext0gsca 34271	The additive neutral element of the scalar field of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    ⇒   ⊢ (𝜑 → (0g‘𝐵) = (0g‘(Scalar‘𝐵)))
Theorem	drgextsubrg 34272	The scalar field is a subring of a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐵))
Theorem	drgextlsp 34273	The scalar field is a subspace of a subring algebra. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑈 ∈ (LSubSp‘𝐵))
Theorem	drgextgsum 34274*	Group sum in a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.)
⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ DivRing)    &   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸))    &   ⊢ 𝐹 = (𝐸 ↾s 𝑈)    &   ⊢ (𝜑 → 𝐹 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)) = (𝐵 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)))
21.4.10.53  Vector Spaces
Theorem	lvecdimfi 34275	Finite version of lvecdim 21441 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18734, which is required in the infinite case. Suggested by Gérard Lang. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
Theorem	exsslsb 34276*	Any finite generating set 𝑆 of a vector space 𝑊 contains a basis. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑆 ∈ Fin)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐾‘𝑆) = 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ 𝐽 𝑠 ⊆ 𝑆)
Theorem	lbslelsp 34277	The size of a basis 𝑋 of a vector space 𝑊 is less than the size of a generating set 𝑌. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝐾 = (LSpan‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LVec)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐾‘𝑌) = 𝐵)    ⇒   ⊢ (𝜑 → (♯‘𝑋) ≤ (♯‘𝑌))
21.4.10.54  Vector Space Dimension
Syntax	cldim 34278	Extend class notation with the dimension of a vector space.
class dim
Definition	df-dim 34279	Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 21441, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.)
⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
Theorem	dimval 34280	The dimension of a vector space 𝐹 is the cardinality of one of its bases. (Contributed by Thierry Arnoux, 6-May-2023.)
⊢ 𝐽 = (LBasis‘𝐹)    ⇒   ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽) → (dim‘𝐹) = (♯‘𝑆))
Theorem	dimvalfi 34281	The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 34280 does not depend on the axiom of choice, but it is limited to the case where the base 𝑆 is finite. (Contributed by Thierry Arnoux, 24-May-2023.)
⊢ 𝐽 = (LBasis‘𝐹)    ⇒   ⊢ ((𝐹 ∈ LVec ∧ 𝑆 ∈ 𝐽 ∧ 𝑆 ∈ Fin) → (dim‘𝐹) = (♯‘𝑆))
Theorem	dimcl 34282	Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*)
Theorem	lmimdim 34283	Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Feb-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝑆 LMIso 𝑇))    &   ⊢ (𝜑 → 𝑆 ∈ LVec)    ⇒   ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇))
Theorem	lmicdim 34284	Module isomorphisms preserve vector space dimensions. (Contributed by Thierry Arnoux, 25-Mar-2025.)
⊢ (𝜑 → 𝑆 ≃𝑚 𝑇)    &   ⊢ (𝜑 → 𝑆 ∈ LVec)    ⇒   ⊢ (𝜑 → (dim‘𝑆) = (dim‘𝑇))
Theorem	lvecdim0i 34285	A vector space of dimension zero is reduced to its identity element. (Contributed by Thierry Arnoux, 31-Jul-2023.)
⊢ 0 = (0g‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ (dim‘𝑉) = 0) → (Base‘𝑉) = { 0 })
Theorem	lvecdim0 34286	A vector space of dimension zero is reduced to its identity element, biconditional version. (Contributed by Thierry Arnoux, 31-Jul-2023.)
⊢ 0 = (0g‘𝑉)    ⇒   ⊢ (𝑉 ∈ LVec → ((dim‘𝑉) = 0 ↔ (Base‘𝑉) = { 0 }))
Theorem	lssdimle 34287	The dimension of a linear subspace is less than or equal to the dimension of the parent vector space. This is corollary 5.4 of [Lang] p. 141. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ (LSubSp‘𝑊)) → (dim‘𝑋) ≤ (dim‘𝑊))
Theorem	dimpropd 34288*	If two structures have the same components (properties), they have the same dimension. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ (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 ∈ LVec)    &   ⊢ (𝜑 → 𝐿 ∈ LVec)    ⇒   ⊢ (𝜑 → (dim‘𝐾) = (dim‘𝐿))
Theorem	rlmdim 34289	The left vector space induced by a ring over itself has dimension 1. (Contributed by Thierry Arnoux, 5-Aug-2023.) Generalize to division rings. (Revised by SN, 22-Mar-2025.)
⊢ 𝑉 = (ringLMod‘𝐹)    ⇒   ⊢ (𝐹 ∈ DivRing → (dim‘𝑉) = 1)
Theorem	frlmdim 34290	Dimension of a free left module. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑉) → (dim‘𝐹) = (♯‘𝐼))
Theorem	tnglvec 34291	Augmenting a structure with a norm conserves left vector spaces. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐺 ∈ LVec ↔ 𝑇 ∈ LVec))
Theorem	tngdim 34292	Dimension of a left vector space augmented with a norm. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    ⇒   ⊢ ((𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉) → (dim‘𝐺) = (dim‘𝑇))
Theorem	rrxdim 34293	Dimension of the generalized Euclidean space. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐻 = (ℝ^‘𝐼)    ⇒   ⊢ (𝐼 ∈ 𝑉 → (dim‘𝐻) = (♯‘𝐼))
Theorem	matdim 34294	Dimension of the space of square matrices. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝐴 = (𝐼 Mat 𝑅)    &   ⊢ 𝑁 = (♯‘𝐼)    ⇒   ⊢ ((𝐼 ∈ Fin ∧ 𝑅 ∈ DivRing) → (dim‘𝐴) = (𝑁 · 𝑁))
Theorem	lbslsat 34295	A nonzero vector 𝑋 is a basis of a line spanned by the singleton 𝑋. Spans of nonzero singletons are sometimes called "atoms", see df-lsatoms 40127 and for example lsatlspsn 40144. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋}))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → {𝑋} ∈ (LBasis‘𝑌))
Theorem	lsatdim 34296	A line, spanned by a nonzero singleton, has dimension 1. (Contributed by Thierry Arnoux, 20-May-2023.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (𝑊 ↾s (𝑁‘{𝑋}))    ⇒   ⊢ ((𝑊 ∈ LVec ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (dim‘𝑌) = 1)
Theorem	drngdimgt0 34297	The dimension of a vector space that is also a division ring is greater than zero. (Contributed by Thierry Arnoux, 29-Jul-2023.)
⊢ ((𝐹 ∈ LVec ∧ 𝐹 ∈ DivRing) → 0 < (dim‘𝐹))
Theorem	lmhmlvec2 34298	A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
Theorem	kerlmhm 34299	The kernel of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐾 = (𝑉 ↾s (◡𝐹 “ { 0 }))    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐾 ∈ LVec)
Theorem	imlmhm 34300	The image of a vector space homomorphism is a vector space itself. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ 𝐼 = (𝑈 ↾s ran 𝐹)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝐼 ∈ LVec)