P. 317 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	grplsmid 31601	The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ ⊕ = (LSSum‘𝐺)    ⇒   ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝐴) → ({𝑋} ⊕ 𝐴) = 𝐴)
20.3.9.25  The quotient map
Theorem	quslsm 31602	Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} ⊕ 𝑆))
Theorem	qusima 31603*	The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝐻 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝐺))    ⇒   ⊢ (𝜑 → (𝐸‘𝐻) = (𝐹 “ 𝐻))
Theorem	nsgqus0 31604	A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    ⇒   ⊢ ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁 ∈ 𝐹)
Theorem	nsgmgclem 31605*	Lemma for nsgmgc 31606. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝑄))    ⇒   ⊢ (𝜑 → {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))
Theorem	nsgmgc 31606*	There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ 𝐽 = (𝑉MGalConn𝑊)    &   ⊢ 𝑉 = (toInc‘𝑆)    &   ⊢ 𝑊 = (toInc‘𝑇)    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐸𝐽𝐹)
Theorem	nsgqusf1olem1 31607*	Lemma for nsgqusf1o 31610. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (((𝜑 ∧ ℎ ∈ (SubGrp‘𝐺)) ∧ 𝑁 ⊆ ℎ) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ 𝑇)
Theorem	nsgqusf1olem2 31608*	Lemma for nsgqusf1o 31610. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → ran 𝐸 = 𝑇)
Theorem	nsgqusf1olem3 31609*	Lemma for nsgqusf1o 31610. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → ran 𝐹 = 𝑆)
Theorem	nsgqusf1o 31610*	The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ}    &   ⊢ 𝑇 = (SubGrp‘𝑄)    &   ⊢ ≤ = (le‘(toInc‘𝑆))    &   ⊢ ≲ = (le‘(toInc‘𝑇))    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ ⊕ = (LSSum‘𝐺)    &   ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)))    &   ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓})    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇)
20.3.9.26  Ideals
Theorem	intlidl 31611	The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → ∩ 𝐶 ∈ (LIdeal‘𝑅))
Theorem	rhmpreimaidl 31612	The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝐼)
Theorem	kerlidl 31613	The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → (◡𝐹 “ { 0 }) ∈ 𝐼)
Theorem	0ringidl 31614	The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }})
Theorem	elrspunidl 31615*	Elementhood to the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝑁 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑆 ⊆ (LIdeal‘𝑅))    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝑁‘∪ 𝑆) ↔ ∃𝑎 ∈ (𝐵 ↑m 𝑆)(𝑎 finSupp 0 ∧ 𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘 ∈ 𝑆 (𝑎‘𝑘) ∈ 𝑘)))
Theorem	lidlincl 31616	Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑈) → (𝐼 ∩ 𝐽) ∈ 𝑈)
Theorem	idlinsubrg 31617	The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    &   ⊢ 𝑉 = (LIdeal‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼 ∈ 𝑈) → (𝐼 ∩ 𝐴) ∈ 𝑉)
Theorem	rhmimaidl 31618	The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 𝑇 = (LIdeal‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑆)    ⇒   ⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵 ∧ 𝐼 ∈ 𝑇) → (𝐹 “ 𝐼) ∈ 𝑈)
20.3.9.27  Prime Ideals
Syntax	cprmidl 31619	Extend class notation with the class of prime ideals.
class PrmIdeal
Definition	df-prmidl 31620*	Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵 ⊆ 𝐼 for ideals 𝐴 and 𝐵, either 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 31625 and isprmidlc 31632. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
⊢ PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑟)𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	prmidlval 31621*	The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))})
Theorem	isprmidl 31622*	The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
Theorem	prmidlnr 31623	A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ≠ 𝐵)
Theorem	prmidl 31624*	The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥 ∈ 𝐼 ∀𝑦 ∈ 𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	prmidl2 31625*	A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 36237 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))
Theorem	idlmulssprm 31626	Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
⊢ × = (LSSum‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)    ⇒   ⊢ (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
Theorem	pridln1 31627	A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ¬ 1 ∈ 𝐼)
Theorem	prmidlidl 31628	A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))
Theorem	prmidlssidl 31629	Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ (𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))
Theorem	lidlnsg 31630	An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
Theorem	cringm4 31631	Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))
Theorem	isprmidlc 31632*	The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃 ≠ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥 ∈ 𝑃 ∨ 𝑦 ∈ 𝑃)))))
Theorem	prmidlc 31633	Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ 𝐵 ∧ 𝐽 ∈ 𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼 ∈ 𝑃 ∨ 𝐽 ∈ 𝑃))
Theorem	0ringprmidl 31634	The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅)
Theorem	prmidl0 31635	The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)
Theorem	rhmpreimaprmidl 31636	The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.)
⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (◡𝐹 “ 𝐽) ∈ 𝑃)
Theorem	qsidomlem1 31637	If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
Theorem	qsidomlem2 31638	A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
Theorem	qsidom 31639	An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))
20.3.9.28  Maximal Ideals
Syntax	cmxidl 31640	Extend class notation with the class of maximal ideals.
class MaxIdeal
Definition	df-mxidl 31641*	Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = (Base‘𝑟))))})
Theorem	mxidlval 31642*	The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝐵)))})
Theorem	ismxidl 31643*	The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀 ≠ 𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝐵)))))
Theorem	mxidlidl 31644	A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
Theorem	mxidlnr 31645	A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ 𝐵)
Theorem	mxidlmax 31646	A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝐵))
Theorem	mxidln1 31647	One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1 ∈ 𝑀)
Theorem	mxidlnzr 31648	A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)
Theorem	mxidlprm 31649	Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ × = (LSSum‘(mulGrp‘𝑅))    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))
Theorem	ssmxidllem 31650*	The set 𝑃 used in the proof of ssmxidl 31651 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝 ≠ 𝐵 ∧ 𝐼 ⊆ 𝑝)}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   ⊢ (𝜑 → 𝐼 ≠ 𝐵)    &   ⊢ (𝜑 → 𝑍 ⊆ 𝑃)    &   ⊢ (𝜑 → 𝑍 ≠ ∅)    &   ⊢ (𝜑 → [⊊] Or 𝑍)    ⇒   ⊢ (𝜑 → ∪ 𝑍 ∈ 𝑃)
Theorem	ssmxidl 31651*	Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ 𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼 ⊆ 𝑚)
Theorem	krull 31652*	Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ (𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))
Theorem	mxidlnzrb 31653*	A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
⊢ (𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))
20.3.9.29  The semiring of ideals of a ring
Syntax	cidlsrg 31654	Extend class notation with the semiring of ideals of a ring.
class IDLsrg
Definition	df-idlsrg 31655*	Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ IDLsrg = (𝑟 ∈ V ↦ ⦋(LIdeal‘𝑟) / 𝑏⦌({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝑏, 𝑗 ∈ 𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝑏 ↦ {𝑗 ∈ 𝑏 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏 ∧ 𝑖 ⊆ 𝑗)}⟩}))
Theorem	idlsrgstr 31656	A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ≤ ⟩})    ⇒   ⊢ 𝑊 Struct ⟨1, ;10⟩
Theorem	idlsrgval 31657*	Lemma for idlsrgbas 31658 through idlsrgtset 31662. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ⊗ = (LSSum‘𝐺)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⊕ ⟩, ⟨(.r‘ndx), (𝑖 ∈ 𝐼, 𝑗 ∈ 𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼 ∧ 𝑖 ⊆ 𝑗)}⟩}))
Theorem	idlsrgbas 31658	Baae of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐼 = (Base‘𝑆))
Theorem	idlsrgplusg 31659	Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → ⊕ = (+g‘𝑆))
Theorem	idlsrg0g 31660	The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → { 0 } = (0g‘𝑆))
Theorem	idlsrgmulr 31661*	Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ ⊗ = (LSSum‘𝐺)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑖 ∈ 𝐵, 𝑗 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 ⊗ 𝑗))) = (.r‘𝑆))
Theorem	idlsrgtset 31662*	Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐼 = (LIdeal‘𝑅)    &   ⊢ 𝐽 = ran (𝑖 ∈ 𝐼 ↦ {𝑗 ∈ 𝐼 ∣ ¬ 𝑖 ⊆ 𝑗})    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐽 = (TopSet‘𝑆))
Theorem	idlsrgmulrval 31663	Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (LSSum‘𝐺)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽)))
Theorem	idlsrgmulrcl 31664	Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ∈ 𝐵)
Theorem	idlsrgmulrss1 31665	In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐼)
Theorem	idlsrgmulrss2 31666	The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ 𝐽)
Theorem	idlsrgmulrssin 31667	In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    &   ⊢ 𝐵 = (LIdeal‘𝑅)    &   ⊢ ⊗ = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼 ⊗ 𝐽) ⊆ (𝐼 ∩ 𝐽))
Theorem	idlsrgmnd 31668	The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑆 ∈ Mnd)
Theorem	idlsrgcmnd 31669	The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝑆 = (IDLsrg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑆 ∈ CMnd)
20.3.9.30  Unique factorization domains
Syntax	cufd 31670	Class of unique factorization domains.
class UFD
Definition	df-ufd 31671*	Define the class of unique factorization domains. A unique factorization domain (UFD for short), is a commutative ring with an absolute value (from abvtriv 20110 this is equivalent to being a domain) such that every prime ideal contains a prime element (this is a characterization due to Irving Kaplansky). A UFD is sometimes also called a "factorial ring" following the terminology of Bourbaki. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ UFD = {𝑟 ∈ CRing ∣ ((AbsVal‘𝑟) ≠ ∅ ∧ ∀𝑖 ∈ (PrmIdeal‘𝑟)(𝑖 ∩ (RPrime‘𝑟)) ≠ ∅)}
Theorem	isufd 31672*	The property of being a Unique Factorization Domain. (Contributed by Thierry Arnoux, 1-Jun-2024.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐼 = (PrmIdeal‘𝑅)    &   ⊢ 𝑃 = (RPrime‘𝑅)    ⇒   ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ CRing ∧ (𝐴 ≠ ∅ ∧ ∀𝑖 ∈ 𝐼 (𝑖 ∩ 𝑃) ≠ ∅)))
Theorem	rprmval 31673*	The prime elements of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (RPrime‘𝑅) = {𝑝 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑝 ∥ (𝑥 · 𝑦) → (𝑝 ∥ 𝑥 ∨ 𝑝 ∥ 𝑦))})
Theorem	isrprm 31674*	Property for 𝑃 to be a prime element in the ring 𝑅. (Contributed by Thierry Arnoux, 1-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑃 ∈ (RPrime‘𝑅) ↔ (𝑃 ∈ (𝐵 ∖ (𝑈 ∪ { 0 })) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑃 ∥ (𝑥 · 𝑦) → (𝑃 ∥ 𝑥 ∨ 𝑃 ∥ 𝑦)))))
20.3.9.31  Associative algebras
Theorem	asclmulg 31675	Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ ∗ = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐾) → (𝐴‘(𝑁 ∗ 𝑋)) = (𝑁 ↑ (𝐴‘𝑋)))
20.3.9.32  Univariate Polynomials
Theorem	fply1 31676	Conditions for a function to be an univariate polynomial. (Contributed by Thierry Arnoux, 19-Aug-2023.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑃 = (Base‘(Poly1‘𝑅))    &   ⊢ (𝜑 → 𝐹:(ℕ0 ↑m 1o)⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑃)
Theorem	ply1scleq 31677	Equality of a constant polynomial is the same as equality of the constant term. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴‘𝐸) = (𝐴‘𝐹) ↔ 𝐸 = 𝐹))
Theorem	ply1chr 31678	The characteristic of a polynomial ring is the characteristic of the underlying ring. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    ⇒   ⊢ (𝑅 ∈ CRing → (chr‘𝑃) = (chr‘𝑅))
Theorem	ply1fermltl 31679	Fermat's little theorem for polynomials. If 𝑃 is prime, Then (𝑋 + 𝐴)↑𝑃 = ((𝑋↑𝑃) + 𝐴) modulo 𝑃. (Contributed by Thierry Arnoux, 24-Jul-2024.)
⊢ 𝑍 = (ℤ/nℤ‘𝑃)    &   ⊢ 𝑊 = (Poly1‘𝑍)    &   ⊢ 𝑋 = (var1‘𝑍)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑁 = (mulGrp‘𝑊)    &   ⊢ ↑ = (.g‘𝑁)    &   ⊢ 𝐶 = (algSc‘𝑊)    &   ⊢ 𝐴 = (𝐶‘((ℤRHom‘𝑍)‘𝐸))    &   ⊢ (𝜑 → 𝑃 ∈ ℙ)    &   ⊢ (𝜑 → 𝐸 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑃 ↑ (𝑋 + 𝐴)) = ((𝑃 ↑ 𝑋) + 𝐴))
20.3.9.33  The subring algebra
Theorem	sra1r 31680	The multiplicative neutral 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	sraring 31681	Condition for a subring algebra to be a ring. (Contributed by Thierry Arnoux, 24-Jul-2023.)
⊢ 𝐴 = ((subringAlg ‘𝑅)‘𝑉)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑉 ⊆ 𝐵) → 𝐴 ∈ Ring)
Theorem	sradrng 31682	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	srasubrg 31683	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 31684	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 31685	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)
20.3.9.34  Division Ring Extensions
Theorem	drgext0g 31686	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 31687	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 31688	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 31689	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 31690	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 31691*	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)))
20.3.9.35  Vector Spaces
Theorem	lvecdimfi 31692	Finite version of lvecdim 20428 which does not require the axiom of choice. The axiom of choice is used in acsmapd 18281, 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)
20.3.9.36  Vector Space Dimension
Syntax	cldim 31693	Extend class notation with the dimension of a vector space.
class dim
Definition	df-dim 31694	Define the dimension of a vector space as the cardinality of its bases. Note that by lvecdim 20428, all bases are equinumerous. (Contributed by Thierry Arnoux, 6-May-2023.)
⊢ dim = (𝑓 ∈ V ↦ ∪ (♯ “ (LBasis‘𝑓)))
Theorem	dimval 31695	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 31696	The dimension of a vector space 𝐹 is the cardinality of one of its bases. This version of dimval 31695 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 31697	Closure of the vector space dimension. (Contributed by Thierry Arnoux, 18-May-2023.)
⊢ (𝑉 ∈ LVec → (dim‘𝑉) ∈ ℕ0*)
Theorem	lvecdim0i 31698	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 31699	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 31700	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))