P. 434 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43301-43400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sn-msqgt0d 43301	A nonzero square is positive. (Contributed by SN, 1-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐴))
Theorem	sn-inelr 43302	inelr 12226 without ax-mulcom 11182. (Contributed by SN, 1-Jun-2024.)
⊢ ¬ i ∈ ℝ
Theorem	sn-itrere 43303	i times a real is real iff the real is zero. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
Theorem	sn-retire 43304	Commuted version of sn-itrere 43303. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
Theorem	cnreeu 43305	The reals in the expression given by cnre 11223 uniquely define a complex number. (Contributed by SN, 27-Jun-2024.)
⊢ (𝜑 → 𝑟 ∈ ℝ)    &   ⊢ (𝜑 → 𝑠 ∈ ℝ)    &   ⊢ (𝜑 → 𝑡 ∈ ℝ)    &   ⊢ (𝜑 → 𝑢 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
Theorem	sn-sup2 43306*	sup2 12189 with exactly the same proof except for using sn-ltp1 43291 instead of ltp1 12073, saving ax-mulcom 11182. (Contributed by SN, 26-Jun-2024.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	sn-sup3d 43307*	sup3 12190 without ax-mulcom 11182, proven trivially from sn-sup2 43306. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	sn-suprcld 43308*	suprcld 12196 without ax-mulcom 11182, proven trivially from sn-sup3d 43307. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
Theorem	sn-suprubd 43309*	suprubd 12195 without ax-mulcom 11182, proven trivially from sn-suprcld 43308. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
21.31.6  Structures
Theorem	sn-base0 43310	Avoid axioms in base0 17299 by using the discouraged df-base 17295. This kind of axiom save is probably not worth it. (Contributed by SN, 16-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∅ = (Base‘∅)
Theorem	nelsubginvcld 43311	The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆))
Theorem	nelsubgcld 43312	A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆))
Theorem	nelsubgsubcld 43313	A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆))
Theorem	rnasclg 43314	The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))
Theorem	frlmfielbas 43315	The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑋 ∈ 𝐵 ↔ 𝑋:𝐼⟶𝑁))
Theorem	frlmfzwrd 43316	A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆)
Theorem	frlmfzowrd 43317	A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆)
Theorem	frlmfzolen 43318	The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁)
Theorem	frlmfzowrdb 43319	The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))
Theorem	frlmfzoccat 43320	The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿))    &   ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀))    &   ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐶 = (Base‘𝑋)    &   ⊢ 𝐷 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑍)    &   ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)
Theorem	frlmvscadiccat 43321	Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿))    &   ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀))    &   ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐶 = (Base‘𝑋)    &   ⊢ 𝐷 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑍)    &   ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐷)    &   ⊢ 𝑂 = ( ·𝑠 ‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝑋)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 ∙ 𝑈) ++ (𝐴 · 𝑉)))
Theorem	grpasscan2d 43322	An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
Theorem	grpcominv1 43323	If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))
Theorem	grpcominv2 43324	If two elements commute, then they commute with each other's inverses (case of the second element commuting with the inverse of the first element). (Contributed by SN, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌))
Theorem	finsubmsubg 43325	A submonoid of a finite group is a subgroup. This does not extend to infinite groups, as the submonoid ℕ0 of the group (ℤ, + ) shows. Note also that the union of a submonoid and its inverses need not be a submonoid, as the submonoid (ℕ0 ∖ {1}) of the group (ℤ, + ) shows: 3 is in that submonoid, -2 is the inverse of 2, but 1 is not in their union. Or simply, the subgroup generated by (ℕ0 ∖ {1}) is ℤ, not (ℤ ∖ {1, -1}). (Contributed by SN, 31-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	opprmndb 43326	A class is a monoid if and only if its opposite (ring) is a monoid. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
Theorem	opprgrpb 43327	A class is a group if and only if its opposite (ring) is a group. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
Theorem	opprablb 43328	A class is an Abelian group if and only if its opposite (ring) is an Abelian group. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Abel ↔ 𝑂 ∈ Abel)
Theorem	imacrhmcl 43329	The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))    ⇒   ⊢ (𝜑 → 𝐶 ∈ CRing)
Theorem	riccrng1 43330	Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ CRing) → 𝑆 ∈ CRing)
Theorem	riccrng 43331	A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ CRing ↔ 𝑆 ∈ CRing))
Theorem	domnexpgn0cl 43332	In a domain, a (nonnegative) power of a nonzero element is nonzero. (Contributed by SN, 6-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐵 ∖ { 0 }))
Theorem	drnginvrn0d 43333	A multiplicative inverse in a division ring is nonzero. (recne0d 12003 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ≠ 0 )
Theorem	drngmullcan 43334	Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11867 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ≠ 0 )    &   ⊢ (𝜑 → (𝑍 · 𝑋) = (𝑍 · 𝑌))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	drngmulrcan 43335	Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11868 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ≠ 0 )    &   ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	drnginvmuld 43336	Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋)))
Theorem	ricdrng1 43337	A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ DivRing) → 𝑆 ∈ DivRing)
Theorem	ricdrng 43338	A ring is a division ring if and only if an isomorphic ring is a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing))
Theorem	ricfld 43339	A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ Field ↔ 𝑆 ∈ Field))
Theorem	asclf1 43340*	Two ways of saying the scalar injection is one-to-one. (Contributed by SN, 3-Jul-2025.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (0g‘𝑆)    &   ⊢ (𝜑 → 𝑊 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → (𝐴:𝐾–1-1→𝐵 ↔ ∀𝑠 ∈ 𝐾 ((𝐴‘𝑠) = 0 → 𝑠 = 𝑁)))
Theorem	abvexp 43341	Move exponentiation in and out of absolute value. (Contributed by SN, 3-Jul-2025.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → (𝐹‘(𝑁 ↑ 𝑋)) = ((𝐹‘𝑋)↑𝑁))
Theorem	fimgmcyclem 43342*	Lemma for fimgmcyc 43343. (Contributed by SN, 7-Jul-2025.)
⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 ≠ 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))    ⇒   ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑞 ∈ ℕ (𝑜 < 𝑞 ∧ (𝑜 · 𝐴) = (𝑞 · 𝐴)))
Theorem	fimgmcyc 43343*	Version of odcl2 19666 for finite magmas: the multiples of an element 𝐴 ∈ 𝐵 are eventually periodic. (Contributed by SN, 3-Jul-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ · = (.g‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ Mgm)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑜 ∈ ℕ ∃𝑝 ∈ ℕ (𝑜 · 𝐴) = ((𝑜 + 𝑝) · 𝐴))
Theorem	fidomncyc 43344*	Version of odcl2 19666 for multiplicative groups of finite domains (that is, a finite monoid where nonzero elements are cancellable): one (1) is a multiple of any nonzero element. (Contributed by SN, 3-Jul-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ (𝑛 ↑ 𝐴) = 1 )
Theorem	fiabv 43345*	In a finite domain (a finite field), the only absolute value is the trivial one (abvtrivg 20973). (Contributed by SN, 3-Jul-2025.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑇 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝐴 = {𝑇})
Theorem	lvecgrp 43346	A vector space is a group. (Contributed by SN, 28-May-2023.)
⊢ (𝑊 ∈ LVec → 𝑊 ∈ Grp)
Theorem	lvecring 43347	The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LVec → 𝐹 ∈ Ring)
Theorem	frlm0vald 43348	All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((0g‘𝐹)‘𝐽) = 0 )
Theorem	frlmsnic 43349*	Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod {𝐼})    &   ⊢ 𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥‘𝐼))    ⇒   ⊢ ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))
Theorem	uvccl 43350	A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ∈ 𝐵)
Theorem	uvcn0 43351	A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) ≠ 0 )
Theorem	psrmnd 43352	The ring of power series is a monoid. (Contributed by SN, 25-Apr-2025.)
⊢ 𝑆 = (𝐼 mPwSer 𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Mnd)    ⇒   ⊢ (𝜑 → 𝑆 ∈ Mnd)
Theorem	mhmcopsr 43353	The composition of a monoid homomorphism and a power series is a power series. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ 𝐹) ∈ 𝐶)
Theorem	mhmcoaddpsr 43354	Show that the ring homomorphism in rhmpsr 43356 preserves addition. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ + = (+g‘𝑃)    &   ⊢ ✚ = (+g‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 MndHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 + 𝐺)) = ((𝐻 ∘ 𝐹) ✚ (𝐻 ∘ 𝐺)))
Theorem	rhmcomulpsr 43355	Show that the ring homomorphism in rhmpsr 43356 preserves multiplication. (Contributed by SN, 18-May-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐶 = (Base‘𝑄)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ∙ = (.r‘𝑄)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 ∘ (𝐹 · 𝐺)) = ((𝐻 ∘ 𝐹) ∙ (𝐻 ∘ 𝐺)))
Theorem	rhmpsr 43356*	Provide a ring homomorphism between two power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (𝐼 mPwSer 𝑅)    &   ⊢ 𝑄 = (𝐼 mPwSer 𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	rhmpsr1 43357*	Provide a ring homomorphism between two univariate power series algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 8-Feb-2025.)
⊢ 𝑃 = (PwSer1‘𝑅)    &   ⊢ 𝑄 = (PwSer1‘𝑆)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))    &   ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ∈ (𝑃 RingHom 𝑄))
Theorem	evl0 43358	The zero polynomial evaluates to zero. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑊 = (𝐼 mPoly 𝑅)    &   ⊢ 𝑂 = (0g‘𝑅)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    ⇒   ⊢ (𝜑 → (𝑄‘ 0 ) = ((𝐵 ↑m 𝐼) × {𝑂}))
Theorem	evlsbagval 43359*	Polynomial evaluation builder for a bag of variables. EDITORIAL: This theorem should stay in my mathbox until there's another use, since 0 and 1 using 𝑈 instead of 𝑆 may not be convenient. (Contributed by SN, 29-Jul-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝑊 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 1 = (1r‘𝑈)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐹 = (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑊 ∧ ((𝑄‘𝐹)‘𝐴) = (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝐵‘𝑣) ↑ (𝐴‘𝑣))))))
Theorem	evlvvvallem 43360*	Lemma for theorems using evlvvval 22321. Version of evlsvvvallem2 22280 using df-evl 22263. (Contributed by SN, 11-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑅))
Theorem	evlselvlem 43361*	Lemma for evlselv 43362. Used to re-index to and from bags of variables in 𝐼 and bags of variables in the subsets 𝐽 and 𝐼 ∖ 𝐽. (Contributed by SN, 10-Mar-2025.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin}    &   ⊢ 𝐶 = {𝑓 ∈ (ℕ0 ↑m (𝐼 ∖ 𝐽)) ∣ (◡𝑓 “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = (𝑐 ∈ 𝐶, 𝑒 ∈ 𝐸 ↦ (𝑐 ∪ 𝑒))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → 𝐻:(𝐶 × 𝐸)–1-1-onto→𝐷)
Theorem	evlselv 43362	Evaluating a selection of variable assignments, then evaluating the rest of the variables, is the same as evaluating with all assignments. (Contributed by SN, 10-Mar-2025.)
⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑈 = ((𝐼 ∖ 𝐽) mPoly 𝑅)    &   ⊢ 𝑇 = (𝐽 mPoly 𝑈)    &   ⊢ 𝐿 = (algSc‘𝑈)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((((𝐼 ∖ 𝐽) eval 𝑅)‘(((𝐽 eval 𝑈)‘(((𝐼 selectVars 𝑅)‘𝐽)‘𝐹))‘(𝐿 ∘ (𝐴 ↾ 𝐽))))‘(𝐴 ↾ (𝐼 ∖ 𝐽))) = (((𝐼 eval 𝑅)‘𝐹)‘𝐴))
Theorem	fsuppind 43363*	Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21943 and frlmplusgval 21951). This theorem is structurally general for polynomial proof usage (see mplelbas 22177 and mpladd 22195). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)    ⇒   ⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)
Theorem	fsuppssindlem1 43364*	Lemma for fsuppssind 43366. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, ((𝐹 ↾ 𝑆)‘𝑥), 0 )))
Theorem	fsuppssindlem2 43365*	Lemma for fsuppssind 43366. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐼)    ⇒   ⊢ (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝑆) ∣ (𝑥 ∈ 𝐼 ↦ if(𝑥 ∈ 𝑆, (𝑓‘𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆⟶𝐵 ∧ (𝐹 ∪ ((𝐼 ∖ 𝑆) × { 0 })) ∈ 𝐻)))
Theorem	fsuppssind 43366*	Induction on functions 𝐹:𝐴⟶𝐵 with finite support (see fsuppind 43363) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐼)    &   ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘f + 𝑦) ∈ 𝐻)    &   ⊢ (𝜑 → 𝑋:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝑋 finSupp 0 )    &   ⊢ (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐻)
Theorem	mhpind 43367*	The homogeneous polynomials of degree 𝑁 are generated by the terms of degree 𝑁 and addition. (Contributed by SN, 28-Jul-2024.)
⊢ 𝐻 = (𝐼 mHomP 𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (𝐼 mPoly 𝑅)    &   ⊢ + = (+g‘𝑃)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝑆 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ (𝜑 → 𝑅 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → (𝐷 × { 0 }) ∈ 𝐺)    &   ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝐵)) → (𝑠 ∈ 𝐷 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐺)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ ((𝐻‘𝑁) ∩ 𝐺) ∧ 𝑦 ∈ ((𝐻‘𝑁) ∩ 𝐺))) → (𝑥 + 𝑦) ∈ 𝐺)    ⇒   ⊢ (𝜑 → 𝑋 ∈ 𝐺)
Theorem	evlsmhpvvval 43368*	Give a formula for the evaluation of a homogeneous polynomial given assignments from variables to values. The difference between this and evlsvvval 22281 is that 𝑏 ∈ 𝐷 is restricted to 𝑏 ∈ 𝐺, that is, we can evaluate an 𝑁-th degree homogeneous polynomial over just the terms where the sum of all variable degrees is 𝑁. (Contributed by SN, 5-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐺 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑀 = (mulGrp‘𝑆)    &   ⊢ ↑ = (.g‘𝑀)    &   ⊢ · = (.r‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝐹)‘𝐴) = (𝑆 Σg (𝑏 ∈ 𝐺 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
Theorem	mhphflem 43369*	Lemma for mhphf 43370. Add several multiples of 𝐿 together, in a case where the total amount of multiplies is 𝑁. (Contributed by SN, 30-Jul-2024.)
⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}    &   ⊢ 𝐻 = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ · = (.g‘𝐺)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ Mnd)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻) → (𝐺 Σg (𝑣 ∈ 𝐼 ↦ ((𝑎‘𝑣) · 𝐿))) = (𝑁 · 𝐿))
Theorem	mhphf 43370	A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf2 43371	A homogeneous polynomial defines a homogeneous function; this is mhphf 43370 with simpler notation in the conclusion in exchange for a complex definition of ∙, which is based on frlmvscafval 21953 but without the finite support restriction (frlmpws 21937, frlmbas 21942) on the assignments 𝐴 from variables to values. TODO?: Polynomials (df-mpl 22098) are defined to have a finite amount of terms (of finite degree). As such, any assignment may be replaced by an assignment with finite support (as only a finite amount of variables matter in a given polynomial, even if the set of variables is infinite). So the finite support restriction can be assumed without loss of generality. (Contributed by SN, 11-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ ∙ = ( ·𝑠 ‘((ringLMod‘𝑆) ↑s 𝐼))    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼))    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf3 43372	A homogeneous polynomial defines a homogeneous function; this is mhphf2 43371 with the finite support restriction (frlmpws 21937, frlmbas 21942) on the assignments 𝐴 from variables to values. See comment of mhphf2 43371. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑈)    &   ⊢ 𝑈 = (𝑆 ↾s 𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐹 = (𝑆 freeLMod 𝐼)    &   ⊢ 𝑀 = (Base‘𝐹)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))    &   ⊢ (𝜑 → 𝐿 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
Theorem	mhphf4 43373	A homogeneous polynomial defines a homogeneous function; this is mhphf3 43372 with evalSub collapsed to eval. (Contributed by SN, 23-Nov-2024.)
⊢ 𝑄 = (𝐼 eval 𝑆)    &   ⊢ 𝐻 = (𝐼 mHomP 𝑆)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝐹 = (𝑆 freeLMod 𝐼)    &   ⊢ 𝑀 = (Base‘𝐹)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑆)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑆))    &   ⊢ (𝜑 → 𝑆 ∈ CRing)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑀)    ⇒   ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))
21.31.7  Projective spaces Looking at a corner in 3D space, one can see three right angles. It is impossible to draw three lines in 2D space such that any two of these lines are perpendicular, but a good enough representation is made by casting lines from the 2D surface. Points along the same cast line are collapsed into one point on the 2D surface. In many cases, the 2D surface is smaller than whatever needs to be represented. If the lines cast were perpendicular to the 2D surface, then only areas as small as the 2D surface could be represented. To fix this, the lines need to get further apart as they go farther from the 2D surface. On the other side of the 2D surface the lines will get closer together and intersect at a point (because it's defined that way). From this perspective, two parallel lines in 3D space will be represented by two lines that seem to intersect at a point "at infinity". Considering all maximal classes of parallel lines on a 2D plane in 3D space, these classes will all appear to intersect at different points at infinity, forming a line at infinity. Therefore the real projective plane can be thought of as the real affine plane together with the line at infinity. The projective plane takes care of some exceptions that may be found in the affine plane. For example, consider the curve that is the zeroes of 𝑦 = 𝑥↑2. Any line connecting the point (0, 1) to the x-axis intersects with the curve twice, except for the vertical line between (0, 1) and (0, 0). In the projective plane, the curve becomes an ellipse and there is no exception. While it may not seem like it, points at infinity and points corresponding to the affine plane are the same type of point. Consider a line going through the origin in 3D (affine) space. Either it intersects the plane 𝑧 = 1 once, or it is entirely within the plane 𝑧 = 0. If it is entirely within the plane 𝑧 = 0, then it corresponds to the point at infinity intersecting all lines on the plane 𝑧 = 1 with the same slope. Else it corresponds to the point in the 2D plane 𝑧 = 1 that it intersects. So there is a bijection between 3D lines through the origin and points on the real projective plane. The concept of projective spaces generalizes the projective plane to any dimension.
Syntax	cprjsp 43374	Extend class notation with the projective space function.
class ℙ𝕣𝕠𝕛
Definition	df-prjsp 43375*	Define the projective space function. In the bijection between 3D lines through the origin and points in the projective plane (see section comment), this is equivalent to making any two 3D points (excluding the origin) equivalent iff one is a multiple of another. This definition does not quite give all the properties needed, since the scalars of a left vector space can be "less dense" than the vectors (for example, making equivalent rational multiples of real numbers). Compare df-lsatoms 39791. (Contributed by BJ and SN, 29-Apr-2023.)
⊢ ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Base‘𝑣) ∖ {(0g‘𝑣)}) / 𝑏⦌(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠 ‘𝑣)𝑦))}))
Theorem	prjspval 43376*	Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}))
Theorem	prjsprel 43377*	Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    ⇒   ⊢ (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ 𝐾 𝑋 = (𝑚 · 𝑌)))
Theorem	prjspertr 43378*	The relation in ℙ𝕣𝕠𝕛 is transitive. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LMod ∧ (𝑋 ∼ 𝑌 ∧ 𝑌 ∼ 𝑍)) → 𝑋 ∼ 𝑍)
Theorem	prjsperref 43379*	The relation in ℙ𝕣𝕠𝕛 is reflexive. (Contributed by Steven Nguyen, 30-Apr-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LMod → (𝑋 ∈ 𝐵 ↔ 𝑋 ∼ 𝑋))
Theorem	prjspersym 43380*	The relation in ℙ𝕣𝕠𝕛 is symmetric. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∼ 𝑌) → 𝑌 ∼ 𝑋)
Theorem	prjsper 43381*	The relation used to define ℙ𝕣𝕠𝕛 is an equivalence relation. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → ∼ Er 𝐵)
Theorem	prjspreln0 43382*	Two nonzero vectors are equivalent by a nonzero scalar. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝑉 ∈ LVec → (𝑋 ∼ 𝑌 ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∃𝑚 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑚 · 𝑌))))
Theorem	prjspvs 43383*	A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵 ∧ 𝑁 ∈ (𝐾 ∖ { 0 })) → (𝑁 · 𝑋) ∼ 𝑋)
Theorem	prjsprellsp 43384*	Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ∼ 𝑌 ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
Theorem	prjspeclsp 43385*	The vectors equivalent to a vector 𝑋 are the nonzero vectors in the span of 𝑋. (Contributed by Steven Nguyen, 6-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝐾 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ {(0g‘𝑉)})    &   ⊢ 𝑆 = (Scalar‘𝑉)    &   ⊢ · = ( ·𝑠 ‘𝑉)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ ((𝑉 ∈ LVec ∧ 𝑋 ∈ 𝐵) → [𝑋] ∼ = ((𝑁‘{𝑋}) ∖ {(0g‘𝑉)}))
Theorem	prjspval2 43386*	Alternate definition of projective space. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ 0 = (0g‘𝑉)    &   ⊢ 𝐵 = ((Base‘𝑉) ∖ { 0 })    &   ⊢ 𝑁 = (LSpan‘𝑉)    ⇒   ⊢ (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = ∪ 𝑧 ∈ 𝐵 {((𝑁‘{𝑧}) ∖ { 0 })})
Syntax	cprjspn 43387	Extend class notation with the n-dimensional projective space function.
class ℙ𝕣𝕠𝕛n
Definition	df-prjspn 43388*	Define the n-dimensional projective space function. A projective space of dimension 1 is a projective line, and a projective space of dimension 2 is a projective plane. Compare df-ehl 25582. This space is considered n-dimensional because the vector space (𝑘 freeLMod (0...𝑛)) is (n+1)-dimensional and the ℙ𝕣𝕠𝕛 function returns equivalence classes with respect to a linear (1-dimensional) relation. (Contributed by BJ and Steven Nguyen, 29-Apr-2023.)
⊢ ℙ𝕣𝕠𝕛n = (𝑛 ∈ ℕ0, 𝑘 ∈ DivRing ↦ (ℙ𝕣𝕠𝕛‘(𝑘 freeLMod (0...𝑛))))
Theorem	prjspnval 43389	Value of the n-dimensional projective space function. (Contributed by Steven Nguyen, 1-May-2023.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (ℙ𝕣𝕠𝕛‘(𝐾 freeLMod (0...𝑁))))
Theorem	prjspnerlem 43390*	A lemma showing that the equivalence relation used in prjspnval2 43391 and the equivalence relation used in prjspval 43376 are equal, but only with the antecedent 𝐾 ∈ DivRing. (Contributed by SN, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ (𝐾 ∈ DivRing → ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑊))𝑥 = (𝑙 · 𝑦))})
Theorem	prjspnval2 43391*	Value of the n-dimensional projective space function, expanded. (Contributed by Steven Nguyen, 15-Jul-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛n𝐾) = (𝐵 / ∼ ))
Theorem	prjspner 43392*	The relation used to define ℙ𝕣𝕠𝕛 (and indirectly ℙ𝕣𝕠𝕛n through df-prjspn 43388) is an equivalence relation. This is a lemma that converts the equivalence relation used in results like prjspertr 43378 and prjspersym 43380 (see prjspnerlem 43390). Several theorems are covered in one thanks to the theorems around df-er 8703. (Contributed by SN, 14-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    ⇒   ⊢ (𝜑 → ∼ Er 𝐵)
Theorem	prjspnvs 43393*	A nonzero multiple of a vector is equivalent to the vector. This converts the equivalence relation used in prjspvs 43383 (see prjspnerlem 43390). (Contributed by SN, 8-Aug-2024.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐶 · 𝑋) ∼ 𝑋)
Theorem	prjspnssbas 43394	A projective point spans a subset of the (nonzero) affine points. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑃 ⊆ 𝒫 𝐵)
Theorem	prjspnn0 43395	A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)
⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑃)    ⇒   ⊢ (𝜑 → 𝐴 ≠ ∅)
Theorem	0prjspnlem 43396	Lemma for 0prjspn 43401. The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ (𝐾 ∈ DivRing → 1 ∈ 𝐵)
Theorem	prjspnfv01 43397*	Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the value of the zeroth coordinate). (Contributed by SN, 13-Aug-2023.)
⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 1 = (1r‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑋)‘0) = if((𝑋‘0) = 0 , 0 , 1 ))
Theorem	prjspner01 43398*	Any vector is equivalent to a vector whose zeroth coordinate is 0 or 1 (proof of the equivalence). (Contributed by SN, 13-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑋 ∼ (𝐹‘𝑋))
Theorem	prjspner1 43399*	Two vectors whose zeroth coordinate is nonzero are equivalent if and only if they have the same representative in the (n-1)-dimensional affine subspace { x0 = 1 } . For example, vectors in 3D space whose 𝑥 coordinate is nonzero are equivalent iff they intersect at the plane 𝑥 = 1 at the same point (also see section header). (Contributed by SN, 13-Aug-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐹 = (𝑏 ∈ 𝐵 ↦ if((𝑏‘0) = 0 , 𝑏, ((𝐼‘(𝑏‘0)) · 𝑏)))    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 0 = (0g‘𝐾)    &   ⊢ 𝐼 = (invr‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ DivRing)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋‘0) ≠ 0 )    &   ⊢ (𝜑 → (𝑌‘0) ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑋 ∼ 𝑌 ↔ (𝐹‘𝑋) = (𝐹‘𝑌)))
Theorem	0prjspnrel 43400*	In the zero-dimensional projective space, all vectors are equivalent to the unit vector. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ 𝑆 𝑥 = (𝑙 · 𝑦))}    &   ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0g‘𝑊)})    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ 𝑊 = (𝐾 freeLMod (0...0))    &   ⊢ 1 = ((𝐾 unitVec (0...0))‘0)    ⇒   ⊢ ((𝐾 ∈ DivRing ∧ 𝑋 ∈ 𝐵) → 𝑋 ∼ 1 )