P. 205 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	issubdrg 20401*	Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼‘𝑥) ∈ 𝐴))
10.4.2  Sub-division rings
Syntax	csdrg 20402	Syntax for subfields (sub-division-rings).
class SubDRing
Definition	df-sdrg 20403*	Define the function associating with a ring the set of its sub-division-rings. A sub-division-ring of a ring is a subset of its base set which is a division ring when equipped with the induced structure (sum, multiplication, zero, and unity). If a ring is commutative (resp., a field), then its sub-division-rings are commutative (resp., are fields) (fldsdrgfld 20414), so we do not make a specific definition for subfields. (Contributed by Stefan O'Rear, 3-Oct-2015.) TODO: extend this definition to a function with domain V or at least Ring and not only DivRing.
⊢ SubDRing = (𝑤 ∈ DivRing ↦ {𝑠 ∈ (SubRing‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ DivRing})
Theorem	issdrg 20404	Property of a division subring. (Contributed by Stefan O'Rear, 3-Oct-2015.)
⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing))
Theorem	sdrgrcl 20405	Reverse closure for a sub-division-ring predicate. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑅 ∈ DivRing)
Theorem	sdrgdrng 20406	A sub-division-ring is a division ring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝑆 ∈ DivRing)
Theorem	sdrgsubrg 20407	A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))
Theorem	sdrgid 20408	Every division ring is a division subring of itself. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 𝐵 ∈ (SubDRing‘𝑅))
Theorem	sdrgss 20409	A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵)
Theorem	sdrgbas 20410	Base set of a sub-division-ring structure. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	issdrg2 20411*	Property of a division subring (closure version). (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ ∀𝑥 ∈ (𝑆 ∖ { 0 })(𝐼‘𝑥) ∈ 𝑆))
Theorem	sdrgunit 20412	A unit of a sub-division-ring is a nonzero element of the subring. (Contributed by SN, 19-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubDRing‘𝑅) → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ 0 )))
Theorem	imadrhmcl 20413	The image of a (nontrivial) division ring homomorphism is a division ring. (Contributed by SN, 17-Feb-2025.)
⊢ 𝑅 = (𝑁 ↾s (𝐹 “ 𝑆))    &   ⊢ 0 = (0g‘𝑁)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   ⊢ (𝜑 → 𝑆 ∈ (SubDRing‘𝑀))    &   ⊢ (𝜑 → ran 𝐹 ≠ { 0 })    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	fldsdrgfld 20414	A sub-division-ring of a field is itself a field, so it is a subfield. We can therefore use SubDRing to express subfields. (Contributed by Thierry Arnoux, 11-Jan-2025.)
⊢ ((𝐹 ∈ Field ∧ 𝐴 ∈ (SubDRing‘𝐹)) → (𝐹 ↾s 𝐴) ∈ Field)
Theorem	acsfn1p 20415*	Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))
Theorem	subrgacs 20416	Closure property of subrings. (Contributed by Stefan O'Rear, 12-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	sdrgacs 20417	Closure property of division subrings. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (SubDRing‘𝑅) ∈ (ACS‘𝐵))
Theorem	cntzsdrg 20418	Centralizers in division rings/fields are subfields. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubDRing‘𝑅))
Theorem	subdrgint 20419*	The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐿 = (𝑅 ↾s ∩ 𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑆 ⊆ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝑆 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑅 ↾s 𝑠) ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝐿 ∈ DivRing)
Theorem	sdrgint 20420	The intersection of a nonempty collection of sub division rings is a sub division ring. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ⊆ (SubDRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubDRing‘𝑅))
Theorem	primefld 20421	The smallest sub division ring of a division ring, here named 𝑃, is a field, called the Prime Field of 𝑅. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝑃 = (𝑅 ↾s ∩ (SubDRing‘𝑅))    ⇒   ⊢ (𝑅 ∈ DivRing → 𝑃 ∈ Field)
Theorem	primefld0cl 20422	The prime field contains the zero element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 0 ∈ ∩ (SubDRing‘𝑅))
Theorem	primefld1cl 20423	The prime field contains the unity element of the division ring. (Contributed by Thierry Arnoux, 22-Aug-2023.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ∈ ∩ (SubDRing‘𝑅))
10.4.3  Absolute value (abstract algebra)
Syntax	cabv 20424	The set of absolute values on a ring.
class AbsVal
Definition	df-abv 20425*	Define the set of absolute values on a ring. An absolute value is a generalization of the usual absolute value function df-abs 15183 to arbitrary rings. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
Theorem	abvfval 20426*	Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
Theorem	isabv 20427*	Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
Theorem	isabvd 20428*	Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → 0 = (0g‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹:𝐵⟶ℝ)    &   ⊢ (𝜑 → (𝐹‘ 0 ) = 0)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 0 < (𝐹‘𝑥))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐴)
Theorem	abvrcl 20429	Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)
Theorem	abvfge0 20430	An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞))
Theorem	abvf 20431	An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶ℝ)
Theorem	abvcl 20432	An absolute value is a function from the ring to the real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ ℝ)
Theorem	abvge0 20433	The absolute value of a number is greater than or equal to zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 0 ≤ (𝐹‘𝑋))
Theorem	abveq0 20434	The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 ))
Theorem	abvne0 20435	The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0)
Theorem	abvgt0 20436	The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 0 < (𝐹‘𝑋))
Theorem	abvmul 20437	An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌)))
Theorem	abvtri 20438	An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))
Theorem	abv0 20439	The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0)
Theorem	abv1z 20440	The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0 ) → (𝐹‘ 1 ) = 1)
Theorem	abv1 20441	The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) → (𝐹‘ 1 ) = 1)
Theorem	abvneg 20442	The absolute value of a negative is the same as that of the positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝑁‘𝑋)) = (𝐹‘𝑋))
Theorem	abvsubtri 20443	An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 − 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))
Theorem	abvrec 20444	The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝐹‘(𝐼‘𝑋)) = (1 / (𝐹‘𝑋)))
Theorem	abvdiv 20445	The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ DivRing ∧ 𝐹 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝐹‘(𝑋 / 𝑌)) = ((𝐹‘𝑋) / (𝐹‘𝑌)))
Theorem	abvdom 20446	Any ring with an absolute value is a domain, which is to say that it contains no zero divisors. (Contributed by Mario Carneiro, 10-Sep-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 )
Theorem	abvres 20447	The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝑆 = (𝑅 ↾s 𝐶)    &   ⊢ 𝐵 = (AbsVal‘𝑆)    ⇒   ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹 ↾ 𝐶) ∈ 𝐵)
Theorem	abvtrivd 20448*	The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1))    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 ) ∧ (𝑧 ∈ 𝐵 ∧ 𝑧 ≠ 0 )) → (𝑦 · 𝑧) ≠ 0 )    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝐴)
Theorem	abvtriv 20449*	The trivial absolute value. (This theorem is true as long as 𝑅 is a domain, but it is not true for rings with zero divisors, which violate the multiplication axiom; abvdom 20446 is the converse of this remark.) (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐴 = (AbsVal‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if(𝑥 = 0 , 0, 1))    ⇒   ⊢ (𝑅 ∈ DivRing → 𝐹 ∈ 𝐴)
Theorem	abvpropd 20450*	If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
10.4.4  Star rings
Syntax	cstf 20451	Extend class notation with the functionalization of the *-ring involution.
class *rf
Syntax	csr 20452	Extend class notation with class of all *-rings.
class *-Ring
Definition	df-staf 20453*	Define the functionalization of the involution in a star ring. This is not strictly necessary but by having *𝑟 as an actual function we can state the principal properties of an involution much more cleanly. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ *rf = (𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑓) ↦ ((*𝑟‘𝑓)‘𝑥)))
Definition	df-srng 20454*	Define class of all star rings. A star ring is a ring with an involution (conjugation) function. Involution (unlike say the ring zero) is not unique and therefore must be added as a new component to the ring. For example, two possible involutions for complex numbers are the identity function and complex conjugation. Definition of involution in [Holland95] p. 204. (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
⊢ *-Ring = {𝑓 ∣ [(*rf‘𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr‘𝑓)) ∧ 𝑖 = ◡𝑖)}
Theorem	staffval 20455*	The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ ∙ = (*rf‘𝑅)    ⇒   ⊢ ∙ = (𝑥 ∈ 𝐵 ↦ ( ∗ ‘𝑥))
Theorem	stafval 20456	The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ ∙ = (*rf‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝐵 → ( ∙ ‘𝐴) = ( ∗ ‘𝐴))
Theorem	staffn 20457	The functionalization is equal to the original function, if it is a function on the right base set. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ ∙ = (*rf‘𝑅)    ⇒   ⊢ ( ∗ Fn 𝐵 → ∙ = ∗ )
Theorem	issrng 20458	The predicate "is a star ring". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ ∗ = (*rf‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring ↔ ( ∗ ∈ (𝑅 RingHom 𝑂) ∧ ∗ = ◡ ∗ ))
Theorem	srngrhm 20459	The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑂 = (oppr‘𝑅)    &   ⊢ ∗ = (*rf‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ∗ ∈ (𝑅 RingHom 𝑂))
Theorem	srngring 20460	A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑅 ∈ *-Ring → 𝑅 ∈ Ring)
Theorem	srngcnv 20461	The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*rf‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ∗ = ◡ ∗ )
Theorem	srngf1o 20462	The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*rf‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ∗ :𝐵–1-1-onto→𝐵)
Theorem	srngcl 20463	The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘𝑋) ∈ 𝐵)
Theorem	srngnvl 20464	The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵) → ( ∗ ‘( ∗ ‘𝑋)) = 𝑋)
Theorem	srngadd 20465	The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 + 𝑌)) = (( ∗ ‘𝑋) + ( ∗ ‘𝑌)))
Theorem	srngmul 20466	The involution function in a star ring distributes over multiplication, with a change in the order of the factors. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ∗ ‘(𝑋 · 𝑌)) = (( ∗ ‘𝑌) · ( ∗ ‘𝑋)))
Theorem	srng1 20467	The conjugate of the ring identity is the identity. (This is sometimes taken as an axiom, and indeed the proof here follows because we defined *𝑟 to be a ring homomorphism, which preserves 1; nevertheless, it is redundant, as can be seen from the proof of issrngd 20469.) (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 )
Theorem	srng0 20468	The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 0 ) = 0 )
Theorem	issrngd 20469*	Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
⊢ (𝜑 → 𝐾 = (Base‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → · = (.r‘𝑅))    &   ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘𝑥) ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 + 𝑦)) = (( ∗ ‘𝑥) + ( ∗ ‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾 ∧ 𝑦 ∈ 𝐾) → ( ∗ ‘(𝑥 · 𝑦)) = (( ∗ ‘𝑦) · ( ∗ ‘𝑥)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐾) → ( ∗ ‘( ∗ ‘𝑥)) = 𝑥)    ⇒   ⊢ (𝜑 → 𝑅 ∈ *-Ring)
Theorem	idsrngd 20470*	A commutative ring is a star ring when the conjugate operation is the identity. (Contributed by Thierry Arnoux, 19-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)    ⇒   ⊢ (𝜑 → 𝑅 ∈ *-Ring)
10.5  Left modules
10.5.1  Definition and basic properties
Syntax	clmod 20471	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 20472	The functionalization of the scalar multiplication operation.
class ·sf
Definition	df-lmod 20473*	Define the class of all left modules, which are generalizations of left vector spaces. A left module over a ring is an (Abelian) group (vectors) together with a ring (scalars) and a left scalar product connecting them. (Contributed by NM, 4-Nov-2013.)
⊢ LMod = {𝑔 ∈ Grp ∣ [(Base‘𝑔) / 𝑣][(+g‘𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠 ‘𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g‘𝑓) / 𝑝][(.r‘𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞 ∈ 𝑘 ∀𝑟 ∈ 𝑘 ∀𝑥 ∈ 𝑣 ∀𝑤 ∈ 𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r‘𝑓)𝑠𝑤) = 𝑤)))}
Definition	df-scaf 20474*	Define the functionalization of the ·𝑠 operator. This restricts the value of ·𝑠 to the stated domain, which is necessary when working with restricted structures, whose operations may be defined on a larger set than the true base. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ ·sf = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑔)), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥( ·𝑠 ‘𝑔)𝑦)))
Theorem	islmod 20475*	The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑟 · 𝑤) ∈ 𝑉 ∧ (𝑟 · (𝑤 + 𝑥)) = ((𝑟 · 𝑤) + (𝑟 · 𝑥)) ∧ ((𝑞 ⨣ 𝑟) · 𝑤) = ((𝑞 · 𝑤) + (𝑟 · 𝑤))) ∧ (((𝑞 × 𝑟) · 𝑤) = (𝑞 · (𝑟 · 𝑤)) ∧ ( 1 · 𝑤) = 𝑤))))
Theorem	lmodlema 20476	Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝑅 · 𝑌) ∈ 𝑉 ∧ (𝑅 · (𝑌 + 𝑋)) = ((𝑅 · 𝑌) + (𝑅 · 𝑋)) ∧ ((𝑄 ⨣ 𝑅) · 𝑌) = ((𝑄 · 𝑌) + (𝑅 · 𝑌))) ∧ (((𝑄 × 𝑅) · 𝑌) = (𝑄 · (𝑅 · 𝑌)) ∧ ( 1 · 𝑌) = 𝑌)))
Theorem	islmodd 20477*	Properties that determine a left module. See note in isgrpd2 18842 regarding the 𝜑 on hypotheses that name structure components. (Contributed by Mario Carneiro, 22-Jun-2014.)
⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → + = (+g‘𝑊))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐹))    &   ⊢ (𝜑 → ⨣ = (+g‘𝐹))    &   ⊢ (𝜑 → × = (.r‘𝐹))    &   ⊢ (𝜑 → 1 = (1r‘𝐹))    &   ⊢ (𝜑 → 𝐹 ∈ Ring)    &   ⊢ (𝜑 → 𝑊 ∈ Grp)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑉) → (𝑥 · 𝑦) ∈ 𝑉)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 ⨣ 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 × 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 1 · 𝑥) = 𝑥)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LMod)
Theorem	lmodgrp 20478	A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp)
Theorem	lmodring 20479	The scalar component of a left module is a ring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring)
Theorem	lmodfgrp 20480	The scalar component of a left module is an additive group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp)
Theorem	lmodgrpd 20481	A left module is a group. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → 𝑊 ∈ Grp)
Theorem	lmodbn0 20482	The base set of a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐵 = (Base‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅)
Theorem	lmodacl 20483	Closure of ring addition for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Theorem	lmodmcl 20484	Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
Theorem	lmodsn0 20485	The set of scalars in a left module is nonempty. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ LMod → 𝐵 ≠ ∅)
Theorem	lmodvacl 20486	Closure of vector addition for a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
Theorem	lmodass 20487	Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Theorem	lmodlcan 20488	Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))
Theorem	lmodvscl 20489	Closure of scalar product for a left module. (hvmulcl 30266 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉)
Theorem	scaffval 20490*	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ∙ = ( ·sf ‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑦 ∈ 𝐵 ↦ (𝑥 · 𝑦))
Theorem	scafval 20491	The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ∙ = ( ·sf ‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∙ 𝑌) = (𝑋 · 𝑌))
Theorem	scafeq 20492	If the scalar multiplication operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ∙ = ( ·sf ‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ( · Fn (𝐾 × 𝐵) → ∙ = · )
Theorem	scaffn 20493	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ∙ = ( ·sf ‘𝑊)    ⇒   ⊢ ∙ Fn (𝐾 × 𝐵)
Theorem	lmodscaf 20494	The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ∙ = ( ·sf ‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → ∙ :(𝐾 × 𝐵)⟶𝐵)
Theorem	lmodvsdi 20495	Distributive law for scalar product (left-distributivity). (ax-hvdistr1 30261 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))
Theorem	lmodvsdir 20496	Distributive law for scalar product (right-distributivity). (ax-hvdistr1 30261 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 ⨣ 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Theorem	lmodvsass 20497	Associative law for scalar product. (ax-hvmulass 30260 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ × = (.r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Theorem	lmod0cl 20498	The ring zero in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ (𝑊 ∈ LMod → 0 ∈ 𝐾)
Theorem	lmod1cl 20499	The ring unity in a left module belongs to the set of scalars. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ (𝑊 ∈ LMod → 1 ∈ 𝐾)
Theorem	lmodvs1 20500	Scalar product with the ring unity. (ax-hvmulid 30259 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋)