P. 190 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18901-19000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	srngcnv 18901	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 18902	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 18903	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 18904	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 18905	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 18906	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 18907	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 18909.) (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ *-Ring → ( ∗ ‘ 1 ) = 1 )
Theorem	srng0 18908	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 18909*	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 18910*	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)
10.6  Left modules
10.6.1  Definition and basic properties
Syntax	clmod 18911	Extend class notation with class of all left modules.
class LMod
Syntax	cscaf 18912	The functionalization of the scalar multiplication operation.
class ·sf
Definition	df-lmod 18913*	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 18914*	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 18915*	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 18916	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 18917*	Properties that determine a left module. See note in isgrpd2 17489 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 18918	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 18919	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 18920	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	lmodbn0 18921	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 18922	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 18923	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 18924	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 18925	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 18926	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 18927	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 18928	Closure of scalar product for a left module. (hvmulcl 27998 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 18929*	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))
Theorem	scafval 18930	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 18931	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 18932	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 18933	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 18934	Distributive law for scalar product (left-distributivity). (ax-hvdistr1 27993 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 18935	Distributive law for scalar product (right-distributivity). (ax-hvdistr1 27993 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 18936	Associative law for scalar product. (ax-hvmulass 27992 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 18937	The ring zero in a left module belongs to the ring base set. (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 18938	The ring unit in a left module belongs to the ring base set. (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 18939	Scalar product with ring unit. (ax-hvmulid 27991 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)
Theorem	lmod0vcl 18940	The zero vector is a vector. (ax-hv0cl 27988 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 0 ∈ 𝑉)
Theorem	lmod0vlid 18941	Left identity law for the zero vector. (hvaddid2 28008 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋)
Theorem	lmod0vrid 18942	Right identity law for the zero vector. (ax-hvaddid 27989 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + 0 ) = 𝑋)
Theorem	lmod0vid 18943	Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋))
Theorem	lmod0vs 18944	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 27995 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑂 = (0g‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑂 · 𝑋) = 0 )
Theorem	lmodvs0 18945	Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28009 analog.) (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾) → (𝑋 · 0 ) = 0 )
Theorem	lmodvsmmulgdi 18946	Distributive law for a group multiple of a scalar multiplication. (Contributed by AV, 2-Sep-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝐶 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑁 ↑ (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))
Theorem	lmodfopnelem1 18947	Lemma 1 for lmodfopne 18949. (Contributed by AV, 2-Oct-2021.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
Theorem	lmodfopnelem2 18948	Lemma 2 for lmodfopne 18949. (Contributed by AV, 2-Oct-2021.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ + = · ) → ( 0 ∈ 𝑉 ∧ 1 ∈ 𝑉))
Theorem	lmodfopne 18949	The (functionalized) operations of a left module (over a nonzero ring) cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 2-Oct-2021.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 1 ≠ 0 ) → + ≠ · )
Theorem	lcomf 18950	A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐾)    &   ⊢ (𝜑 → 𝐻:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐻):𝐼⟶𝐵)
Theorem	lcomfsupp 18951	A linear-combination sum is finitely supported if the coefficients are. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by AV, 15-Jul-2019.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐺:𝐼⟶𝐾)    &   ⊢ (𝜑 → 𝐻:𝐼⟶𝐵)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑌 = (0g‘𝐹)    &   ⊢ (𝜑 → 𝐺 finSupp 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 ∘𝑓 · 𝐻) finSupp 0 )
Theorem	lmodvnegcl 18952	Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) ∈ 𝑉)
Theorem	lmodvnegid 18953	Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑋 + (𝑁‘𝑋)) = 0 )
Theorem	lmodvneg1 18954	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 𝑀 = (invg‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑀‘ 1 ) · 𝑋) = (𝑁‘𝑋))
Theorem	lmodvsneg 18955	Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑀 = (invg‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀‘𝑅) · 𝑋))
Theorem	lmodvsubcl 18956	Closure of vector subtraction. (hvsubcl 28002 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
Theorem	lmodcom 18957	Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	lmodabl 18958	A left module is an abelian group (of vectors, under addition). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel)
Theorem	lmodcmn 18959	A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
Theorem	lmodnegadd 18960	Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼‘𝐴) · 𝑋) + ((𝐼‘𝐵) · 𝑌)))
Theorem	lmod4 18961	Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈)))
Theorem	lmodvsubadd 18962	Relationship between vector subtraction and addition. (hvsubadd 28062 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	lmodvaddsub4 18963	Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 − 𝐶) = (𝐷 − 𝐵)))
Theorem	lmodvpncan 18964	Addition/subtraction cancellation law for vectors. (hvpncan 28024 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 + 𝐵) − 𝐵) = 𝐴)
Theorem	lmodvnpcan 18965	Cancellation law for vector subtraction (npcan 10328 analog). (Contributed by NM, 19-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) + 𝐵) = 𝐴)
Theorem	lmodvsubval2 18966	Value of vector subtraction in terms of addition. (hvsubval 28001 analog.) (Contributed by NM, 31-Mar-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + ((𝑁‘ 1 ) · 𝐵)))
Theorem	lmodsubvs 18967	Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑁 = (invg‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 − (𝐴 · 𝑌)) = (𝑋 + ((𝑁‘𝐴) · 𝑌)))
Theorem	lmodsubdi 18968	Scalar multiplication distributive law for subtraction. (hvsubdistr1 28034 analogue, with longer proof since our scalar multiplication is not commutative.) (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴 · (𝑋 − 𝑌)) = ((𝐴 · 𝑋) − (𝐴 · 𝑌)))
Theorem	lmodsubdir 18969	Scalar multiplication distributive law for subtraction. (hvsubdistr2 28035 analog.) (Contributed by NM, 2-Jul-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝑆 = (-g‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴𝑆𝐵) · 𝑋) = ((𝐴 · 𝑋) − (𝐵 · 𝑋)))
Theorem	lmodsubeq0 18970	If the difference between two vectors is zero, they are equal. (hvsubeq0 28053 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
Theorem	lmodsubid 18971	Subtraction of a vector from itself. (hvsubid 28011 analog.) (Contributed by NM, 16-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ − = (-g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝑉) → (𝐴 − 𝐴) = 0 )
Theorem	lmodvsghm 18972*	Scalar multiplication of the vector space by a fixed scalar is an automorphism of the additive group of vectors. (Contributed by Mario Carneiro, 5-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾) → (𝑥 ∈ 𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))
Theorem	lmodprop2d 18973*	If two structures have the same components (properties), one is a left module iff the other one is. This version of lmodpropd 18974 also breaks up the components of the scalar ring. (Contributed by Mario Carneiro, 27-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ 𝐹 = (Scalar‘𝐾)    &   ⊢ 𝐺 = (Scalar‘𝐿)    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑃 = (Base‘𝐺))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(.r‘𝐹)𝑦) = (𝑥(.r‘𝐺)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
Theorem	lmodpropd 18974*	If two structures have the same components (properties), one is a left module iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 27-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))    &   ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))    &   ⊢ 𝑃 = (Base‘𝐹)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))
Theorem	gsumvsmul 18975*	Pull a scalar multiplication out of a sum of vectors. This theorem properly generalizes gsummulc2 18653, since every ring is a left module over itself. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) (Revised by AV, 10-Jul-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑆 = (Scalar‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ LMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑌) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = (𝑋 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑌))))
Theorem	mptscmfsupp0 18976*	A mapping to a scalar product is finitely supported if the mapping to the scalar is finitely supported. (Contributed by AV, 5-Oct-2019.)
⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑄 ∈ LMod)    &   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑄))    &   ⊢ 𝐾 = (Base‘𝑄)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑆 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐷) → 𝑊 ∈ 𝐾)    &   ⊢ 0 = (0g‘𝑄)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ ∗ = ( ·𝑠 ‘𝑄)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ 𝑆) finSupp 𝑍)    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝐷 ↦ (𝑆 ∗ 𝑊)) finSupp 0 )
Theorem	mptscmfsuppd 18977*	A function mapping to a scalar product in which one factor is finitely supported is finitely supported. Formerly part of proof for ply1coe 19714. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by AV, 8-Aug-2019.) (Proof shortened by AV, 18-Oct-2019.)
⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝑆 = (Scalar‘𝑃)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ (𝜑 → 𝑃 ∈ LMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴:𝑋⟶𝑌)    &   ⊢ (𝜑 → 𝐴 finSupp (0g‘𝑆))    ⇒   ⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) · 𝑍)) finSupp (0g‘𝑃))
Theorem	rmodislmodlem 18978*	Lemma for rmodislmod 18979. This is the part of the proof of rmodislmod 18979 which requires the scalar ring to be commutative. (Contributed by AV, 3-Dec-2021.)
⊢ 𝑉 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ 𝐹 = (Scalar‘𝑅)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &   ⊢ ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   ⊢ 𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   ⊢ ((𝐹 ∈ CRing ∧ (𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾 ∧ 𝑐 ∈ 𝑉)) → ((𝑎 × 𝑏) ∗ 𝑐) = (𝑎 ∗ (𝑏 ∗ 𝑐)))
Theorem	rmodislmod 18979*	The right module 𝑅 induces a left module 𝐿 by replacing the scalar multiplication with a reversed multiplication if the scalar ring is commutative. The hypothesis "rmodislmod.r" is a definition of a right module analogous to the definition df-lmod 18913 of a left module, see also islmod 18915. (Contributed by AV, 3-Dec-2021.)
⊢ 𝑉 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑅)    &   ⊢ 𝐹 = (Scalar‘𝑅)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ⨣ = (+g‘𝐹)    &   ⊢ × = (.r‘𝐹)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞 ∈ 𝐾 ∀𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 ∀𝑤 ∈ 𝑉 (((𝑤 · 𝑟) ∈ 𝑉 ∧ ((𝑤 + 𝑥) · 𝑟) = ((𝑤 · 𝑟) + (𝑥 · 𝑟)) ∧ (𝑤 · (𝑞 ⨣ 𝑟)) = ((𝑤 · 𝑞) + (𝑤 · 𝑟))) ∧ ((𝑤 · (𝑞 × 𝑟)) = ((𝑤 · 𝑞) · 𝑟) ∧ (𝑤 · 1 ) = 𝑤)))    &   ⊢ ∗ = (𝑠 ∈ 𝐾, 𝑣 ∈ 𝑉 ↦ (𝑣 · 𝑠))    &   ⊢ 𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ∗ ⟩)    ⇒   ⊢ (𝐹 ∈ CRing → 𝐿 ∈ LMod)
10.6.2  Subspaces and spans in a left module
Syntax	clss 18980	Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp
Definition	df-lss 18981*	Define the set of linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
⊢ LSubSp = (𝑤 ∈ V ↦ {𝑠 ∈ (𝒫 (Base‘𝑤) ∖ {∅}) ∣ ∀𝑥 ∈ (Base‘(Scalar‘𝑤))∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥( ·𝑠 ‘𝑤)𝑎)(+g‘𝑤)𝑏) ∈ 𝑠})
Theorem	lssset 18982*	The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ 𝑋 → 𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})
Theorem	islss 18983*	The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑎 ∈ 𝑈 ∀𝑏 ∈ 𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
Theorem	islssd 18984*	Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ (𝜑 → 𝐹 = (Scalar‘𝑊))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐹))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑊))    &   ⊢ (𝜑 → + = (+g‘𝑊))    &   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊))    &   ⊢ (𝜑 → 𝑆 = (LSubSp‘𝑊))    &   ⊢ (𝜑 → 𝑈 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑈 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝑈 ∈ 𝑆)
Theorem	lssss 18985	A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
Theorem	lssel 18986	A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉)
Theorem	lss1 18987	The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → 𝑉 ∈ 𝑆)
Theorem	lssuni 18988	The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    ⇒   ⊢ (𝜑 → ∪ 𝑆 = 𝑉)
Theorem	lssn0 18989	A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅)
Theorem	00lss 18990	The empty structure has no subspaces (for use with fvco4i 6315). (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ ∅ = (LSubSp‘∅)
Theorem	lsscl 18991	Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)
Theorem	lssvsubcl 18992	Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ − = (-g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → (𝑋 − 𝑌) ∈ 𝑈)
Theorem	lssvancl1 18993	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 19184. Can it be used along with lspsnne1 19165, lspsnne2 19166 to shorten this proof? (Contributed by NM, 14-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈)
Theorem	lssvancl2 18994	Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ¬ (𝑌 + 𝑋) ∈ 𝑈)
Theorem	lss0cl 18995	The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈)
Theorem	lsssn0 18996	The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆)
Theorem	lss0ss 18997	The zero subspace is included in every subspace. (sh0le 28427 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆) → { 0 } ⊆ 𝑋)
Theorem	lssle0 18998	No subspace is smaller than the zero subspace. (shle0 28429 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑋 ⊆ { 0 } ↔ 𝑋 = { 0 }))
Theorem	lssne0 18999*	A nonzero subspace has a nonzero vector. (shne0i 28435 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ (𝑋 ∈ 𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 ))
Theorem	lssneln0 19000	A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ LMod)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))