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

Theoremsrngcnv 18901 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*rf𝑅)       (𝑅 ∈ *-Ring → = )

Theoremsrngf1o 18902 The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*rf𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ *-Ring → :𝐵1-1-onto𝐵)

Theoremsrngcl 18903 The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵) → ( 𝑋) ∈ 𝐵)

Theoremsrngnvl 18904 The involution function in a star ring is an involution. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵) → ( ‘( 𝑋)) = 𝑋)

Theoremsrngadd 18905 The involution function in a star ring distributes over addition. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 + 𝑌)) = (( 𝑋) + ( 𝑌)))

Theoremsrngmul 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.)
= (*𝑟𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ *-Ring ∧ 𝑋𝐵𝑌𝐵) → ( ‘(𝑋 · 𝑌)) = (( 𝑌) · ( 𝑋)))

Theoremsrng1 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.)
= (*𝑟𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ *-Ring → ( 1 ) = 1 )

Theoremsrng0 18908 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
= (*𝑟𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ *-Ring → ( 0 ) = 0 )

Theoremissrngd 18909* Properties that determine a star ring. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2015.)
(𝜑𝐾 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑 = (*𝑟𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑥𝐾) → ( 𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 + 𝑦)) = (( 𝑥) + ( 𝑦)))    &   ((𝜑𝑥𝐾𝑦𝐾) → ( ‘(𝑥 · 𝑦)) = (( 𝑦) · ( 𝑥)))    &   ((𝜑𝑥𝐾) → ( ‘( 𝑥)) = 𝑥)       (𝜑𝑅 ∈ *-Ring)

Theoremidsrngd 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)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)       (𝜑𝑅 ∈ *-Ring)

10.6  Left modules

10.6.1  Definition and basic properties

Syntaxclmod 18911 Extend class notation with class of all left modules.
class LMod

Syntaxcscaf 18912 The functionalization of the scalar multiplication operation.
class ·sf

Definitiondf-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][(+g𝑔) / 𝑎][(Scalar‘𝑔) / 𝑓][( ·𝑠𝑔) / 𝑠][(Base‘𝑓) / 𝑘][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡](𝑓 ∈ Ring ∧ ∀𝑞𝑘𝑟𝑘𝑥𝑣𝑤𝑣 (((𝑟𝑠𝑤) ∈ 𝑣 ∧ (𝑟𝑠(𝑤𝑎𝑥)) = ((𝑟𝑠𝑤)𝑎(𝑟𝑠𝑥)) ∧ ((𝑞𝑝𝑟)𝑠𝑤) = ((𝑞𝑠𝑤)𝑎(𝑟𝑠𝑤))) ∧ (((𝑞𝑡𝑟)𝑠𝑤) = (𝑞𝑠(𝑟𝑠𝑤)) ∧ ((1r𝑓)𝑠𝑤) = 𝑤)))}

Definitiondf-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( ·𝑠𝑔)𝑦)))

Theoremislmod 18915* The predicate "is a left module". (Contributed by NM, 4-Nov-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)       (𝑊 ∈ LMod ↔ (𝑊 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((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))))

Theoremlmodlema 18916 Lemma for properties of a left module. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾) ∧ (𝑋𝑉𝑌𝑉)) → (((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)))

Theoremislmodd 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.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑 = (+g𝐹))    &   (𝜑× = (.r𝐹))    &   (𝜑1 = (1r𝐹))    &   (𝜑𝐹 ∈ Ring)    &   (𝜑𝑊 ∈ Grp)    &   ((𝜑𝑥𝐵𝑦𝑉) → (𝑥 · 𝑦) ∈ 𝑉)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝑉𝑧𝑉)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝑉)) → ((𝑥 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝑉)) → ((𝑥 × 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧)))    &   ((𝜑𝑥𝑉) → ( 1 · 𝑥) = 𝑥)       (𝜑𝑊 ∈ LMod)

Theoremlmodgrp 18918 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
(𝑊 ∈ LMod → 𝑊 ∈ Grp)

Theoremlmodring 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)

Theoremlmodfgrp 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)

Theoremlmodbn0 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 ≠ ∅)

Theoremlmodacl 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)    &    + = (+g𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)

Theoremlmodmcl 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)    &    · = (.r𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)

Theoremlmodsn0 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 ≠ ∅)

Theoremlmodvacl 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)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) ∈ 𝑉)

Theoremlmodass 18926 Left module vector sum is associative. (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theoremlmodlcan 18927 Left cancellation law for vector sum. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉𝑍𝑉)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Theoremlmodvscl 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝑉) → (𝑅 · 𝑋) ∈ 𝑉)

Theoremscaffval 18929* The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)        = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))

Theoremscafval 18930 The scalar multiplication operation as a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       ((𝑋𝐾𝑌𝐵) → (𝑋 𝑌) = (𝑋 · 𝑌))

Theoremscafeq 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)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)       ( · Fn (𝐾 × 𝐵) → = · )

Theoremscaffn 18932 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)        Fn (𝐾 × 𝐵)

Theoremlmodscaf 18933 The scalar multiplication operation is a function. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = ( ·sf𝑊)       (𝑊 ∈ LMod → :(𝐾 × 𝐵)⟶𝐵)

Theoremlmodvsdi 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)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ (𝑅𝐾𝑋𝑉𝑌𝑉)) → (𝑅 · (𝑋 + 𝑌)) = ((𝑅 · 𝑋) + (𝑅 · 𝑌)))

Theoremlmodvsdir 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)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))

Theoremlmodvsass 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    × = (.r𝐹)       ((𝑊 ∈ LMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 × 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))

Theoremlmod0cl 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 = (0g𝐹)       (𝑊 ∈ LMod → 0𝐾)

Theoremlmod1cl 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 = (1r𝐹)       (𝑊 ∈ LMod → 1𝐾)

Theoremlmodvs1 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)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ( 1 · 𝑋) = 𝑋)

Theoremlmod0vcl 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 = (0g𝑊)       (𝑊 ∈ LMod → 0𝑉)

Theoremlmod0vlid 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)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ( 0 + 𝑋) = 𝑋)

Theoremlmod0vrid 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)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)

Theoremlmod0vid 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)    &    + = (+g𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑋 + 𝑋) = 𝑋0 = 𝑋))

Theoremlmod0vs 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)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑂 · 𝑋) = 0 )

Theoremlmodvs0 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝐾) → (𝑋 · 0 ) = 0 )

Theoremlmodvsmmulgdi 18946 Distributive law for a group multiple of a scalar multiplication. (Contributed by AV, 2-Sep-2019.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    = (.g𝑊)    &   𝐸 = (.g𝐹)       ((𝑊 ∈ LMod ∧ (𝐶𝐾𝑁 ∈ ℕ0𝑋𝑉)) → (𝑁 (𝐶 · 𝑋)) = ((𝑁𝐸𝐶) · 𝑋))

Theoremlmodfopnelem1 18947 Lemma 1 for lmodfopne 18949. (Contributed by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)

Theoremlmodfopnelem2 18948 Lemma 2 for lmodfopne 18949. (Contributed by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝑊 ∈ LMod ∧ + = · ) → ( 0𝑉1𝑉))

Theoremlmodfopne 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.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    1 = (1r𝑆)       ((𝑊 ∈ LMod ∧ 10 ) → +· )

Theoremlcomf 18950 A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺:𝐼𝐾)    &   (𝜑𝐻:𝐼𝐵)    &   (𝜑𝐼𝑉)       (𝜑 → (𝐺𝑓 · 𝐻):𝐼𝐵)

Theoremlcomfsupp 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)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐺:𝐼𝐾)    &   (𝜑𝐻:𝐼𝐵)    &   (𝜑𝐼𝑉)    &    0 = (0g𝑊)    &   𝑌 = (0g𝐹)    &   (𝜑𝐺 finSupp 𝑌)       (𝜑 → (𝐺𝑓 · 𝐻) finSupp 0 )

Theoremlmodvnegcl 18952 Closure of vector negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁𝑋) ∈ 𝑉)

Theoremlmodvnegid 18953 Addition of a vector with its negative. (Contributed by NM, 18-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑋 + (𝑁𝑋)) = 0 )

Theoremlmodvneg1 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 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    1 = (1r𝐹)    &   𝑀 = (invg𝐹)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑀1 ) · 𝑋) = (𝑁𝑋))

Theoremlmodvsneg 18955 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑀 = (invg𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = ((𝑀𝑅) · 𝑋))

Theoremlmodvsubcl 18956 Closure of vector subtraction. (hvsubcl 28002 analog.) (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 𝑌) ∈ 𝑉)

Theoremlmodcom 18957 Left module vector sum is commutative. (Contributed by Gérard Lang, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremlmodabl 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)

Theoremlmodcmn 18959 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
(𝑊 ∈ LMod → 𝑊 ∈ CMnd)

Theoremlmodnegadd 18960 Distribute negation through addition of scalar products. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝑅 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑅)    &   𝐼 = (invg𝑅)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘((𝐴 · 𝑋) + (𝐵 · 𝑌))) = (((𝐼𝐴) · 𝑋) + ((𝐼𝐵) · 𝑌)))

Theoremlmod4 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)    &    + = (+g𝑊)       ((𝑊 ∈ LMod ∧ (𝑋𝑉𝑌𝑉) ∧ (𝑍𝑉𝑈𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈)))

Theoremlmodvsubadd 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)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))

Theoremlmodvaddsub4 18963 Vector addition/subtraction law. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ (𝐴𝑉𝐵𝑉) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 𝐶) = (𝐷 𝐵)))

Theoremlmodvpncan 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)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 + 𝐵) 𝐵) = 𝐴)

Theoremlmodvnpcan 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)    &    + = (+g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 𝐵) + 𝐵) = 𝐴)

Theoremlmodvsubval2 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)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝐹)    &    1 = (1r𝐹)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → (𝐴 𝐵) = (𝐴 + ((𝑁1 ) · 𝐵)))

Theoremlmodsubvs 18967 Subtraction of a scalar product in terms of addition. (Contributed by NM, 9-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑁 = (invg𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 (𝐴 · 𝑌)) = (𝑋 + ((𝑁𝐴) · 𝑌)))

Theoremlmodsubdi 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)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝐴 · (𝑋 𝑌)) = ((𝐴 · 𝑋) (𝐴 · 𝑌)))

Theoremlmodsubdir 18969 Scalar multiplication distributive law for subtraction. (hvsubdistr2 28035 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝑆𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))

Theoremlmodsubeq0 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 = (0g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 𝐵) = 0𝐴 = 𝐵))

Theoremlmodsubid 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 = (0g𝑊)    &    = (-g𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑉) → (𝐴 𝐴) = 0 )

Theoremlmodvsghm 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)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾) → (𝑥𝑉 ↦ (𝑅 · 𝑥)) ∈ (𝑊 GrpHom 𝑊))

Theoremlmodprop2d 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑𝑃 = (Base‘𝐹))    &   (𝜑𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g𝐹)𝑦) = (𝑥(+g𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(.r𝐹)𝑦) = (𝑥(.r𝐺)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Theoremlmodpropd 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.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))       (𝜑 → (𝐾 ∈ LMod ↔ 𝐿 ∈ LMod))

Theoremgsumvsmul 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 = (0g𝑅)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   (𝜑𝑅 ∈ LMod)    &   (𝜑𝐴𝑉)    &   (𝜑𝑋𝐾)    &   ((𝜑𝑘𝐴) → 𝑌𝐵)    &   (𝜑 → (𝑘𝐴𝑌) finSupp 0 )       (𝜑 → (𝑅 Σg (𝑘𝐴 ↦ (𝑋 · 𝑌))) = (𝑋 · (𝑅 Σg (𝑘𝐴𝑌))))

Theoremmptscmfsupp0 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.)
(𝜑𝐷𝑉)    &   (𝜑𝑄 ∈ LMod)    &   (𝜑𝑅 = (Scalar‘𝑄))    &   𝐾 = (Base‘𝑄)    &   ((𝜑𝑘𝐷) → 𝑆𝐵)    &   ((𝜑𝑘𝐷) → 𝑊𝐾)    &    0 = (0g𝑄)    &   𝑍 = (0g𝑅)    &    = ( ·𝑠𝑄)    &   (𝜑 → (𝑘𝐷𝑆) finSupp 𝑍)       (𝜑 → (𝑘𝐷 ↦ (𝑆 𝑊)) finSupp 0 )

Theoremmptscmfsuppd 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)    &    · = ( ·𝑠𝑃)    &   (𝜑𝑃 ∈ LMod)    &   (𝜑𝑋𝑉)    &   ((𝜑𝑘𝑋) → 𝑍𝐵)    &   (𝜑𝐴:𝑋𝑌)    &   (𝜑𝐴 finSupp (0g𝑆))       (𝜑 → (𝑘𝑋 ↦ ((𝐴𝑘) · 𝑍)) finSupp (0g𝑃))

Theoremrmodislmodlem 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)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((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)))    &    = (𝑠𝐾, 𝑣𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ⟩)       ((𝐹 ∈ CRing ∧ (𝑎𝐾𝑏𝐾𝑐𝑉)) → ((𝑎 × 𝑏) 𝑐) = (𝑎 (𝑏 𝑐)))

Theoremrmodislmod 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)    &    + = (+g𝑅)    &    · = ( ·𝑠𝑅)    &   𝐹 = (Scalar‘𝑅)    &   𝐾 = (Base‘𝐹)    &    = (+g𝐹)    &    × = (.r𝐹)    &    1 = (1r𝐹)    &   (𝑅 ∈ Grp ∧ 𝐹 ∈ Ring ∧ ∀𝑞𝐾𝑟𝐾𝑥𝑉𝑤𝑉 (((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)))    &    = (𝑠𝐾, 𝑣𝑉 ↦ (𝑣 · 𝑠))    &   𝐿 = (𝑅 sSet ⟨( ·𝑠 ‘ndx), ⟩)       (𝐹 ∈ CRing → 𝐿 ∈ LMod)

10.6.2  Subspaces and spans in a left module

Syntaxclss 18980 Extend class notation with linear subspaces of a left module or left vector space.
class LSubSp

Definitiondf-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))∀𝑎𝑠𝑏𝑠 ((𝑥( ·𝑠𝑤)𝑎)(+g𝑤)𝑏) ∈ 𝑠})

Theoremlssset 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)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊𝑋𝑆 = {𝑠 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∀𝑥𝐵𝑎𝑠𝑏𝑠 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑠})

Theoremislss 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)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑈𝑆 ↔ (𝑈𝑉𝑈 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))

Theoremislssd 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.)
(𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝑆 = (LSubSp‘𝑊))    &   (𝜑𝑈𝑉)    &   (𝜑𝑈 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑎𝑈𝑏𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)       (𝜑𝑈𝑆)

Theoremlssss 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)       (𝑈𝑆𝑈𝑉)

Theoremlssel 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)       ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)

Theoremlss1 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 → 𝑉𝑆)

Theoremlssuni 18988 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 𝑆 = 𝑉)

Theoremlssn0 18989 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈 ≠ ∅)

Theorem00lss 18990 The empty structure has no subspaces (for use with fvco4i 6315). (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (LSubSp‘∅)

Theoremlsscl 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)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑈𝑆 ∧ (𝑍𝐵𝑋𝑈𝑌𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)

Theoremlssvsubcl 18992 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
= (-g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝑈𝑌𝑈)) → (𝑋 𝑌) ∈ 𝑈)

Theoremlssvancl1 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)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈)

Theoremlssvancl2 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)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑌 + 𝑋) ∈ 𝑈)

Theoremlss0cl 18995 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 0𝑈)

Theoremlsssn0 18996 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → { 0 } ∈ 𝑆)

Theoremlss0ss 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 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → { 0 } ⊆ 𝑋)

Theoremlssle0 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 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → (𝑋 ⊆ { 0 } ↔ 𝑋 = { 0 }))

Theoremlssne0 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 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))

Theoremlssneln0 19000 A vector which doesn't belong to a subspace is nonzero. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))

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