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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremabvfval 19701* Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
 
Theoremisabv 19702* Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
 
Theoremisabvd 19703* Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)
(𝜑𝐴 = (AbsVal‘𝑅))    &   (𝜑𝐵 = (Base‘𝑅))    &   (𝜑+ = (+g𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐹:𝐵⟶ℝ)    &   (𝜑 → (𝐹0 ) = 0)    &   ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))       (𝜑𝐹𝐴)
 
Theoremabvrcl 19704 Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)       (𝐹𝐴𝑅 ∈ Ring)
 
Theoremabvfge0 19705 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)       (𝐹𝐴𝐹:𝐵⟶(0[,)+∞))
 
Theoremabvf 19706 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)       (𝐹𝐴𝐹:𝐵⟶ℝ)
 
Theoremabvcl 19707 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)       ((𝐹𝐴𝑋𝐵) → (𝐹𝑋) ∈ ℝ)
 
Theoremabvge0 19708 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)       ((𝐹𝐴𝑋𝐵) → 0 ≤ (𝐹𝑋))
 
Theoremabveq0 19709 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 = (0g𝑅)       ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
 
Theoremabvne0 19710 The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴𝑋𝐵𝑋0 ) → (𝐹𝑋) ≠ 0)
 
Theoremabvgt0 19711 The absolute value of a nonzero number is strictly positive. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴𝑋𝐵𝑋0 ) → 0 < (𝐹𝑋))
 
Theoremabvmul 19712 An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))
 
Theoremabvtri 19713 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    + = (+g𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))
 
Theoremabv0 19714 The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    0 = (0g𝑅)       (𝐹𝐴 → (𝐹0 ) = 0)
 
Theoremabv1z 19715 The absolute value of one is one in a non-trivial ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝐹𝐴10 ) → (𝐹1 ) = 1)
 
Theoremabv1 19716 The absolute value of one is one in a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ DivRing ∧ 𝐹𝐴) → (𝐹1 ) = 1)
 
Theoremabvneg 19717 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 = (invg𝑅)       ((𝐹𝐴𝑋𝐵) → (𝐹‘(𝑁𝑋)) = (𝐹𝑋))
 
Theoremabvsubtri 19718 An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    = (-g𝑅)       ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 𝑌)) ≤ ((𝐹𝑋) + (𝐹𝑌)))
 
Theoremabvrec 19719 The absolute value distributes under reciprocal. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       (((𝑅 ∈ DivRing ∧ 𝐹𝐴) ∧ (𝑋𝐵𝑋0 )) → (𝐹‘(𝐼𝑋)) = (1 / (𝐹𝑋)))
 
Theoremabvdiv 19720 The absolute value distributes under division. (Contributed by Mario Carneiro, 10-Sep-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    / = (/r𝑅)       (((𝑅 ∈ DivRing ∧ 𝐹𝐴) ∧ (𝑋𝐵𝑌𝐵𝑌0 )) → (𝐹‘(𝑋 / 𝑌)) = ((𝐹𝑋) / (𝐹𝑌)))
 
Theoremabvdom 19721 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 = (0g𝑅)    &    · = (.r𝑅)       ((𝐹𝐴 ∧ (𝑋𝐵𝑋0 ) ∧ (𝑌𝐵𝑌0 )) → (𝑋 · 𝑌) ≠ 0 )
 
Theoremabvres 19722 The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐴 = (AbsVal‘𝑅)    &   𝑆 = (𝑅s 𝐶)    &   𝐵 = (AbsVal‘𝑆)       ((𝐹𝐴𝐶 ∈ (SubRing‘𝑅)) → (𝐹𝐶) ∈ 𝐵)
 
Theoremabvtrivd 19723* The trivial absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐹 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, 1))    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑦𝐵𝑦0 ) ∧ (𝑧𝐵𝑧0 )) → (𝑦 · 𝑧) ≠ 0 )       (𝜑𝐹𝐴)
 
Theoremabvtriv 19724* 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 19721 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 = (0g𝑅)    &   𝐹 = (𝑥𝐵 ↦ if(𝑥 = 0 , 0, 1))       (𝑅 ∈ DivRing → 𝐹𝐴)
 
Theoremabvpropd 19725* If two structures have the same ring components, they have the same collection of absolute values. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (AbsVal‘𝐾) = (AbsVal‘𝐿))
 
10.4.4  Star rings
 
Syntaxcstf 19726 Extend class notation with the functionalization of the *-ring involution.
class *rf
 
Syntaxcsr 19727 Extend class notation with class of all *-rings.
class *-Ring
 
Definitiondf-staf 19728* 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) ↦ ((*𝑟𝑓)‘𝑥)))
 
Definitiondf-srng 19729* 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[(*rf𝑓) / 𝑖](𝑖 ∈ (𝑓 RingHom (oppr𝑓)) ∧ 𝑖 = 𝑖)}
 
Theoremstaffval 19730* The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &    = (*rf𝑅)        = (𝑥𝐵 ↦ ( 𝑥))
 
Theoremstafval 19731 The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝐵 = (Base‘𝑅)    &    = (*𝑟𝑅)    &    = (*rf𝑅)       (𝐴𝐵 → ( 𝐴) = ( 𝐴))
 
Theoremstaffn 19732 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)    &    = (*𝑟𝑅)    &    = (*rf𝑅)       ( Fn 𝐵 = )
 
Theoremissrng 19733 The predicate "is a star ring". (Contributed by NM, 22-Sep-2011.) (Revised by Mario Carneiro, 6-Oct-2015.)
𝑂 = (oppr𝑅)    &    = (*rf𝑅)       (𝑅 ∈ *-Ring ↔ ( ∈ (𝑅 RingHom 𝑂) ∧ = ))
 
Theoremsrngrhm 19734 The involution function in a star ring is an antiautomorphism. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑂 = (oppr𝑅)    &    = (*rf𝑅)       (𝑅 ∈ *-Ring → ∈ (𝑅 RingHom 𝑂))
 
Theoremsrngring 19735 A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑅 ∈ *-Ring → 𝑅 ∈ Ring)
 
Theoremsrngcnv 19736 The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*rf𝑅)       (𝑅 ∈ *-Ring → = )
 
Theoremsrngf1o 19737 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 19738 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 19739 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 19740 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 19741 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 19742 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 19744.) (Contributed by Mario Carneiro, 6-Oct-2015.)
= (*𝑟𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ *-Ring → ( 1 ) = 1 )
 
Theoremsrng0 19743 The conjugate of the ring zero is zero. (Contributed by Mario Carneiro, 7-Oct-2015.)
= (*𝑟𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ *-Ring → ( 0 ) = 0 )
 
Theoremissrngd 19744* 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 19745* 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)
 
10.5  Left modules
 
10.5.1  Definition and basic properties
 
Syntaxclmod 19746 Extend class notation with class of all left modules.
class LMod
 
Syntaxcscaf 19747 The functionalization of the scalar multiplication operation.
class ·sf
 
Definitiondf-lmod 19748* 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 19749* 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 19750* 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 19751 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 19752* Properties that determine a left module. See note in isgrpd2 18234 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 19753 A left module is a group. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 25-Jun-2014.)
(𝑊 ∈ LMod → 𝑊 ∈ Grp)
 
Theoremlmodring 19754 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 19755 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 19756 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 19757 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 19758 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 19759 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 19760 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 19761 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 19762 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 19763 Closure of scalar product for a left module. (hvmulcl 28940 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 19764* 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)    &    = ( ·sf𝑊)    &    · = ( ·𝑠𝑊)        = (𝑥𝐾, 𝑦𝐵 ↦ (𝑥 · 𝑦))
 
Theoremscafval 19765 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 19766 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 19767 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 19768 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 19769 Distributive law for scalar product (left-distributivity). (ax-hvdistr1 28935 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 19770 Distributive law for scalar product (right-distributivity). (ax-hvdistr1 28935 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 19771 Associative law for scalar product. (ax-hvmulass 28934 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 19772 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 19773 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 19774 Scalar product with ring unit. (ax-hvmulid 28933 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 19775 The zero vector is a vector. (ax-hv0cl 28930 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ LMod → 0𝑉)
 
Theoremlmod0vlid 19776 Left identity law for the zero vector. (hvaddid2 28950 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 19777 Right identity law for the zero vector. (ax-hvaddid 28931 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 19778 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 19779 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (ax-hvmul0 28937 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 19780 Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (hvmul0 28951 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 19781 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 19782 Lemma 1 for lmodfopne 19784. (Contributed by AV, 2-Oct-2021.)
· = ( ·sf𝑊)    &    + = (+𝑓𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       ((𝑊 ∈ LMod ∧ + = · ) → 𝑉 = 𝐾)
 
Theoremlmodfopnelem2 19783 Lemma 2 for lmodfopne 19784. (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 19784 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 19785 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 19786 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 19787 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 19788 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 19789 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 19790 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 19791 Closure of vector subtraction. (hvsubcl 28944 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 19792 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 19793 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 19794 A left module is a commutative monoid under addition. (Contributed by NM, 7-Jan-2015.)
(𝑊 ∈ LMod → 𝑊 ∈ CMnd)
 
Theoremlmodnegadd 19795 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 19796 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 19797 Relationship between vector subtraction and addition. (hvsubadd 29004 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 19798 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 19799 Addition/subtraction cancellation law for vectors. (hvpncan 28966 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 19800 Cancellation law for vector subtraction (npcan 10966 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)
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