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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlveclmod 19501 A left vector space is a left module. (Contributed by NM, 9-Dec-2013.)
(𝑊 ∈ LVec → 𝑊 ∈ LMod)

Theoremlsslvec 19502 A vector subspace is a vector space. (Contributed by NM, 14-Mar-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LVec ∧ 𝑈𝑆) → 𝑋 ∈ LVec)

Theoremlvecvs0or 19503 If a scalar product is zero, one of its factors must be zero. (hvmul0or 28454 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴 · 𝑋) = 0 ↔ (𝐴 = 𝑂𝑋 = 0 )))

Theoremlvecvsn0 19504 A scalar product is nonzero iff both of its factors are nonzero. (Contributed by NM, 3-Jan-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴 · 𝑋) ≠ 0 ↔ (𝐴𝑂𝑋0 )))

Theoremlssvs0or 19505 If a scalar product belongs to a subspace, either the scalar component is zero or the vector component also belongs to the subspace. (Contributed by NM, 5-Apr-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝐴𝐾)       (𝜑 → ((𝐴 · 𝑋) ∈ 𝑈 ↔ (𝐴 = 0𝑋𝑈)))

Theoremlvecvscan 19506 Cancellation law for scalar multiplication. (hvmulcan 28501 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴0 )       (𝜑 → ((𝐴 · 𝑋) = (𝐴 · 𝑌) ↔ 𝑋 = 𝑌))

Theoremlvecvscan2 19507 Cancellation law for scalar multiplication. (hvmulcan2 28502 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑋0 )       (𝜑 → ((𝐴 · 𝑋) = (𝐵 · 𝑋) ↔ 𝐴 = 𝐵))

Theoremlvecinv 19508 Invert coefficient of scalar product. (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝐼 = (invr𝐹)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐴 ∈ (𝐾 ∖ { 0 }))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 = (𝐴 · 𝑌) ↔ 𝑌 = ((𝐼𝐴) · 𝑋)))

Theoremlspsnvs 19509 A nonzero scalar product does not change the span of a singleton. (spansncol 28999 analog.) (Contributed by NM, 23-Apr-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    0 = (0g𝐹)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LVec ∧ (𝑅𝐾𝑅0 ) ∧ 𝑋𝑉) → (𝑁‘{(𝑅 · 𝑋)}) = (𝑁‘{𝑋}))

Theoremlspsneleq 19510 Membership relation that implies equality of spans. (spansneleq 29001 analog.) (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))    &   (𝜑𝑌0 )       (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))

Theoremlspsncmp 19511 Comparable spans of nonzero singletons are equal. (Contributed by NM, 27-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ (𝑁‘{𝑋}) = (𝑁‘{𝑌})))

Theoremlspsnne1 19512 Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))

Theoremlspsnne2 19513 Two ways to express that vectors have different spans. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))       (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

Theoremlspsnnecom 19514 Swap two vectors with different spans. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))       (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋}))

Theoremlspabs2 19515 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Theoremlspabs3 19516 Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑋 + 𝑌) ≠ 0 )    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))       (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))

Theoremlspsneq 19517* Equal spans of singletons must have proportional vectors. See lspsnss2 19400 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ (𝐾 ∖ { 0 })𝑋 = (𝑘 · 𝑌)))

Theoremlspsneu 19518* Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝑂 = (0g𝑆)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ((𝑁‘{𝑋}) = (𝑁‘{𝑌}) ↔ ∃!𝑘 ∈ (𝐾 ∖ {𝑂})𝑋 = (𝑘 · 𝑌)))

Theoremlspsnel4 19519 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn4 29004 analog.) (Contributed by NM, 4-Jul-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))    &   (𝜑𝑌0 )       (𝜑 → (𝑋𝑈𝑌𝑈))

Theoremlspdisj 19520 The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑 → ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 })

Theoremlspdisjb 19521 A nonzero vector is not in a subspace iff its span is disjoint with the subspace. (Contributed by NM, 23-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (¬ 𝑋𝑈 ↔ ((𝑁‘{𝑋}) ∩ 𝑈) = { 0 }))

Theoremlspdisj2 19522 Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌})) = { 0 })

Theoremlspfixed 19523* Show membership in the span of the sum of two vectors, one of which (𝑌) is fixed in advance. (Contributed by NM, 27-May-2015.) (Revised by AV, 12-Jul-2022.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

TheoremlspfixedOLD 19524* Obsolete version of lspfixed 19523 as of 12-Jul-2022. (Contributed by NM, 27-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ∃𝑧 ∈ ((𝑁‘{𝑍}) ∖ { 0 })𝑋 ∈ (𝑁‘{(𝑌 + 𝑧)}))

Theoremlspexch 19525 Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 19526 versus lspexchn2 19527); look for lspexch 19525 and prcom 4499 in same proof. TODO: would a hypothesis of ¬ 𝑋 ∈ (𝑁‘{𝑍}) instead of (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) be better overall? This would be shorter and also satisfy the 𝑋0 condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))    &   (𝜑𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑𝑌 ∈ (𝑁‘{𝑋, 𝑍}))

Theoremlspexchn1 19526 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 19525 to see if this will shorten proofs. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))

Theoremlspexchn2 19527 Exchange property for span of a pair with negated membership. TODO: look at uses of lspexch 19525 to see if this will shorten proofs. (Contributed by NM, 24-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍}))    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌}))       (𝜑 → ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋}))

Theoremlspindpi 19528 Partial independence property. (Contributed by NM, 23-Apr-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))       (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))

Theoremlspindp1 19529 Alternate way to say 3 vectors are mutually independent (swap 1st and 2nd). (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, 𝑌})))

Theoremlspindp2l 19530 Alternate way to say 3 vectors are mutually independent (rotate left). (Contributed by NM, 10-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ((𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))

Theoremlspindp2 19531 Alternate way to say 3 vectors are mutually independent (rotate right). (Contributed by NM, 12-Apr-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑍𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑍, 𝑋})))

Theoremlspindp3 19532 Independence of 2 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑋 + 𝑌)}))

Theoremlspindp4 19533 (Partial) independence of 3 vectors is preserved by vector sum. (Contributed by NM, 26-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑋 + 𝑌)}))

Theoremlvecindp 19534 Compute the 𝑋 coefficient in a sum with an independent vector 𝑋 (first conjunct), which can then be removed to continue with the remaining vectors summed in expressions 𝑌 and 𝑍 (second conjunct). Typically, 𝑈 is the span of the remaining vectors. (Contributed by NM, 5-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 19-Jul-2022.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑 → ((𝐴 · 𝑋) + 𝑌) = ((𝐵 · 𝑋) + 𝑍))       (𝜑 → (𝐴 = 𝐵𝑌 = 𝑍))

Theoremlvecindp2 19535 Sums of independent vectors must have equal coefficients. (Contributed by NM, 22-Mar-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝐶𝐾)    &   (𝜑𝐷𝐾)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   (𝜑 → ((𝐴 · 𝑋) + (𝐵 · 𝑌)) = ((𝐶 · 𝑋) + (𝐷 · 𝑌)))       (𝜑 → (𝐴 = 𝐶𝐵 = 𝐷))

Theoremlspsnsubn0 19536 Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 𝑌) ≠ 0 )

Theoremlsmcv 19537 Subspace sum has the covering property (using spans of singletons to represent atoms). Similar to Exercise 5 of [Kalmbach] p. 153. (spansncvi 29083 analog.) TODO: ugly proof; can it be shortened? (Contributed by NM, 2-Oct-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑇𝑆)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)       ((𝜑𝑇𝑈𝑈 ⊆ (𝑇 (𝑁‘{𝑋}))) → 𝑈 = (𝑇 (𝑁‘{𝑋})))

Theoremlspsolvlem 19538* Lemma for lspsolv 19539. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑄 = {𝑧𝑉 ∣ ∃𝑟𝐵 (𝑧 + (𝑟 · 𝑌)) ∈ (𝑁𝐴)}    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋 ∈ (𝑁‘(𝐴 ∪ {𝑌})))       (𝜑 → ∃𝑟𝐵 (𝑋 + (𝑟 · 𝑌)) ∈ (𝑁𝐴))

Theoremlspsolv 19539 If 𝑋 is in the span of 𝐴 ∪ {𝑌} but not 𝐴, then 𝑌 is in the span of 𝐴 ∪ {𝑋}. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LVec ∧ (𝐴𝑉𝑌𝑉𝑋 ∈ ((𝑁‘(𝐴 ∪ {𝑌})) ∖ (𝑁𝐴)))) → 𝑌 ∈ (𝑁‘(𝐴 ∪ {𝑋})))

Theoremlssacsex 19540* In a vector space, subspaces form an algebraic closure system whose closure operator has the exchange property. Strengthening of lssacs 19362 by lspsolv 19539. (Contributed by David Moews, 1-May-2017.)
𝐴 = (LSubSp‘𝑊)    &   𝑁 = (mrCls‘𝐴)    &   𝑋 = (Base‘𝑊)       (𝑊 ∈ LVec → (𝐴 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))))

Theoremlspsnat 19541 There is no subspace strictly between the zero subspace and the span of a vector (i.e. a 1-dimensional subspace is an atom). (h1datomi 29012 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (((𝑊 ∈ LVec ∧ 𝑈𝑆𝑋𝑉) ∧ 𝑈 ⊆ (𝑁‘{𝑋})) → (𝑈 = (𝑁‘{𝑋}) ∨ 𝑈 = { 0 }))

Theoremlspsncv0 19542* The span of a singleton covers the zero subspace, using Definition 3.2.18 of [PtakPulmannova] p. 68 for "covers".) (Contributed by NM, 12-Aug-2014.) (Revised by AV, 13-Jul-2022.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)       (𝜑 → ¬ ∃𝑦𝑆 ({ 0 } ⊊ 𝑦𝑦 ⊊ (𝑁‘{𝑋})))

Theoremlspsncv0OLD 19543* Obsolete version of lspsncv0 19542 as of 13-Jul-2022. (Contributed by NM, 12-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑋𝑉)    &   (𝜑𝑋0 )       (𝜑 → ¬ ∃𝑦𝑆 ({ 0 } ⊊ 𝑦𝑦 ⊊ (𝑁‘{𝑋})))

Theoremlsppratlem1 19544 Lemma for lspprat 19550. Let 𝑥 ∈ (𝑈 ∖ {0}) (if there is no such 𝑥 then 𝑈 is the zero subspace), and let 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) (assuming the conclusion is false). The goal is to write 𝑋, 𝑌 in terms of 𝑥, 𝑦, which would normally be done by solving the system of linear equations. The span equivalent of this process is lspsolv 19539 (hence the name), which we use extensively below. In this lemma, we show that since 𝑥 ∈ (𝑁‘{𝑋, 𝑌}), either 𝑥 ∈ (𝑁‘{𝑌}) or 𝑋 ∈ (𝑁‘{𝑥, 𝑌}). (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)    &   (𝜑𝑥 ∈ (𝑈 ∖ { 0 }))    &   (𝜑𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))       (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})))

Theoremlsppratlem2 19545 Lemma for lspprat 19550. Show that if 𝑋 and 𝑌 are both in (𝑁‘{𝑥, 𝑦}) (which will be our goal for each of the two cases above), then (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈, contradicting the hypothesis for 𝑈. (Contributed by NM, 29-Aug-2014.) (Revised by Mario Carneiro, 5-Sep-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)    &   (𝜑𝑥 ∈ (𝑈 ∖ { 0 }))    &   (𝜑𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   (𝜑𝑋 ∈ (𝑁‘{𝑥, 𝑦}))    &   (𝜑𝑌 ∈ (𝑁‘{𝑥, 𝑦}))       (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)

Theoremlsppratlem3 19546 Lemma for lspprat 19550. In the first case of lsppratlem1 19544, since 𝑥 ∉ (𝑁‘∅), also 𝑌 ∈ (𝑁‘{𝑥}), and since 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑥}) and 𝑦 ∉ (𝑁‘{𝑥}), we have 𝑋 ∈ (𝑁‘{𝑥, 𝑦}) as desired. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)    &   (𝜑𝑥 ∈ (𝑈 ∖ { 0 }))    &   (𝜑𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   (𝜑𝑥 ∈ (𝑁‘{𝑌}))       (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Theoremlsppratlem4 19547 Lemma for lspprat 19550. In the second case of lsppratlem1 19544, 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑌}) and 𝑦 ∉ (𝑁‘{𝑥}) implies 𝑌 ∈ (𝑁‘{𝑥, 𝑦}) and thus 𝑋 ∈ (𝑁‘{𝑥, 𝑌}) ⊆ (𝑁‘{𝑥, 𝑦}) as well. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)    &   (𝜑𝑥 ∈ (𝑈 ∖ { 0 }))    &   (𝜑𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))    &   (𝜑𝑋 ∈ (𝑁‘{𝑥, 𝑌}))       (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Theoremlsppratlem5 19548 Lemma for lspprat 19550. Combine the two cases and show a contradiction to 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}) under the assumptions on 𝑥 and 𝑦. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)    &   (𝜑𝑥 ∈ (𝑈 ∖ { 0 }))    &   (𝜑𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))       (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)

Theoremlsppratlem6 19549 Lemma for lspprat 19550. Negating the assumption on 𝑦, we arrive close to the desired conclusion. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))    &    0 = (0g𝑊)       (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥})))

Theoremlspprat 19550* A proper subspace of the span of a pair of vectors is the span of a singleton (an atom) or the zero subspace (if 𝑧 is zero). Proof suggested by Mario Carneiro, 28-Aug-2014. (Contributed by NM, 29-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))       (𝜑 → ∃𝑧𝑉 𝑈 = (𝑁‘{𝑧}))

Theoremislbs2 19551* An equivalent formulation of the basis predicate in a vector space: a subset is a basis iff no element is in the span of the rest of the set. (Contributed by Mario Carneiro, 14-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LVec → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵 ¬ 𝑥 ∈ (𝑁‘(𝐵 ∖ {𝑥})))))

Theoremislbs3 19552* An equivalent formulation of the basis predicate: a subset is a basis iff it is a minimal spanning set. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LVec → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑠(𝑠𝐵 → (𝑁𝑠) ⊊ 𝑉))))

Theoremlbsacsbs 19553 Being a basis in a vector space is equivalent to being a basis in the associated algebraic closure system. Equivalent to islbs2 19551. (Contributed by David Moews, 1-May-2017.)
𝐴 = (LSubSp‘𝑊)    &   𝑁 = (mrCls‘𝐴)    &   𝑋 = (Base‘𝑊)    &   𝐼 = (mrInd‘𝐴)    &   𝐽 = (LBasis‘𝑊)       (𝑊 ∈ LVec → (𝑆𝐽 ↔ (𝑆𝐼 ∧ (𝑁𝑆) = 𝑋)))

Theoremlvecdim 19554 The dimension theorem for vector spaces: any two bases of the same vector space are equinumerous. Proven by using lssacsex 19540 and lbsacsbs 19553 to show that being a basis for a vector space is equivalent to being a basis for the associated algebraic closure system, and then using acsexdimd 17569. (Contributed by David Moews, 1-May-2017.)
𝐽 = (LBasis‘𝑊)       ((𝑊 ∈ LVec ∧ 𝑆𝐽𝑇𝐽) → 𝑆𝑇)

Theoremlbsextlem1 19555* Lemma for lbsext 19560. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}       (𝜑𝑆 ≠ ∅)

Theoremlbsextlem2 19556* Lemma for lbsext 19560. Since 𝐴 is a chain (actually, we only need it to be closed under binary union), the union 𝑇 of the spans of each individual element of 𝐴 is a subspace, and it contains all of 𝐴 (except for our target vector 𝑥- we are trying to make 𝑥 a linear combination of all the other vectors in some set from 𝐴). (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   𝑃 = (LSubSp‘𝑊)    &   (𝜑𝐴𝑆)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → [] Or 𝐴)    &   𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))       (𝜑 → (𝑇𝑃 ∧ ( 𝐴 ∖ {𝑥}) ⊆ 𝑇))

Theoremlbsextlem3 19557* Lemma for lbsext 19560. A chain in 𝑆 has an upper bound in 𝑆. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   𝑃 = (LSubSp‘𝑊)    &   (𝜑𝐴𝑆)    &   (𝜑𝐴 ≠ ∅)    &   (𝜑 → [] Or 𝐴)    &   𝑇 = 𝑢𝐴 (𝑁‘(𝑢 ∖ {𝑥}))       (𝜑 𝐴𝑆)

Theoremlbsextlem4 19558* Lemma for lbsext 19560. lbsextlem3 19557 satisfies the conditions for the application of Zorn's lemma zorn 9664 (thus invoking AC), and so there is a maximal linearly independent set extending 𝐶. Here we prove that such a set is a basis. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐶𝑉)    &   (𝜑 → ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))    &   𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶𝑧 ∧ ∀𝑥𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))}    &   (𝜑 → 𝒫 𝑉 ∈ dom card)       (𝜑 → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsextg 19559* For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (((𝑊 ∈ LVec ∧ 𝒫 𝑉 ∈ dom card) ∧ 𝐶𝑉 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsext 19560* For any linearly independent subset 𝐶 of 𝑉, there is a basis containing the vectors in 𝐶. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LVec ∧ 𝐶𝑉 ∧ ∀𝑥𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) → ∃𝑠𝐽 𝐶𝑠)

Theoremlbsexg 19561 Every vector space has a basis. This theorem is an AC equivalent; this is the forward implication. (Contributed by Mario Carneiro, 17-May-2015.)
𝐽 = (LBasis‘𝑊)       ((CHOICE𝑊 ∈ LVec) → 𝐽 ≠ ∅)

Theoremlbsex 19562 Every vector space has a basis. This theorem is an AC equivalent. (Contributed by Mario Carneiro, 25-Jun-2014.)
𝐽 = (LBasis‘𝑊)       (𝑊 ∈ LVec → 𝐽 ≠ ∅)

Theoremlvecprop2d 19563* If two structures have the same components (properties), one is a left vector space iff the other one is. This version of lvecpropd 19564 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 ∈ LVec ↔ 𝐿 ∈ LVec))

Theoremlvecpropd 19564* If two structures have the same components (properties), one is a left vector space iff the other one is. (Contributed 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 ∈ LVec ↔ 𝐿 ∈ LVec))

10.8  Ideals

10.8.1  The subring algebra; ideals

Syntaxcsra 19565 Extend class notation with the subring algebra generator.
class subringAlg

Syntaxcrglmod 19566 Extend class notation with the left module induced by a ring over itself.
class ringLMod

Syntaxclidl 19567 Ring left-ideal function.
class LIdeal

Syntaxcrsp 19568 Ring span function.
class RSpan

Definitiondf-sra 19569* Any ring can be regarded as a left algebra over any of its subrings. The function subringAlg associates with any ring and any of its subrings the left algebra consisting in the ring itself regarded as a left algebra over the subring. It has an inner product which is simply the ring product. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
subringAlg = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ (((𝑤 sSet ⟨(Scalar‘ndx), (𝑤s 𝑠)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑤)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑤)⟩)))

Definitiondf-rgmod 19570 Any ring can be regarded as a left algebra over itself. The function ringLMod associates with any ring the left algebra consisting in the ring itself regarded as a left algebra over itself. It has an inner product which is simply the ring product. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod = (𝑤 ∈ V ↦ ((subringAlg ‘𝑤)‘(Base‘𝑤)))

Definitiondf-lidl 19571 Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
LIdeal = (LSubSp ∘ ringLMod)

Definitiondf-rsp 19572 Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
RSpan = (LSpan ∘ ringLMod)

Theoremsraval 19573 Lemma for srabase 19575 through sravsca 19579. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Thierry Arnoux, 16-Jun-2019.)
((𝑊𝑉𝑆 ⊆ (Base‘𝑊)) → ((subringAlg ‘𝑊)‘𝑆) = (((𝑊 sSet ⟨(Scalar‘ndx), (𝑊s 𝑆)⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Theoremsralem 19574 Lemma for srabase 19575 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   (𝑁 < 5 ∨ 8 < 𝑁)       (𝜑 → (𝐸𝑊) = (𝐸𝐴))

Theoremsrabase 19575 Base set of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (Base‘𝑊) = (Base‘𝐴))

Theoremsraaddg 19576 Additive operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (+g𝑊) = (+g𝐴))

Theoremsramulr 19577 Multiplicative operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (.r𝐴))

Theoremsrasca 19578 The set of scalars of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (𝑊s 𝑆) = (Scalar‘𝐴))

Theoremsravsca 19579 The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = ( ·𝑠𝐴))

Theoremsraip 19580 The inner product operation of a subring algebra. (Contributed by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (.r𝑊) = (·𝑖𝐴))

Theoremsratset 19581 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopSet‘𝑊) = (TopSet‘𝐴))

Theoremsratopn 19582 Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (TopOpen‘𝑊) = (TopOpen‘𝐴))

Theoremsrads 19583 Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑 → (dist‘𝑊) = (dist‘𝐴))

Theoremsralmod 19584 The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Theoremsralmod0 19585 The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.)
(𝜑𝐴 = ((subringAlg ‘𝑊)‘𝑆))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝑆 ⊆ (Base‘𝑊))       (𝜑0 = (0g𝐴))

Theoremissubrngd2 19586* Prove a subring by closure (definition version). (Contributed by Stefan O'Rear, 7-Dec-2014.)
(𝜑𝑆 = (𝐼s 𝐷))    &   (𝜑0 = (0g𝐼))    &   (𝜑+ = (+g𝐼))    &   (𝜑𝐷 ⊆ (Base‘𝐼))    &   (𝜑0𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 + 𝑦) ∈ 𝐷)    &   ((𝜑𝑥𝐷) → ((invg𝐼)‘𝑥) ∈ 𝐷)    &   (𝜑1 = (1r𝐼))    &   (𝜑· = (.r𝐼))    &   (𝜑1𝐷)    &   ((𝜑𝑥𝐷𝑦𝐷) → (𝑥 · 𝑦) ∈ 𝐷)    &   (𝜑𝐼 ∈ Ring)       (𝜑𝐷 ∈ (SubRing‘𝐼))

Theoremrlmfn 19587 ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.)
ringLMod Fn V

Theoremrlmval 19588 Value of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(ringLMod‘𝑊) = ((subringAlg ‘𝑊)‘(Base‘𝑊))

Theoremlidlval 19589 Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(LIdeal‘𝑊) = (LSubSp‘(ringLMod‘𝑊))

Theoremrspval 19590 Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
(RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊))

Theoremrlmval2 19591 Value of the ring module extended. (Contributed by AV, 2-Dec-2018.) (Revised by Thierry Arnoux, 16-Jun-2019.)
(𝑊𝑋 → (ringLMod‘𝑊) = (((𝑊 sSet ⟨(Scalar‘ndx), 𝑊⟩) sSet ⟨( ·𝑠 ‘ndx), (.r𝑊)⟩) sSet ⟨(·𝑖‘ndx), (.r𝑊)⟩))

Theoremrlmbas 19592 Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(Base‘𝑅) = (Base‘(ringLMod‘𝑅))

Theoremrlmplusg 19593 Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(+g𝑅) = (+g‘(ringLMod‘𝑅))

Theoremrlm0 19594 Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
(0g𝑅) = (0g‘(ringLMod‘𝑅))

Theoremrlmsub 19595 Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
(-g𝑅) = (-g‘(ringLMod‘𝑅))

Theoremrlmmulr 19596 Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(.r𝑅) = (.r‘(ringLMod‘𝑅))

Theoremrlmsca 19597 Scalars in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
(𝑅𝑋𝑅 = (Scalar‘(ringLMod‘𝑅)))

Theoremrlmsca2 19598 Scalars in the ring module. (Contributed by Stefan O'Rear, 1-Apr-2015.)
( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅))

Theoremrlmvsca 19599 Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
(.r𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅))

Theoremrlmtopn 19600 Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(TopOpen‘𝑅) = (TopOpen‘(ringLMod‘𝑅))

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43657
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