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Theorem List for Metamath Proof Explorer - 44001-44100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlindepsnlininds 44001 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Theoremislindeps 44002* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))

Theoremlincext1 44003* Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸𝑚 𝑆))

Theoremlincext2 44004* Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 )

Theoremlincext3 44005* Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋}))) ∧ (𝐺 finSupp 0 ∧ (𝑌( ·𝑠𝑀)𝑋) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})))) → (𝐹( linC ‘𝑀)𝑆) = 𝑍)

Theoremlindslinindsimp1 44006* Implication 1 for lindslininds 44013. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))

Theoremlindslinindimp2lem1 44007* Lemma 1 for lindslinindsimp2 44012. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝑌𝐵)

Theoremlindslinindimp2lem2 44008* Lemma 2 for lindslinindsimp2 44012. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))

Theoremlindslinindimp2lem3 44009* Lemma 3 for lindslinindsimp2 44012. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )

Theoremlindslinindimp2lem4 44010* Lemma 4 for lindslinindsimp2 44012. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) (Proof shortened by II, 16-Feb-2023.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))

Theoremlindslinindsimp2lem5 44011* Lemma 5 for lindslinindsimp2 44012. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))

Theoremlindslinindsimp2 44012* Implication 2 for lindslininds 44013. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))

Theoremlindslininds 44013 Equivalence of definitions df-linds 20633 and df-lininds 43991 for (linear) independence for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑆𝑉𝑀 ∈ LMod) → (𝑆 linIndS 𝑀𝑆 ∈ (LIndS‘𝑀)))

Theoremlinds0 44014 The empty set is always a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
(𝑀𝑉 → ∅ linIndS 𝑀)

Theoremel0ldep 44015 A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
(((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Theoremel0ldepsnzr 44016 A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
(((𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ NzRing) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Theoremlindsrng01 44017 Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 19337), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)       ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Theoremlindszr 44018 Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀)

Theoremsnlindsntorlem 44019* Lemma for snlindsntor 44020. (Contributed by AV, 15-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))

Theoremsnlindsntor 44020* A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (𝑟 · 𝑚) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists 𝑎𝑅, 𝑎 ≠ 0, such that 𝑎 · 𝑥 = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Theoremldepsprlem 44021 Lemma for ldepspr 44022. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝐴𝑆)) → (𝑋 = (𝐴 · 𝑌) → (( 1 · 𝑋)(+g𝑀)((𝑁𝐴) · 𝑌)) = 𝑍))

Theoremldepspr 44022 If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → ((𝐴𝑆𝑋 = (𝐴 · 𝑌)) → {𝑋, 𝑌} linDepS 𝑀))

Theoremlincresunit3lem3 44023 Lemma 3 for lincresunit3 44030. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &    · = ( ·𝑠𝑀)       (((𝑀 ∈ LMod ∧ 𝑋𝐵𝑌𝐵) ∧ 𝐴𝑈) → (((𝑁𝐴) · 𝑋) = ((𝑁𝐴) · 𝑌) ↔ 𝑋 = 𝑌))

Theoremlincresunitlem1 44024 Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹𝑋))) ∈ 𝐸)

Theoremlincresunitlem2 44025 Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑌𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑌)) ∈ 𝐸)

Theoremlincresunit1 44026* Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))

Theoremlincresunit2 44027* Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )

Theoremlincresunit3lem1 44028* Lemma 1 for lincresunit3 44030. (Contributed by AV, 17-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)((𝐺𝑧)( ·𝑠𝑀)𝑧)) = ((𝐹𝑧)( ·𝑠𝑀)𝑧))

Theoremlincresunit3lem2 44029* Lemma 2 for lincresunit3 44030. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))

Theoremlincresunit3 44030* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)

Theoremlincreslvec3 44031* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LVec ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ≠ 0𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)

Theoremislindeps2 44032* Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (∃𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀))

Theoremislininds2 44033* Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)))

Theoremisldepslvec2 44034* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 44032 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LVec ∧ 𝑆 ∈ 𝒫 𝐵) → (∃𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ 𝑆 linDepS 𝑀))

Theoremlindssnlvec 44035 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g𝑀)) → {𝑆} linIndS 𝑀)

20.38.20.4  Simple left modules and the ` ZZ `-module

Theoremlmod1lem1 44036* Lemma 1 for lmod1 44041. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼})

Theoremlmod1lem2 44037* Lemma 2 for lmod1 44041. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem3 44038* Lemma 3 for lmod1 44041. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem4 44039* Lemma 4 for lmod1 44041. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem5 44040* Lemma 5 for lmod1 44041. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)

Theoremlmod1 44041* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Theoremlmod1zr 44042 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∈ LMod)

Theoremlmod1zrnlvec 44043 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∉ LVec)

Theoremlmodn0 44044 Left modules exist. (Contributed by AV, 29-Apr-2019.)
LMod ≠ ∅

Theoremzlmodzxzequa 44045 Example of an equation within the -module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    = ( ·𝑠𝑍)    &    = (-g𝑍)    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       ((2 𝐴) (3 𝐵)) = 0

Theoremzlmodzxznm 44046 Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    = ( ·𝑠𝑍)    &    = (-g𝑍)    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       𝑖 ∈ ℤ ((𝑖 𝐴) ≠ 𝐵 ∧ (𝑖 𝐵) ≠ 𝐴)

Theoremzlmodzxzldeplem 44047 A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       𝐴𝐵

Theoremzlmodzxzequap 44048 Example of an equation within the -module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    + = (+g𝑍)    &    = ( ·𝑠𝑍)       ((2 𝐴) + (-3 𝐵)) = 0

Theoremzlmodzxzldeplem1 44049 Lemma 1 for zlmodzxzldep 44053. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵})

Theoremzlmodzxzldeplem2 44050 Lemma 2 for zlmodzxzldep 44053. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       𝐹 finSupp 0

Theoremzlmodzxzldeplem3 44051 Lemma 3 for zlmodzxzldep 44053. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       (𝐹( linC ‘𝑍){𝐴, 𝐵}) = (0g𝑍)

Theoremzlmodzxzldeplem4 44052* Lemma 4 for zlmodzxzldep 44053. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       𝑦 ∈ {𝐴, 𝐵} (𝐹𝑦) ≠ 0

Theoremzlmodzxzldep 44053 { A , B } is a linearly dependent set within the -module ℤ × ℤ (see example in [Roman] p. 112). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       {𝐴, 𝐵} linDepS 𝑍

Theoremldepsnlinclem1 44054 Lemma 1 for ldepsnlinc 44057. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       (𝐹 ∈ ((Base‘ℤring) ↑𝑚 {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)

Theoremldepsnlinclem2 44055 Lemma 2 for ldepsnlinc 44057. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       (𝐹 ∈ ((Base‘ℤring) ↑𝑚 {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)

20.38.20.5  Differences between (left) modules and (left) vector spaces

Theoremlvecpsslmod 44056 The class of all (left) vector spaces is a proper subclass of the class of all (left) modules. Although it is obvious (and proven by lveclmod 19568) that every left vector space is a left module, there is (at least) one left module which is no left vector space, for example the zero module over the zero ring, see lmod1zrnlvec 44043. (Contributed by AV, 29-Apr-2019.)
LVec ⊊ LMod

Theoremldepsnlinc 44057* The reverse implication of islindeps2 44032 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 44045 and zlmodzxznm 44046. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣))

Theoremldepslinc 44058* For (left) vector spaces, isldepslvec2 44034 provides an alternative definition of being a linearly dependent subset, whereas ldepsnlinc 44057 indicates that there is not an analogous alternative definition for arbitrary (left) modules. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)
(∀𝑚 ∈ LVec ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)) ∧ ¬ ∀𝑚 ∈ LMod ∀𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ↔ ∃𝑣𝑠𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 (𝑠 ∖ {𝑣}))(𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = 𝑣)))

20.38.21  Complexity theory

20.38.21.1  Auxiliary theorems

Theoremoffval0 44059* Value of an operation applied to two functions. (Contributed by AV, 15-May-2020.)
((𝐹𝑉𝐺𝑊) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))

Theoremsuppdm 44060 If the range of a function does not contain the zero, the support of the function equals its domain. (Contributed by AV, 20-May-2020.)
(((Fun 𝐹𝐹𝑉𝑍𝑊) ∧ 𝑍 ∉ ran 𝐹) → (𝐹 supp 𝑍) = dom 𝐹)

Theoremeluz2cnn0n1 44061 An integer greater than 1 is a complex number not equal to 0 or 1. (Contributed by AV, 23-May-2020.)
(𝐵 ∈ (ℤ‘2) → 𝐵 ∈ (ℂ ∖ {0, 1}))

Theoremdivge1b 44062 The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴𝐵 ↔ 1 ≤ (𝐵 / 𝐴)))

Theoremdivgt1b 44063 The ratio of a real number to a positive real number is greater than 1 iff the divisor (the positive real number) is less than the dividend (the real number). (Contributed by AV, 30-May-2020.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 1 < (𝐵 / 𝐴)))

Theoremltsubaddb 44064 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶) < (𝐵𝐷) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶)))

Theoremltsubsubb 44065 Equivalence for the "less than" relation between differences. (Contributed by AV, 6-Jun-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐴𝐶) < (𝐵𝐷) ↔ (𝐴𝐵) < (𝐶𝐷)))

Theoremltsubadd2b 44066 Equivalence for the "less than" relation between differences and sums. (Contributed by AV, 6-Jun-2020.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ)) → ((𝐷𝐶) < (𝐵𝐴) ↔ (𝐴 + 𝐷) < (𝐵 + 𝐶)))

Theoremdivsub1dir 44067 Distribution of division over subtraction by 1. (Contributed by AV, 6-Jun-2020.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) − 1) = ((𝐴𝐵) / 𝐵))

Theoremexpnegico01 44068 An integer greater than 1 to the power of a negative integer is in the closed-below, open-above interval between 0 and 1. (Contributed by AV, 24-May-2020.)
((𝐵 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ ∧ 𝑁 < 0) → (𝐵𝑁) ∈ (0[,)1))

Theoremelfzolborelfzop1 44069 An element of a half-open integer interval is either equal to the left bound of the interval or an element of a half-open integer interval with a lower bound increased by 1. (Contributed by AV, 2-Jun-2020.)
(𝐾 ∈ (𝑀..^𝑁) → (𝐾 = 𝑀𝐾 ∈ ((𝑀 + 1)..^𝑁)))

Theorempw2m1lepw2m1 44070 2 to the power of a positive integer decreased by 1 is less than or equal to 2 to the power of the integer minus 1. (Contributed by AV, 30-May-2020.)
(𝐼 ∈ ℕ → (2↑(𝐼 − 1)) ≤ ((2↑𝐼) − 1))

Theoremzgtp1leeq 44071 If an integer is between another integer and its predecessor, the integer is equal to the other integer. (Contributed by AV, 7-Jun-2020.)
((𝐼 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (((𝐴 − 1) < 𝐼𝐼𝐴) → 𝐼 = 𝐴))

Theoremflsubz 44072 An integer can be moved in and out of the floor of a difference. (Contributed by AV, 29-May-2020.)
((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ) → (⌊‘(𝐴𝑁)) = ((⌊‘𝐴) − 𝑁))

20.38.21.2  The modulo (remainder) operation (extension)

Theoremfldivmod 44073 Expressing the floor of a division by the modulo operator. (Contributed by AV, 6-Jun-2020.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (⌊‘(𝐴 / 𝐵)) = ((𝐴 − (𝐴 mod 𝐵)) / 𝐵))

Theoremmod0mul 44074* If an integer is 0 modulo a positive integer, this integer must be the product of another integer and the modulus. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) = 0 → ∃𝑥 ∈ ℤ 𝐴 = (𝑥 · 𝑁)))

Theoremmodn0mul 44075* If an integer is not 0 modulo a positive integer, this integer must be the sum of the product of another integer and the modulus and a positive integer less than the modulus. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod 𝑁) ≠ 0 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (1..^𝑁)𝐴 = ((𝑥 · 𝑁) + 𝑦)))

Theoremm1modmmod 44076 An integer decreased by 1 modulo a positive integer minus the integer modulo the same modulus is either -1 or the modulus minus 1. (Contributed by AV, 7-Jun-2020.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) mod 𝑁) − (𝐴 mod 𝑁)) = if((𝐴 mod 𝑁) = 0, (𝑁 − 1), -1))

Theoremdifmodm1lt 44077 The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd 𝐴 and 𝑁 = 2, since ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) would be (1 − 0) = 1 which is not less than (𝑁 − 1) = 1. (Contributed by AV, 6-Jun-2012.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ 2 < 𝑁) → ((𝐴 mod 𝑁) − ((𝐴 − 1) mod 𝑁)) < (𝑁 − 1))

20.38.21.3  Even and odd integers

Theoremnn0onn0ex 44078* For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1))

Theoremnn0enn0ex 44079* For each even nonnegative integer there is a nonnegative integer which, multiplied by 2, results in the even nonnegative integer. (Contributed by AV, 30-May-2020.)
((𝑁 ∈ ℕ0 ∧ (𝑁 / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = (2 · 𝑚))

Theoremnnennex 44080* For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.)
((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚))

Theoremnneop 44081 A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 + 1) / 2) ∈ ℕ))

Theoremnneom 44082 A positive integer is even or odd. (Contributed by AV, 30-May-2020.)
(𝑁 ∈ ℕ → ((𝑁 / 2) ∈ ℕ ∨ ((𝑁 − 1) / 2) ∈ ℕ0))

Theoremnn0eo 44083 A nonnegative integer is even or odd. (Contributed by AV, 27-May-2020.)
(𝑁 ∈ ℕ0 → ((𝑁 / 2) ∈ ℕ0 ∨ ((𝑁 + 1) / 2) ∈ ℕ0))

Theoremnnpw2even 44084 2 to the power of a positive integer is even. (Contributed by AV, 2-Jun-2020.)
(𝑁 ∈ ℕ → ((2↑𝑁) / 2) ∈ ℕ)

Theoremzefldiv2 44085 The floor of an even integer divided by 2 is equal to the integer divided by 2. (Contributed by AV, 7-Jun-2020.)
((𝑁 ∈ ℤ ∧ (𝑁 / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = (𝑁 / 2))

Theoremzofldiv2 44086 The floor of an odd integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.)
((𝑁 ∈ ℤ ∧ ((𝑁 + 1) / 2) ∈ ℤ) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))

Theoremnn0ofldiv2 44087 The floor of an odd nonnegative integer divided by 2 is equal to the integer first decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.) (Proof shortened by AV, 7-Jun-2020.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))

Theoremflnn0div2ge 44088 The floor of a positive integer divided by 2 is greater than or equal to the integer decreased by 1 and then divided by 2. (Contributed by AV, 1-Jun-2020.)
(𝑁 ∈ ℕ0 → ((𝑁 − 1) / 2) ≤ (⌊‘(𝑁 / 2)))

Theoremflnn0ohalf 44089 The floor of the half of an odd positive integer is equal to the floor of the half of the integer decreased by 1. (Contributed by AV, 5-Jun-2012.)
((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → (⌊‘(𝑁 / 2)) = (⌊‘((𝑁 − 1) / 2)))

20.38.21.4  The natural logarithm on complex numbers (extension)

Theoremlogcxp0 44090 Logarithm of a complex power. Generalization of logcxp 24933. (Contributed by AV, 22-May-2020.)
((𝐴 ∈ (ℂ ∖ {0}) ∧ 𝐵 ∈ ℂ ∧ (𝐵 · (log‘𝐴)) ∈ ran log) → (log‘(𝐴𝑐𝐵)) = (𝐵 · (log‘𝐴)))

Theoremregt1loggt0 44091 The natural logarithm for a real number greater than 1 is greater than 0. (Contributed by AV, 25-May-2020.)
(𝐵 ∈ (1(,)+∞) → 0 < (log‘𝐵))

20.38.21.5  Division of functions

Syntaxcfdiv 44092 Extend class notation with the division operator of two functions.
class /f

Definitiondf-fdiv 44093* Define the division of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
/f = (𝑓 ∈ V, 𝑔 ∈ V ↦ ((𝑓𝑓 / 𝑔) ↾ (𝑔 supp 0)))

Theoremfdivval 44094 The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)
((𝐹𝑉𝐺𝑊) → (𝐹 /f 𝐺) = ((𝐹𝑓 / 𝐺) ↾ (𝐺 supp 0)))

Theoremfdivmpt 44095* The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴𝑉) → (𝐹 /f 𝐺) = (𝑥 ∈ (𝐺 supp 0) ↦ ((𝐹𝑥) / (𝐺𝑥))))

Theoremfdivmptf 44096 The quotient of two functions into the complex numbers is a function into the complex numbers. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℂ)

Theoremrefdivmptf 44097 The quotient of two functions into the real numbers is a function into the real numbers. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴𝑉) → (𝐹 /f 𝐺):(𝐺 supp 0)⟶ℝ)

Theoremfdivpm 44098 The quotient of two functions into the complex numbers is a partial function. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴𝑉) → (𝐹 /f 𝐺) ∈ (ℂ ↑pm 𝐴))

Theoremrefdivpm 44099 The quotient of two functions into the real numbers is a partial function. (Contributed by AV, 16-May-2020.)
((𝐹:𝐴⟶ℝ ∧ 𝐺:𝐴⟶ℝ ∧ 𝐴𝑉) → (𝐹 /f 𝐺) ∈ (ℝ ↑pm 𝐴))

Theoremfdivmptfv 44100 The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.)
(((𝐹:𝐴⟶ℂ ∧ 𝐺:𝐴⟶ℂ ∧ 𝐴𝑉) ∧ 𝑋 ∈ (𝐺 supp 0)) → ((𝐹 /f 𝐺)‘𝑋) = ((𝐹𝑋) / (𝐺𝑋)))

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