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Theorem List for Metamath Proof Explorer - 31001-31100   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.3.9.22  The ring of integers modulo ` N `

Theoremznfermltl 31001 Fermat's little theorem in ℤ/n. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝑍 = (ℤ/nℤ‘𝑃)    &   𝐵 = (Base‘𝑍)    &    = (.g‘(mulGrp‘𝑍))       ((𝑃 ∈ ℙ ∧ 𝐴𝐵) → (𝑃 𝐴) = 𝐴)

20.3.9.23  Independent sets and families

Theoremislinds5 31002* A set is linearly independent if and only if it has no non-trivial representations of zero. (Contributed by Thierry Arnoux, 18-May-2023.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝐹)       ((𝑊 ∈ LMod ∧ 𝑉𝐵) → (𝑉 ∈ (LIndS‘𝑊) ↔ ∀𝑎 ∈ (𝐾m 𝑉)((𝑎 finSupp 0 ∧ (𝑊 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))) = 𝑂) → 𝑎 = (𝑉 × { 0 }))))

Theoremellspds 31003* Variation on ellspd 20500. (Contributed by Thierry Arnoux, 18-May-2023.)
𝑁 = (LSpan‘𝑀)    &   𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝑉𝐵)       (𝜑 → (𝑋 ∈ (𝑁𝑉) ↔ ∃𝑎 ∈ (𝐾m 𝑉)(𝑎 finSupp 0𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))))))

Theorem0ellsp 31004 Zero is in all spans. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0g𝑊)    &   𝐵 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑆𝐵) → 0 ∈ (𝑁𝑆))

Theorem0nellinds 31005 The group identity cannot be an element of an independent set. (Contributed by Thierry Arnoux, 8-May-2023.)
0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝐹 ∈ (LIndS‘𝑊)) → ¬ 0𝐹)

Theoremrspsnel 31006* Membership in a principal ideal. Analogous to lspsnel 19776. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝐼 ∈ (𝐾‘{𝑋}) ↔ ∃𝑥𝐵 𝐼 = (𝑥 · 𝑋)))

Theoremrspsnid 31007 A principal ideal contains the element that generates it. (Contributed by Thierry Arnoux, 15-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ∈ (𝐾‘{𝐺}))

Theoremelrsp 31008* Write the elements of a ring span as finite linear combinations. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)       (𝜑 → (𝑋 ∈ (𝑁𝐼) ↔ ∃𝑎 ∈ (𝐵m 𝐼)(𝑎 finSupp 0𝑋 = (𝑅 Σg (𝑖𝐼 ↦ ((𝑎𝑖) · 𝑖))))))

Theoremrspidlid 31009 The ideal span of an ideal is the ideal itself. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐾𝐼) = 𝐼)

Theorempidlnz 31010 A principal ideal generated by a nonzero element is not the zero ideal. (Contributed by Thierry Arnoux, 11-Apr-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐾 = (RSpan‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑋0 ) → (𝐾‘{𝑋}) ≠ { 0 })

Theoremlbslsp 31011* Any element of a left module 𝑀 can be expressed as a linear combination of the elements of a basis 𝑉 of 𝑀. (Contributed by Thierry Arnoux, 3-Aug-2023.)
𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝑉 ∈ (LBasis‘𝑀))       (𝜑 → (𝑋𝐵 ↔ ∃𝑎 ∈ (𝐾m 𝑉)(𝑎 finSupp 0𝑋 = (𝑀 Σg (𝑣𝑉 ↦ ((𝑎𝑣) · 𝑣))))))

Theoremlindssn 31012 Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.)
𝐵 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ LVec ∧ 𝑋𝐵𝑋0 ) → {𝑋} ∈ (LIndS‘𝑊))

Theoremlindflbs 31013 Conditions for an independent family to be a basis. (Contributed by Thierry Arnoux, 21-Jul-2023.)
𝐵 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑆 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑂 = (0g𝑊)    &    0 = (0g𝑆)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑆 ∈ NzRing)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼1-1𝐵)       (𝜑 → (ran 𝐹 ∈ (LBasis‘𝑊) ↔ (𝐹 LIndF 𝑊 ∧ (𝑁‘ran 𝐹) = 𝐵)))

Theoremlinds2eq 31014 Deduce equality of elements in an independent set. (Contributed by Thierry Arnoux, 18-Jul-2023.)
𝐹 = (Base‘(Scalar‘𝑊))    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &    0 = (0g‘(Scalar‘𝑊))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐵 ∈ (LIndS‘𝑊))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝐾𝐹)    &   (𝜑𝐿𝐹)    &   (𝜑𝐾0 )    &   (𝜑 → (𝐾 · 𝑋) = (𝐿 · 𝑌))       (𝜑 → (𝑋 = 𝑌𝐾 = 𝐿))

Theoremlindfpropd 31015* Property deduction for linearly independent families. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)    &   (𝜑𝑋𝐴)       (𝜑 → (𝑋 LIndF 𝐾𝑋 LIndF 𝐿))

Theoremlindspropd 31016* Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (Base‘(Scalar‘𝐾)) = (Base‘(Scalar‘𝐿)))    &   (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) ∈ (Base‘𝐾))    &   ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐾)) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝐾𝑉)    &   (𝜑𝐿𝑊)       (𝜑 → (LIndS‘𝐾) = (LIndS‘𝐿))

20.3.9.24  Subgroup sum / Sumset / Minkowski sum

The sumset (also called the Minkowski sum) of two subsets 𝐴 and 𝐵, is defined to be the set of all sums of an element from 𝐴 with an element from 𝐵.

The sumset operation can be used for both group (additive) operations and ring (multiplicative) operations.

Theoremelgrplsmsn 31017* Membership in a sumset with a singleton for a group operation. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝐺𝑉)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑍 ∈ (𝐴 {𝑋}) ↔ ∃𝑥𝐴 𝑍 = (𝑥 + 𝑋)))

Theoremlsmsnorb 31018* The sumset of a group with a single element is the element's orbit by the group action. See gaorb 18437. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑔 + 𝑥) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐴 {𝑋}) = [𝑋] )

Theoremlsmsnorb2 31019* The sumset of a single element with a group is the element's orbit by the group action. See gaorb 18437. (Contributed by Thierry Arnoux, 24-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &    = (LSSum‘𝐺)    &    = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝐵 ∧ ∃𝑔𝐴 (𝑥 + 𝑔) = 𝑦)}    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝐵)    &   (𝜑𝑋𝐵)       (𝜑 → ({𝑋} 𝐴) = [𝑋] )

Theoremelringlsm 31020* Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥𝐸𝑦𝐹 𝑍 = (𝑥 · 𝑦)))

Theoremelringlsmd 31021 Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌𝐹)       (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹))

Theoremringlsmss 31022 Closure of the product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐸 × 𝐹) ⊆ 𝐵)

Theoremringlsmss1 31023 The product of an ideal 𝐼 of a commutative ring 𝑅 with some set E is a subset of the ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 × 𝐸) ⊆ 𝐼)

Theoremringlsmss2 31024 The product with an ideal of a ring is a subset of that ideal. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐸𝐵)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐸 × 𝐼) ⊆ 𝐼)

Theoremlsmsnpridl 31025 The product of the ring with a single element is equal to the principal ideal generated by that element. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) = (𝐾‘{𝑋}))

Theoremlsmsnidl 31026 The product of the ring with a single element is a principal ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝐵 × {𝑋}) ∈ (LPIdeal‘𝑅))

Theoremlsmidllsp 31027 The sum of two ideals is the ideal generated by their union. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) = (𝐾‘(𝐼𝐽)))

Theoremlsmidl 31028 The sum of two ideals is an ideal. (Contributed by Thierry Arnoux, 21-Jan-2024.)
𝐵 = (Base‘𝑅)    &    = (LSSum‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))       (𝜑 → (𝐼 𝐽) ∈ (LIdeal‘𝑅))

Theoremlsmssass 31029 Group sum is associative, subset version (see lsmass 18795). (Contributed by Thierry Arnoux, 1-Jun-2024.)
= (LSSum‘𝐺)    &   𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝑅𝐵)    &   (𝜑𝑇𝐵)    &   (𝜑𝑈𝐵)       (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))

Theoremgrplsm0l 31030 Sumset with the identity singleton is the original set. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝐴𝐵𝐴 ≠ ∅) → ({ 0 } 𝐴) = 𝐴)

Theoremgrplsmid 31031 The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
= (LSSum‘𝐺)       ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)

20.3.9.25  The quotient map

Theoremquslsm 31032 Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &    = (LSSum‘𝐺)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋𝐵)       (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))

Theoremqusima 31033* The image of a subgroup by the natural map from elements to their cosets. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑥𝐵 ↦ [𝑥](𝐺 ~QG 𝑁))    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐻𝑆)    &   (𝜑𝑆 ⊆ (SubGrp‘𝐺))       (𝜑 → (𝐸𝐻) = (𝐹𝐻))

Theoremnsgqus0 31034 A normal subgroup 𝑁 is a member of all subgroups 𝐹 of the quotient group by 𝑁. In fact, it is the identity element of the quotient group. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))       ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝐹 ∈ (SubGrp‘𝑄)) → 𝑁𝐹)

Theoremnsgmgclem 31035* Lemma for nsgmgc 31036. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))    &   (𝜑𝐹 ∈ (SubGrp‘𝑄))       (𝜑 → {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝐹} ∈ (SubGrp‘𝐺))

Theoremnsgmgc 31036* There is a monotone Galois connection between the lattice of subgroups of a group 𝐺 containing a normal subgroup 𝑁 and the lattice of subgroups of the quotient group 𝐺 / 𝑁. This is sometimes called the lattice theorem. (Contributed by Thierry Arnoux, 27-Jul-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &   𝐽 = (𝑉MGalConn𝑊)    &   𝑉 = (toInc‘𝑆)    &   𝑊 = (toInc‘𝑇)    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸𝐽𝐹)

Theoremnsgqusf1olem1 31037* Lemma for nsgqusf1o 31040. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)

Theoremnsgqusf1olem2 31038* Lemma for nsgqusf1o 31040. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐸 = 𝑇)

Theoremnsgqusf1olem3 31039* Lemma for nsgqusf1o 31040. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑 → ran 𝐹 = 𝑆)

Theoremnsgqusf1o 31040* The canonical projection homomorphism 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.)
𝐵 = (Base‘𝐺)    &   𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}    &   𝑇 = (SubGrp‘𝑄)    &    = (le‘(toInc‘𝑆))    &    = (le‘(toInc‘𝑇))    &   𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &    = (LSSum‘𝐺)    &   𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))    &   𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})    &   (𝜑𝑁 ∈ (NrmSGrp‘𝐺))       (𝜑𝐸:𝑆1-1-onto𝑇)

20.3.9.26  Ideals

Theoremintlidl 31041 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Thierry Arnoux, 8-Jun-2024.)
((𝑅 ∈ Ring ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (LIdeal‘𝑅)) → 𝐶 ∈ (LIdeal‘𝑅))

Theoremrhmpreimaidl 31042 The preimage of an ideal by a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝐼 = (LIdeal‘𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐽 ∈ (LIdeal‘𝑆)) → (𝐹𝐽) ∈ 𝐼)

Theoremkerlidl 31043 The kernel of a ring homomorphism is an ideal. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐼 = (LIdeal‘𝑅)    &    0 = (0g𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ { 0 }) ∈ 𝐼)

Theorem0ringidl 31044 The zero ideal is the only ideal of the trivial ring. (Contributed by Thierry Arnoux, 1-Jul-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (LIdeal‘𝑅) = {{ 0 }})

Theoremelrspunidl 31045* Elementhood to the span of a union of ideals. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝑁 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑆 ⊆ (LIdeal‘𝑅))       (𝜑 → (𝑋 ∈ (𝑁 𝑆) ↔ ∃𝑎 ∈ (𝐵m 𝑆)(𝑎 finSupp 0𝑋 = (𝑅 Σg 𝑎) ∧ ∀𝑘𝑆 (𝑎𝑘) ∈ 𝑘)))

Theoremlidlincl 31046 Ideals are closed under intersection. (Contributed by Thierry Arnoux, 5-Jul-2024.)
𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐽𝑈) → (𝐼𝐽) ∈ 𝑈)

Theoremidlinsubrg 31047 The intersection between an ideal and a subring is an ideal of the subring. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (LIdeal‘𝑅)    &   𝑉 = (LIdeal‘𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐼𝑈) → (𝐼𝐴) ∈ 𝑉)

Theoremrhmimaidl 31048 The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
𝐵 = (Base‘𝑆)    &   𝑇 = (LIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)

20.3.9.27  Prime Ideals

Syntaxcprmidl 31049 Extend class notation with the class of prime ideals.
class PrmIdeal

Definitiondf-prmidl 31050* Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵𝐼 for ideals 𝐴 and 𝐵, either 𝐴𝐼 or 𝐵𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see prmidl2 31055 and isprmidlc 31062. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 14-Jan-2024.)
PrmIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑎 ∈ (LIdeal‘𝑟)∀𝑏 ∈ (LIdeal‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(.r𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremprmidlval 31051* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (PrmIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Theoremisprmidl 31052* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑎 ∈ (LIdeal‘𝑅)∀𝑏 ∈ (LIdeal‘𝑅)(∀𝑥𝑎𝑦𝑏 (𝑥 · 𝑦) ∈ 𝑃 → (𝑎𝑃𝑏𝑃)))))

Theoremprmidlnr 31053 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃𝐵)

Theoremprmidl 31054* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐽 ∈ (LIdeal‘𝑅))) ∧ ∀𝑥𝐼𝑦𝐽 (𝑥 · 𝑦) ∈ 𝑃) → (𝐼𝑃𝐽𝑃))

Theoremprmidl2 31055* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 35548 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ Ring ∧ 𝑃 ∈ (LIdeal‘𝑅)) ∧ (𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))) → 𝑃 ∈ (PrmIdeal‘𝑅))

Theoremidlmulssprm 31056 Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼𝑃 or 𝐽𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
× = (LSSum‘(mulGrp‘𝑅))    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑃 ∈ (PrmIdeal‘𝑅))    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)       (𝜑 → (𝐼𝑃𝐽𝑃))

Theorempridln1 31057 A proper ideal cannot contain the ring unity. (Contributed by Thierry Arnoux, 9-Apr-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ¬ 1𝐼)

Theoremprmidlidl 31058 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) → 𝑃 ∈ (LIdeal‘𝑅))

Theoremprmidlssidl 31059 Prime ideals as a subset of ideals. (Contributed by Thierry Arnoux, 2-Jun-2024.)
(𝑅 ∈ Ring → (PrmIdeal‘𝑅) ⊆ (LIdeal‘𝑅))

Theoremlidlnsg 31060 An ideal is a normal subgroup. (Contributed by Thierry Arnoux, 14-Jan-2024.)
((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))

Theoremcringm4 31061 Commutative/associative law for commutative ring. (Contributed by Thierry Arnoux, 14-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ (𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵)) → ((𝑋 · 𝑌) · (𝑍 · 𝑊)) = ((𝑋 · 𝑍) · (𝑌 · 𝑊)))

Theoremisprmidlc 31062* The predicate "is prime ideal" for commutative rings. Alternate definition for commutative rings. See definition in [Lang] p. 92. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ CRing → (𝑃 ∈ (PrmIdeal‘𝑅) ↔ (𝑃 ∈ (LIdeal‘𝑅) ∧ 𝑃𝐵 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) ∈ 𝑃 → (𝑥𝑃𝑦𝑃)))))

Theoremprmidlc 31063 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 12-Jan-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (((𝑅 ∈ CRing ∧ 𝑃 ∈ (PrmIdeal‘𝑅)) ∧ (𝐼𝐵𝐽𝐵 ∧ (𝐼 · 𝐽) ∈ 𝑃)) → (𝐼𝑃𝐽𝑃))

Theorem0ringprmidl 31064 The trivial ring does not have any prime ideal. (Contributed by Thierry Arnoux, 30-Jun-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → (PrmIdeal‘𝑅) = ∅)

Theoremprmidl0 31065 The zero ideal of a commutative ring 𝑅 is a prime ideal if and only if 𝑅 is an integral domain. (Contributed by Thierry Arnoux, 30-Jun-2024.)
0 = (0g𝑅)       ((𝑅 ∈ CRing ∧ { 0 } ∈ (PrmIdeal‘𝑅)) ↔ 𝑅 ∈ IDomn)

Theoremrhmpreimaprmidl 31066 The preimage of a prime ideal by a ring homomorphism is a prime ideal. (Contributed by Thierry Arnoux, 29-Jun-2024.)
𝑃 = (PrmIdeal‘𝑅)       (((𝑆 ∈ CRing ∧ 𝐹 ∈ (𝑅 RingHom 𝑆)) ∧ 𝐽 ∈ (PrmIdeal‘𝑆)) → (𝐹𝐽) ∈ 𝑃)

Theoremqsidomlem1 31067 If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))

Theoremqsidomlem2 31068 A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Theoremqsidom 31069 An ideal 𝐼 in the commutative ring 𝑅 is prime if and only if the factor ring 𝑄 is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))       ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑄 ∈ IDomn ↔ 𝐼 ∈ (PrmIdeal‘𝑅)))

20.3.9.28  Maximal Ideals

Syntaxcmxidl 31070 Extend class notation with the class of maximal ideals.
class MaxIdeal

Definitiondf-mxidl 31071* Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
MaxIdeal = (𝑟 ∈ Ring ↦ {𝑖 ∈ (LIdeal‘𝑟) ∣ (𝑖 ≠ (Base‘𝑟) ∧ ∀𝑗 ∈ (LIdeal‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = (Base‘𝑟))))})

Theoremmxidlval 31072* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})

Theoremismxidl 31073* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))

Theoremmxidlidl 31074 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))

Theoremmxidlnr 31075 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀𝐵)

Theoremmxidlmax 31076 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) ∧ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝐵))

Theoremmxidln1 31077 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → ¬ 1𝑀)

Theoremmxidlnzr 31078 A ring with a maximal ideal is a nonzero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑅 ∈ NzRing)

Theoremmxidlprm 31079 Every maximal ideal is prime. Statement in [Lang] p. 92. (Contributed by Thierry Arnoux, 21-Jan-2024.)
× = (LSSum‘(mulGrp‘𝑅))       ((𝑅 ∈ CRing ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (PrmIdeal‘𝑅))

Theoremssmxidllem 31080* The set 𝑃 used in the proof of ssmxidl 31081 satisfies the condition of Zorn's Lemma. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)    &   𝑃 = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑𝐼𝐵)    &   (𝜑𝑍𝑃)    &   (𝜑𝑍 ≠ ∅)    &   (𝜑 → [] Or 𝑍)       (𝜑 𝑍𝑃)

Theoremssmxidl 31081* Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)

Theoremkrull 31082* Krull's theorem: Any nonzero ring has at least one maximal ideal. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ NzRing → ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅))

Theoremmxidlnzrb 31083* A ring is nonzero if and only if it has maximal ideals. (Contributed by Thierry Arnoux, 10-Apr-2024.)
(𝑅 ∈ Ring → (𝑅 ∈ NzRing ↔ ∃𝑚 𝑚 ∈ (MaxIdeal‘𝑅)))

20.3.9.29  The semiring of ideals of a ring

Syntaxcidlsrg 31084 Extend class notation with the semiring of ideals of a ring.
class IDLsrg

Definitiondf-idlsrg 31085* Define a structure for the ideals of a ring. (Contributed by Thierry Arnoux, 21-Jan-2024.)
IDLsrg = (𝑟 ∈ V ↦ (LIdeal‘𝑟) / 𝑏({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (LSSum‘𝑟)⟩, ⟨(.r‘ndx), (𝑖𝑏, 𝑗𝑏 ↦ ((RSpan‘𝑟)‘(𝑖(LSSum‘(mulGrp‘𝑟))𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝑏 ↦ {𝑗𝑏 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝑏𝑖𝑗)}⟩}))

Theoremidlsrgstr 31086 A constructed semiring of ideals is a structure. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑊 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(.r‘ndx), · ⟩} ∪ {⟨(TopSet‘ndx), 𝐽⟩, ⟨(le‘ndx), ⟩})       𝑊 Struct ⟨1, 10⟩

Theoremidlsrgval 31087* Lemma for idlsrgbas 31088 through idlsrgtset 31092. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝐼 = (LIdeal‘𝑅)    &    = (LSSum‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (LSSum‘𝐺)       (𝑅𝑉 → (IDLsrg‘𝑅) = ({⟨(Base‘ndx), 𝐼⟩, ⟨(+g‘ndx), ⟩, ⟨(.r‘ndx), (𝑖𝐼, 𝑗𝐼 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗)))⟩} ∪ {⟨(TopSet‘ndx), ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})⟩, ⟨(le‘ndx), {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐼𝑖𝑗)}⟩}))

Theoremidlsrgbas 31088 Baae of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐼 = (LIdeal‘𝑅)       (𝑅𝑉𝐼 = (Base‘𝑆))

Theoremidlsrgplusg 31089 Additive operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &    = (LSSum‘𝑅)       (𝑅𝑉 = (+g𝑆))

Theoremidlsrg0g 31090 The zero ideal is the additive identity of the semiring of ideals. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } = (0g𝑆))

Theoremidlsrgmulr 31091* Multiplicative operation of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    = (LSSum‘𝐺)       (𝑅𝑉 → (𝑖𝐵, 𝑗𝐵 ↦ ((RSpan‘𝑅)‘(𝑖 𝑗))) = (.r𝑆))

Theoremidlsrgtset 31092* Topology component of the ideals of a ring. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐼 = (LIdeal‘𝑅)    &   𝐽 = ran (𝑖𝐼 ↦ {𝑗𝐼 ∣ ¬ 𝑖𝑗})       (𝑅𝑉𝐽 = (TopSet‘𝑆))

Theoremidlsrgmulrval 31093 Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   𝐺 = (mulGrp‘𝑅)    &    · = (LSSum‘𝐺)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽)))

Theoremidlsrgmulrcl 31094 Ideals of a ring 𝑅 are closed under multiplication. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ∈ 𝐵)

Theoremidlsrgmulrss1 31095 In a commutative ring, the product of two ideals is a subset of the first one. (Contributed by Thierry Arnoux, 16-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ 𝐼)

Theoremidlsrgmulrss2 31096 The product of two ideals is a subset of the second one. (Contributed by Thierry Arnoux, 2-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ 𝐽)

Theoremidlsrgmulrssin 31097 In a commutative ring, the product of two ideals is a subset of their intersection. (Contributed by Thierry Arnoux, 17-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)    &   𝐵 = (LIdeal‘𝑅)    &    = (.r𝑆)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐼𝐵)    &   (𝜑𝐽𝐵)       (𝜑 → (𝐼 𝐽) ⊆ (𝐼𝐽))

Theoremidlsrgmnd 31098 The ideals of a ring form a monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ Mnd)

Theoremidlsrgcmnd 31099 The ideals of a ring form a commutative monoid. (Contributed by Thierry Arnoux, 1-Jun-2024.)
𝑆 = (IDLsrg‘𝑅)       (𝑅 ∈ Ring → 𝑆 ∈ CMnd)

20.3.9.30  Unique factorization domains

Syntaxcufd 31100 Class of unique factorization domains.
class UFD

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