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Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremopelxpii 39701 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)

Theoremiunsn 39702* Indexed union of a singleton. Compare dfiun2 4920 and rnmpt 5794. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}

Theoremimaopab 39703* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}

Theoremfnsnbt 39704 A function's domain is a singleton iff the function is a singleton. (Contributed by Steven Nguyen, 18-Aug-2023.)
(𝐴 ∈ V → (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩}))

Theoremfnimasnd 39705 The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝑆𝐴)       (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})

Theoremfvmptd4 39706* Deduction version of fvmpt 6757 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   (𝜑𝐹 = (𝑥𝐷𝐵))    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)

Theoremofun 39707 A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
(𝜑𝐴 Fn 𝑀)    &   (𝜑𝐵 Fn 𝑀)    &   (𝜑𝐶 Fn 𝑁)    &   (𝜑𝐷 Fn 𝑁)    &   (𝜑𝑀𝑉)    &   (𝜑𝑁𝑊)    &   (𝜑 → (𝑀𝑁) = ∅)       (𝜑 → ((𝐴𝐶) ∘f 𝑅(𝐵𝐷)) = ((𝐴f 𝑅𝐵) ∪ (𝐶f 𝑅𝐷)))

Theoremdfqs2 39708* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)

Theoremdfqs3 39709* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}

Theoremqseq12d 39710 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))

Theoremqsalrel 39711* The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑥 𝑦)    &   (𝜑 Er 𝐴)    &   (𝜑𝑁𝐴)       (𝜑 → (𝐴 / ) = {𝐴})

Theoremelmapdd 39712 Deduction associated with elmapd 8428. (Contributed by SN, 29-Jul-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶:𝐵𝐴)       (𝜑𝐶 ∈ (𝐴m 𝐵))

Theoremisfsuppd 39713 Deduction form of isfsupp 8860. (Contributed by SN, 29-Jul-2024.)
(𝜑𝑅𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑 → Fun 𝑅)    &   (𝜑 → (𝑅 supp 𝑍) ∈ Fin)       (𝜑𝑅 finSupp 𝑍)

Theoremfzosumm1 39714* Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑 → (𝑁 − 1) ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))

Theoremccatcan2d 39715 Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
(𝜑𝐴 ∈ Word 𝑉)    &   (𝜑𝐵 ∈ Word 𝑉)    &   (𝜑𝐶 ∈ Word 𝑉)       (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))

Theoremacos1half 39716 The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
(arccos‘(1 / 2)) = (π / 3)

20.26.2  Structures

Theoremnelsubginvcld 39717 The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝜑 → (𝑁𝑋) ∈ (𝐵𝑆))

Theoremnelsubgcld 39718 A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    + = (+g𝐺)       (𝜑 → (𝑋 + 𝑌) ∈ (𝐵𝑆))

Theoremnelsubgsubcld 39719 A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐺 ∈ Grp)    &   (𝜑𝑆 ∈ (SubGrp‘𝐺))    &   (𝜑𝑋 ∈ (𝐵𝑆))    &   𝐵 = (Base‘𝐺)    &   (𝜑𝑌𝑆)    &    = (-g𝐺)       (𝜑 → (𝑋 𝑌) ∈ (𝐵𝑆))

Theoremrnasclg 39720 The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
𝐴 = (algSc‘𝑊)    &    1 = (1r𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))

Theoremselvval2lem1 39721 𝑇 is an associative algebra. For simplicity, 𝐼 stands for (𝐼𝐽) and we have 𝐽𝑊 instead of 𝐽𝐼. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝑇 ∈ AssAlg)

Theoremselvval2lem2 39722 𝐷 is a ring homomorphism. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑𝐷 ∈ (𝑅 RingHom 𝑇))

Theoremselvval2lem3 39723 The third argument passed to evalSub is in the domain. (Contributed by SN, 15-Dec-2023.)
𝑈 = (𝐼 mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)       (𝜑 → ran 𝐷 ∈ (SubRing‘𝑇))

Theoremselvval2lemn 39724 A lemma to illustrate the purpose of selvval2lem3 39723 and the value of 𝑄. Will be renamed in the future when this section is moved to main. (Contributed by SN, 5-Nov-2023.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑄 = ((𝐼 evalSub 𝑇)‘ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑆 = (𝑇s ran 𝐷)    &   𝑋 = (𝑇s (𝐵m 𝐼))    &   𝐵 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝑄 ∈ (𝑊 RingHom 𝑋))

Theoremselvval2lem4 39725 The fourth argument passed to evalSub is in the domain (a polynomial in (𝐼 mPoly (𝐽 mPoly ((𝐼𝐽) mPoly 𝑅)))). (Contributed by SN, 5-Nov-2023.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐷 = (𝐶 ∘ (algSc‘𝑈))    &   𝑆 = (𝑇s ran 𝐷)    &   𝑊 = (𝐼 mPoly 𝑆)    &   𝑋 = (Base‘𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (𝐷𝐹) ∈ 𝑋)

Theoremselvval2lem5 39726* The fifth argument passed to evalSub is in the domain (a function 𝐼𝐸). (Contributed by SN, 22-Feb-2024.)
𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐶 = (algSc‘𝑇)    &   𝐸 = (Base‘𝑇)    &   𝐹 = (𝑥𝐼 ↦ if(𝑥𝐽, ((𝐽 mVar 𝑈)‘𝑥), (𝐶‘(((𝐼𝐽) mVar 𝑅)‘𝑥))))    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)       (𝜑𝐹 ∈ (𝐸m 𝐼))

Theoremselvcl 39727 Closure of the "variable selection" function. (Contributed by SN, 22-Feb-2024.)
𝑃 = (𝐼 mPoly 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝑈 = ((𝐼𝐽) mPoly 𝑅)    &   𝑇 = (𝐽 mPoly 𝑈)    &   𝐸 = (Base‘𝑇)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝐽𝐼)    &   (𝜑𝐹𝐵)       (𝜑 → (((𝐼 selectVars 𝑅)‘𝐽)‘𝐹) ∈ 𝐸)

Theoremfrlmfielbas 39728 The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑋𝐵𝑋:𝐼𝑁))

Theoremfrlmfzwrd 39729 A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0...𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzowrd 39730 A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       (𝑋𝐵𝑋 ∈ Word 𝑆)

Theoremfrlmfzolen 39731 The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝑁 ∈ ℕ0𝑋𝐵) → (♯‘𝑋) = 𝑁)

Theoremfrlmfzowrdb 39732 The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝑆 = (Base‘𝐾)       ((𝐾𝑉𝑁 ∈ ℕ0) → (𝑋𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))

Theoremfrlmfzoccat 39733 The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾𝑍)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)

Theoremfrlmvscadiccat 39734 Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-2023.)
𝑊 = (𝐾 freeLMod (0..^𝐿))    &   𝑋 = (𝐾 freeLMod (0..^𝑀))    &   𝑌 = (𝐾 freeLMod (0..^𝑁))    &   𝐵 = (Base‘𝑊)    &   𝐶 = (Base‘𝑋)    &   𝐷 = (Base‘𝑌)    &   (𝜑𝐾𝑍)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)    &   𝑂 = ( ·𝑠𝑊)    &    = ( ·𝑠𝑋)    &    · = ( ·𝑠𝑌)    &   𝑆 = (Base‘𝐾)    &   (𝜑𝐴𝑆)       (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 𝑈) ++ (𝐴 · 𝑉)))

Theoremismhmd 39735* Deduction version of ismhm 18014. (Contributed by SN, 27-Jul-2024.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑇)    &   (𝜑𝑆 ∈ Mnd)    &   (𝜑𝑇 ∈ Mnd)    &   (𝜑𝐹:𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))    &   (𝜑 → (𝐹0 ) = 𝑍)       (𝜑𝐹 ∈ (𝑆 MndHom 𝑇))

Theoremgrpcld 39736 Closure of the operation of a group. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 + 𝑌) ∈ 𝐵)

Theoremablcmnd 39737 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝐺 ∈ Abel)       (𝜑𝐺 ∈ CMnd)

Theoremringcld 39738 Closure of the multiplication operation of a ring. (Contributed by SN, 29-Jul-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 · 𝑌) ∈ 𝐵)

Theoremringassd 39739 Associative law for multiplication in a ring. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)       (𝜑 → ((𝑋 · 𝑌) · 𝑍) = (𝑋 · (𝑌 · 𝑍)))

Theoremringlidmd 39740 The unit element of a ring is a left multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → ( 1 · 𝑋) = 𝑋)

Theoremringridmd 39741 The unit element of a ring is a right multiplicative identity. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑋 · 1 ) = 𝑋)

Theoremringabld 39742 A ring is an Abelian group. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ Abel)

Theoremringcmnd 39743 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ CMnd)

Theoremdrngringd 39744 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Ring)

Theoremdrnggrpd 39745 A division ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Grp)

Theoremdrnginvrcld 39746 Closure of the multiplicative inverse in a division ring. (reccld 11437 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )       (𝜑 → (𝐼𝑋) ∈ 𝐵)

Theoremdrnginvrn0d 39747 A multiplicative inverse in a division ring is nonzero. (recne0d 11438 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )       (𝜑 → (𝐼𝑋) ≠ 0 )

Theoremdrnginvrld 39748 Property of the multiplicative inverse in a division ring. (recid2d 11440 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )       (𝜑 → ((𝐼𝑋) · 𝑋) = 1 )

Theoremdrnginvrrd 39749 Property of the multiplicative inverse in a division ring. (recidd 11439 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑋0 )       (𝜑 → (𝑋 · (𝐼𝑋)) = 1 )

Theoremdrngmulcanad 39750 Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11303 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑍0 )    &   (𝜑 → (𝑍 · 𝑋) = (𝑍 · 𝑌))       (𝜑𝑋 = 𝑌)

Theoremdrngmulcan2ad 39751 Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11304 analog). (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝑍0 )    &   (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))       (𝜑𝑋 = 𝑌)

Theoremdrnginvmuld 39752 Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   𝐼 = (invr𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )       (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼𝑌) · (𝐼𝑋)))

Theoremlmodgrpd 39753 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LMod)       (𝜑𝑊 ∈ Grp)

Theoremlvecgrp 39754 A vector space is a group. (Contributed by SN, 28-May-2023.)
(𝑊 ∈ LVec → 𝑊 ∈ Grp)

Theoremlveclmodd 39755 A vector space is a left module. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ LMod)

Theoremlvecgrpd 39756 A vector space is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ Grp)

Theoremlvecring 39757 The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ Ring)

Theoremlmhmlvec 39758 The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec))

Theoremfrlm0vald 39759 All coordinates of the zero vector are zero. (Contributed by SN, 14-Aug-2024.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)       (𝜑 → ((0g𝐹)‘𝐽) = 0 )

Theoremfrlmsnic 39760* Given a free module with a singleton as the index set, that is, a free module of one-dimensional vectors, the function that maps each vector to its coordinate is a module isomorphism from that module to its ring of scalars seen as a module. (Contributed by Steven Nguyen, 18-Aug-2023.)
𝑊 = (𝐾 freeLMod {𝐼})    &   𝐹 = (𝑥 ∈ (Base‘𝑊) ↦ (𝑥𝐼))       ((𝐾 ∈ Ring ∧ 𝐼 ∈ V) → 𝐹 ∈ (𝑊 LMIso (ringLMod‘𝐾)))

Theoremuvccl 39761 A unit vector is a vector. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐼𝑊𝐽𝐼) → (𝑈𝐽) ∈ 𝐵)

Theoremuvcn0 39762 A unit vector is nonzero. (Contributed by Steven Nguyen, 16-Jul-2023.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑌)       ((𝑅 ∈ NzRing ∧ 𝐼𝑊𝐽𝐼) → (𝑈𝐽) ≠ 0 )

Theorempwselbasr 39763 The reverse direction of pwselbasb 16809: a function between the index and base set of a structure is an element of the structure power. (Contributed by SN, 29-Jul-2024.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝑌)    &   (𝜑𝑅𝑊)    &   (𝜑𝐼𝑍)    &   (𝜑𝑋:𝐼𝐵)       (𝜑𝑋𝑉)

Theorempwspjmhmmgpd 39764* The projection given by pwspjmhm 18050 is also a monoid homomorphism between the respective multiplicative groups. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝐴𝐼)       (𝜑 → (𝑥𝐵 ↦ (𝑥𝐴)) ∈ (𝑀 MndHom 𝑇))

Theorempwsexpg 39765 Value of a group exponentiation in a structure power. Compare pwsmulg 18329. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &    = (.g𝑀)    &    · = (.g𝑇)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)    &   (𝜑𝐴𝐼)       (𝜑 → ((𝑁 𝑋)‘𝐴) = (𝑁 · (𝑋𝐴)))

Theorempwsgprod 39766* Finite products in a power structure are taken componentwise. Compare pwsgsum 19160. (Contributed by SN, 30-Jul-2024.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑌)    &   𝑀 = (mulGrp‘𝑌)    &   𝑇 = (mulGrp‘𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑 ∧ (𝑥𝐼𝑦𝐽)) → 𝑈𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 1 )       (𝜑 → (𝑀 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑇 Σg (𝑦𝐽𝑈))))

Theoremevlsval3 39767* Give a formula for the polynomial evaluation homomorphism. (Contributed by SN, 26-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝐵 = (Base‘𝑃)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐾 = (Base‘𝑆)    &   𝑈 = (𝑆s 𝑅)    &   𝑇 = (𝑆s (𝐾m 𝐼))    &   𝑀 = (mulGrp‘𝑇)    &    = (.g𝑀)    &    · = (.r𝑇)    &   𝐸 = (𝑝𝐵 ↦ (𝑇 Σg (𝑏𝐷 ↦ ((𝐹‘(𝑝𝑏)) · (𝑀 Σg (𝑏f 𝐺))))))    &   𝐹 = (𝑥𝑅 ↦ ((𝐾m 𝐼) × {𝑥}))    &   𝐺 = (𝑥𝐼 ↦ (𝑎 ∈ (𝐾m 𝐼) ↦ (𝑎𝑥)))    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑𝑄 = 𝐸)

Theoremevlsscaval 39768 Polynomial evaluation builder for a scalar. Compare evl1scad 21044. Note that scalar multiplication by 𝑋 is the same as vector multiplication by (𝐴𝑋) by asclmul1 20638. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   𝐴 = (algSc‘𝑃)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)    &   (𝜑𝐿 ∈ (𝐾m 𝐼))       (𝜑 → ((𝐴𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝐴𝑋))‘𝐿) = 𝑋))

Theoremevlsvarval 39769 Polynomial evaluation builder for a variable. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑉 = (𝐼 mVar 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝐼)    &   (𝜑𝐴 ∈ (𝐾m 𝐼))       (𝜑 → ((𝑉𝑋) ∈ 𝐵 ∧ ((𝑄‘(𝑉𝑋))‘𝐴) = (𝐴𝑋)))

Theoremevlsbagval 39770* Polynomial evaluation builder for a bag of variables. EDITORIAL: This theorem should stay in my mathbox until there's another use, since 0 and 1 using 𝑈 instead of 𝑆 is convenient for its sole use case mhphf 39780, but may not be convenient for other uses. (Contributed by SN, 29-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Base‘𝑃)    &   𝐾 = (Base‘𝑆)    &   𝑀 = (mulGrp‘𝑆)    &    = (.g𝑀)    &    0 = (0g𝑈)    &    1 = (1r𝑈)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐹 = (𝑠𝐷 ↦ if(𝑠 = 𝐵, 1 , 0 ))    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐴 ∈ (𝐾m 𝐼))    &   (𝜑𝐵𝐷)       (𝜑 → (𝐹𝑊 ∧ ((𝑄𝐹)‘𝐴) = (𝑀 Σg (𝑣𝐼 ↦ ((𝐵𝑣) (𝐴𝑣))))))

Theoremevlsexpval 39771 Polynomial evaluation builder for exponentiation. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑍)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐴 ∈ (𝐾m 𝐼))    &   (𝜑 → (𝑀𝐵 ∧ ((𝑄𝑀)‘𝐴) = 𝑉))    &    = (.g‘(mulGrp‘𝑃))    &    = (.g‘(mulGrp‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑁 𝑀) ∈ 𝐵 ∧ ((𝑄‘(𝑁 𝑀))‘𝐴) = (𝑁 𝑉)))

𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑍)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐴 ∈ (𝐾m 𝐼))    &   (𝜑 → (𝑀𝐵 ∧ ((𝑄𝑀)‘𝐴) = 𝑉))    &   (𝜑 → (𝑁𝐵 ∧ ((𝑄𝑁)‘𝐴) = 𝑊))    &    = (+g𝑃)    &    + = (+g𝑆)       (𝜑 → ((𝑀 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 𝑁))‘𝐴) = (𝑉 + 𝑊)))

Theoremevlsmulval 39773 Polynomial evaluation builder for multiplication. (Contributed by SN, 27-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝑃 = (𝐼 mPoly 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑃)    &   (𝜑𝐼𝑍)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐴 ∈ (𝐾m 𝐼))    &   (𝜑 → (𝑀𝐵 ∧ ((𝑄𝑀)‘𝐴) = 𝑉))    &   (𝜑 → (𝑁𝐵 ∧ ((𝑄𝑁)‘𝐴) = 𝑊))    &    = (.r𝑃)    &    · = (.r𝑆)       (𝜑 → ((𝑀 𝑁) ∈ 𝐵 ∧ ((𝑄‘(𝑀 𝑁))‘𝐴) = (𝑉 · 𝑊)))

Theoremfsuppind 39774* Induction on functions 𝐹:𝐴𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 20511 and frlmplusgval 20519). This theorem is structurally general for polynomial proof usage (see mplelbas 20748 and mpladd 20762). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ((𝜑 ∧ (𝑎𝐼𝑏𝐵)) → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)       ((𝜑 ∧ (𝑋:𝐼𝐵𝑋 finSupp 0 )) → 𝑋𝐻)

Theoremfsuppssindlem1 39775* Lemma for fsuppssind 39777. Functions are zero outside of their support. (Contributed by SN, 15-Jul-2024.)
(𝜑0𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑 → (𝐹 supp 0 ) ⊆ 𝑆)       (𝜑𝐹 = (𝑥𝐼 ↦ if(𝑥𝑆, ((𝐹𝑆)‘𝑥), 0 )))

Theoremfsuppssindlem2 39776* Lemma for fsuppssind 39777. Write a function as a union. (Contributed by SN, 15-Jul-2024.)
(𝜑𝐵𝑊)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝐼)       (𝜑 → (𝐹 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑥𝐼 ↦ if(𝑥𝑆, (𝑓𝑥), 0 )) ∈ 𝐻} ↔ (𝐹:𝑆𝐵 ∧ (𝐹 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))

Theoremfsuppssind 39777* Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 39774) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐼𝑉)    &   (𝜑𝑆𝐼)    &   (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)    &   ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)    &   ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)    &   (𝜑𝑋:𝐼𝐵)    &   (𝜑𝑋 finSupp 0 )    &   (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)       (𝜑𝑋𝐻)

Theoremmhpind 39778* The homogeneous polynomials of degree 𝑁 are generated by the terms of degree 𝑁 and addition. (Contributed by SN, 28-Jul-2024.)
𝐻 = (𝐼 mHomP 𝑅)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑃 = (𝐼 mPoly 𝑅)    &    + = (+g𝑃)    &   𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝑆 = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Grp)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))    &   (𝜑 → (𝐷 × { 0 }) ∈ 𝐺)    &   ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝐷 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐺)    &   ((𝜑 ∧ (𝑥 ∈ ((𝐻𝑁) ∩ 𝐺) ∧ 𝑦 ∈ ((𝐻𝑁) ∩ 𝐺))) → (𝑥 + 𝑦) ∈ 𝐺)       (𝜑𝑋𝐺)

Theoremmhphflem 39779* Lemma for mhphf 39780. Add several multiples of 𝐿 together, in a case where the total amount of multiplies is 𝑁. (Contributed by SN, 30-Jul-2024.)
𝐷 = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}    &   𝐻 = {𝑔𝐷 ∣ ((ℂflds0) Σg 𝑔) = 𝑁}    &   𝐵 = (Base‘𝐺)    &    · = (.g𝐺)    &   (𝜑𝐼𝑉)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐿𝐵)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑎𝐻) → (𝐺 Σg (𝑣𝐼 ↦ ((𝑎𝑣) · 𝐿))) = (𝑁 · 𝐿))

Theoremmhphf 39780 A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.)
𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)    &   𝐻 = (𝐼 mHomP 𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐾 = (Base‘𝑆)    &    · = (.r𝑆)    &    = (.g‘(mulGrp‘𝑆))    &   (𝜑𝐼𝑉)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝐿𝑅)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋 ∈ (𝐻𝑁))    &   (𝜑𝐴 ∈ (𝐾m 𝐼))       (𝜑 → ((𝑄𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 𝐿) · ((𝑄𝑋)‘𝐴)))

20.26.3  Arithmetic theorems

Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results.

For example, ax-1rid 10635 is used in mulid1 10667 related theorems, so one could trade off the extra axioms in mulid1 10667 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 10622; in the other direction, real number closure laws can be avoided by using ax-resscn 10622 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number).

The natural numbers are especially amenable to axiom reductions, as the set is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This makes adding natural numbers conveniently only require the rearrangement of parentheses, as shown with the following:

(4 + 3) = 7

((3 + 1) + (2 + 1)) = (6 + 1)

((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) =

((((((1 + 1) + 1) + 1) + 1) + 1) + 1)

This only requires ax-addass 10630, ax-1cn 10623, and ax-addcl 10625. (And in practice, the expression isn't completely expanded into ones.)

Multiplication by 1 requires either mulid2i 10674 or (ax-1rid 10635 and 1re 10669) as seen in 1t1e1 11826 and 1t1e1ALT 39784. Multiplying with greater natural numbers uses ax-distr 10632. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = 10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 10629 (see readdid1 39879 and readdid2 39873).

Theoremc0exALT 39781 Alternate proof of c0ex 10663 using more set theory axioms but fewer complex number axioms (add ax-10 2143, ax-11 2159, ax-13 2380, ax-nul 5174, and remove ax-1cn 10623, ax-icn 10624, ax-addcl 10625, and ax-mulcl 10627). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V

Theorem0cnALT3 39782 Alternate proof of 0cn 10661 using ax-resscn 10622, ax-addrcl 10626, ax-rnegex 10636, ax-cnre 10638 instead of ax-icn 10624, ax-addcl 10625, ax-mulcl 10627, ax-i2m1 10633. Version of 0cnALT 10902 using ax-1cn 10623 instead of ax-icn 10624. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ

Theoremelre0re 39783 Specialized version of 0red 10672 without using ax-1cn 10623 and ax-cnre 10638. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ → 0 ∈ ℝ)

Theorem1t1e1ALT 39784 Alternate proof of 1t1e1 11826 using a different set of axioms (add ax-mulrcl 10628, ax-i2m1 10633, ax-1ne0 10634, ax-rrecex 10637 and remove ax-resscn 10622, ax-mulcom 10629, ax-mulass 10631, ax-distr 10632). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 · 1) = 1

Theoremremulcan2d 39785 mulcan2d 11302 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))

Theoremreaddid1addid2d 39786 Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 10842, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐵 + 𝐴) = 𝐵)       ((𝜑𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)

Theoremsn-1ne2 39787 A proof of 1ne2 11872 without using ax-mulcom 10629, ax-mulass 10631, ax-pre-mulgt0 10642. Based on mul02lem2 10845. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2

Theoremnnn1suc 39788* A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)

Theoremnnadd1com 39789 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))

Theoremnnaddcom 39790 Addition is commutative for natural numbers. Uses fewer axioms than addcom 10854. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))

Theoremnnaddcomli 39791 Version of addcomli 10860 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶

Theoremnnadddir 39792 Right-distributivity for natural numbers without ax-mulcom 10629. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))

Theoremnnmul1com 39793 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10629. Since (𝐴 · 1) is 𝐴 by ax-1rid 10635, this is equivalent to remulid2 39902 for natural numbers, but using fewer axioms (avoiding ax-resscn 10622, ax-addass 10630, ax-mulass 10631, ax-rnegex 10636, ax-pre-lttri 10639, ax-pre-lttrn 10640, ax-pre-ltadd 10641). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))

Theoremnnmulcom 39794 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

Theoremmvrrsubd 39795 Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11078. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵𝐶))       (𝜑 → (𝐴 + 𝐶) = 𝐵)

Theoremladdrotrd 39796 Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11079 with a commuted consequent, and of mvrladdd 11081 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑 → (𝐶𝐴) = 𝐵)

Theoremraddcom12d 39797 Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11080 with a commuted consequent, and of mvlraddd 11078 with a commuted hypothesis. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐵 = (𝐴𝐶))

Theoremlsubrotld 39798 Rotate the variables left in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐵 + 𝐶) = 𝐴)

Theoremlsubcom23d 39799 Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐴𝐶) = 𝐵)

Theoremaddsubeq4com 39800 Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴𝐶) = (𝐷𝐵)))

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