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Theorem List for Metamath Proof Explorer - 39401-39500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsn-dtru 39401* dtru 5236 without ax-8 2113 or ax-12 2175. (Contributed by SN, 21-Sep-2023.)
¬ ∀𝑥 𝑥 = 𝑦
 
Theoremdifexd 39402 Existence of a difference. (Contributed by SN, 16-Jul-2024.)
(𝜑𝐴𝑉)       (𝜑 → (𝐴𝐵) ∈ V)
 
Theoremunexd 39403 The union of two sets is a set. (Contributed by SN, 16-Jul-2024.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → (𝐴𝐵) ∈ V)
 
Theorempssexg 39404 The proper subset of a set is also a set. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 
Theorempssn0 39405 A proper superset is nonempty. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵𝐵 ≠ ∅)
 
Theorempsspwb 39406 Classes are proper subclasses if and only if their power classes are proper subclasses. (Contributed by Steven Nguyen, 17-Jul-2022.)
(𝐴𝐵 ↔ 𝒫 𝐴 ⊊ 𝒫 𝐵)
 
Theoremxppss12 39407 Proper subset theorem for Cartesian product. (Contributed by Steven Nguyen, 17-Jul-2022.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ⊊ (𝐵 × 𝐷))
 
Theoremelpwbi 39408 Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)
𝐵 ∈ V       (𝐴𝐵𝐴 ∈ 𝒫 𝐵)
 
Theoremopelxpii 39409 Ordered pair membership in a Cartesian product (implication). (Contributed by Steven Nguyen, 17-Jul-2022.)
𝐴𝐶    &   𝐵𝐷       𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷)
 
Theoremiunsn 39410* Indexed union of a singleton. Compare dfiun2 4920 and rnmpt 5791. (Contributed by Steven Nguyen, 7-Jun-2023.)
𝑥𝐴 {𝐵} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 
Theoremimaopab 39411* The image of a class of ordered pairs. (Contributed by Steven Nguyen, 6-Jun-2023.)
({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝜑}
 
Theoremfnsnbt 39412 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 39413 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)})
 
Theoremofun 39414 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 39415* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = ran (𝑥𝐴 ↦ [𝑥]𝑅)
 
Theoremdfqs3 39416* Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
(𝐴 / 𝑅) = 𝑥𝐴 {[𝑥]𝑅}
 
Theoremqseq12d 39417 Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
 
Theoremqsalrel 39418* 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})
 
Theoremfzosumm1 39419* 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 39420 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))
 
Theoremnelsubginvcld 39421 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 39422 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 39423 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 39424 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 39425 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 39426 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 39427 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 39428 A lemma to illustrate the purpose of selvval2lem3 39427 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 39429 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 39430* 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 39431 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 39432 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 39433 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 39434 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 39435 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 39436 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 39437 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)    &   (𝜑𝐾 ∈ Ring)    &   (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑈𝐶)    &   (𝜑𝑉𝐷)       (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)
 
Theoremfrlmvscadiccat 39438 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)    &   (𝜑𝐾 ∈ Ring)    &   (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)))
 
Theoremgrpmndd 39439 A group is a monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝐺 ∈ Grp)       (𝜑𝐺 ∈ Mnd)
 
Theoremablcmnd 39440 An Abelian group is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝐺 ∈ Abel)       (𝜑𝐺 ∈ CMnd)
 
Theoremringabld 39441 A ring is an Abelian group. EDITORIAL: This cannot be used to shorten ringgrpd 19299 because ringabl 19326 depends on ringgrp 19295. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ Abel)
 
Theoremringcmnd 39442 A ring is a commutative monoid. (Contributed by SN, 1-Jun-2024.)
(𝜑𝑅 ∈ Ring)       (𝜑𝑅 ∈ CMnd)
 
Theoremdrngringd 39443 A division ring is a ring. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Ring)
 
Theoremdrnggrpd 39444 A division ring is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑅 ∈ DivRing)       (𝜑𝑅 ∈ Grp)
 
Theoremlmodgrpd 39445 A left module is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LMod)       (𝜑𝑊 ∈ Grp)
 
Theoremlvecgrp 39446 A vector space is a group. (Contributed by SN, 28-May-2023.)
(𝑊 ∈ LVec → 𝑊 ∈ Grp)
 
Theoremlveclmodd 39447 A vector space is a left module. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ LMod)
 
Theoremlvecgrpd 39448 A vector space is a group. (Contributed by SN, 16-May-2024.)
(𝜑𝑊 ∈ LVec)       (𝜑𝑊 ∈ Grp)
 
Theoremlvecring 39449 The scalar component of a vector space is a ring. (Contributed by SN, 28-May-2023.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → 𝐹 ∈ Ring)
 
Theoremlmhmlvec 39450 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))
 
Theoremfrlmsnic 39451* 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 39452 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 39453 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 )
 
Theoremfsuppind 39454* Induction on functions 𝐹:𝐴𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 20445 and frlmplusgval 20453). This theorem is structurally general for polynomial proof usage (see mplelbas 20668 and mpladd 20680). (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 39455* Lemma for fsuppssind 39457. 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 39456* Lemma for fsuppssind 39457. 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 39457* Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 39454) 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)
 
20.26.2  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 10596 is used in mulid1 10628 related theorems, so one could trade off the extra axioms in mulid1 10628 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 10583; in the other direction, real number closure laws can be avoided by using ax-resscn 10583 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 10591, ax-1cn 10584, and ax-addcl 10586. (And in practice, the expression isn't completely expanded into ones.)

Multiplication by 1 requires either mulid2i 10635 or (ax-1rid 10596 and 1re 10630) as seen in 1t1e1 11787 and 1t1e1ALT 39461. Multiplying with greater natural numbers uses ax-distr 10593. 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 10590 (see readdid1 39545 and readdid2 39539).

 
Theoremc0exALT 39458 Alternate proof of c0ex 10624 using more set theory axioms but fewer complex number axioms (add ax-10 2142, ax-11 2158, ax-13 2379, ax-nul 5174, and remove ax-1cn 10584, ax-icn 10585, ax-addcl 10586, and ax-mulcl 10588). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ V
 
Theorem0cnALT3 39459 Alternate proof of 0cn 10622 using ax-resscn 10583, ax-addrcl 10587, ax-rnegex 10597, ax-cnre 10599 instead of ax-icn 10585, ax-addcl 10586, ax-mulcl 10588, ax-i2m1 10594. Version of 0cnALT 10863 using ax-1cn 10584 instead of ax-icn 10585. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
0 ∈ ℂ
 
Theoremelre0re 39460 Specialized version of 0red 10633 without using ax-1cn 10584 and ax-cnre 10599. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝐴 ∈ ℝ → 0 ∈ ℝ)
 
Theorem1t1e1ALT 39461 Alternate proof of 1t1e1 11787 using a different set of axioms (add ax-mulrcl 10589, ax-i2m1 10594, ax-1ne0 10595, ax-rrecex 10598 and remove ax-resscn 10583, ax-mulcom 10590, ax-mulass 10592, ax-distr 10593). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 · 1) = 1
 
Theoremremulcan2d 39462 mulcan2d 11263 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 39463 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 10803, 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 39464 A proof of 1ne2 11833 without using ax-mulcom 10590, ax-mulass 10592, ax-pre-mulgt0 10603. Based on mul02lem2 10806. (Contributed by SN, 13-Dec-2023.)
1 ≠ 2
 
Theoremnnn1suc 39465* 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 39466 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
 
Theoremnnaddcom 39467 Addition is commutative for natural numbers. Uses fewer axioms than addcom 10815. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremnnaddcomli 39468 Version of addcomli 10821 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶
 
Theoremnnadddir 39469 Right-distributivity for natural numbers without ax-mulcom 10590. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
Theoremnnmul1com 39470 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 10590. Since (𝐴 · 1) is 𝐴 by ax-1rid 10596, this is equivalent to remulid2 39568 for natural numbers, but using fewer axioms (avoiding ax-resscn 10583, ax-addass 10591, ax-mulass 10592, ax-rnegex 10597, ax-pre-lttri 10600, ax-pre-lttrn 10601, ax-pre-ltadd 10602). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
 
Theoremnnmulcom 39471 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremaddsubeq4com 39472 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)))
 
Theoremsqsumi 39473 A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + (𝐵 · 𝐵)) + (2 · (𝐴 · 𝐵)))
 
Theoremnegn0nposznnd 39474 Lemma for dffltz 39613. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴 ≠ 0)    &   (𝜑 → ¬ 0 < 𝐴)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℕ)
 
Theoremsqmid3api 39475 Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.)
𝐴 ∈ ℂ    &   𝑁 ∈ ℂ    &   (𝐴 + 𝑁) = 𝐵    &   (𝐵 + 𝑁) = 𝐶       (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁))
 
Theoremdecaddcom 39476 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴𝐵 + 𝐶) = (𝐴𝐶 + 𝐵)
 
Theoremsqn5i 39477 The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℕ0       (𝐴5 · 𝐴5) = (𝐴 · (𝐴 + 1))25
 
Theoremsqn5ii 39478 The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
𝐴 ∈ ℕ0    &   (𝐴 + 1) = 𝐵    &   (𝐴 · 𝐵) = 𝐶       (𝐴5 · 𝐴5) = 𝐶25
 
Theoremdecpmulnc 39479 Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11080. (Contributed by Steven Nguyen, 9-Dec-2022.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝐸    &   ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   (𝐵 · 𝐷) = 𝐺       (𝐴𝐵 · 𝐶𝐷) = 𝐸𝐹𝐺
 
Theoremdecpmul 39480 Partial products algorithm for two digit multiplication. (Contributed by Steven Nguyen, 10-Dec-2022.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝐸    &   ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   (𝐵 · 𝐷) = 𝐺𝐻    &   (𝐸𝐺 + 𝐹) = 𝐼    &   𝐺 ∈ ℕ0    &   𝐻 ∈ ℕ0       (𝐴𝐵 · 𝐶𝐷) = 𝐼𝐻
 
Theoremsqdeccom12 39481 The square of a number in terms of its digits switched. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0       ((𝐴𝐵 · 𝐴𝐵) − (𝐵𝐴 · 𝐵𝐴)) = (99 · ((𝐴 · 𝐴) − (𝐵 · 𝐵)))
 
Theoremsq3deccom12 39482 Variant of sqdeccom12 39481 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   (𝐴 + 𝐶) = 𝐷       ((𝐴𝐵𝐶 · 𝐴𝐵𝐶) − (𝐷𝐵 · 𝐷𝐵)) = (99 · ((𝐴𝐵 · 𝐴𝐵) − (𝐶 · 𝐶)))
 
Theorem235t711 39483 Calculate a product by long multiplication as a base comparison with other multiplication algorithms.

Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 10639 saving the lower level uses of mulcomli 10639 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12195 are added then this proof would benefit more than ex-decpmul 39484.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 11760 or 8t7e56 12206. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 · 711) = 167085
 
Theoremex-decpmul 39484 Example usage of decpmul 39480. This proof is significantly longer than 235t711 39483. There is more unnecessary carrying compared to 235t711 39483. Although saving 5 visual steps, using mulcomli 10639 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(235 · 711) = 167085
 
20.26.3  Exponents
 
Theoremoexpreposd 39485 Lemma for dffltz 39613. (Contributed by Steven Nguyen, 4-Mar-2023.)
(𝜑𝑁 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)       (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁𝑀)))
 
Theoremcxpgt0d 39486 Exponentiation with a positive mantissa is positive. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℝ)       (𝜑 → 0 < (𝐴𝑐𝑁))
 
Theoremdvdsexpim 39487 dvdssqim 15894 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴𝐵 → (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremnn0rppwr 39488 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 15898 extended to nonnegative integers. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
 
Theoremexpgcd 39489 Exponentiation distributes over GCD. sqgcd 15899 extended to nonnegative exponents. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremnn0expgcd 39490 Exponentiation distributes over GCD. nn0gcdsq 16082 extended to nonnegative exponents. expgcd 39489 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremzexpgcd 39491 Exponentiation distributes over GCD. zgcdsq 16083 extended to nonnegative exponents. nn0expgcd 39490 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremnumdenexp 39492 numdensq 16084 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → ((numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁) ∧ (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁)))
 
Theoremnumexp 39493 numsq 16085 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁))
 
Theoremdenexp 39494 densq 16086 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁))
 
Theoremexp11d 39495 sq11d 13617 for positive real bases and nonzero exponents. (Contributed by Steven Nguyen, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)    &   (𝜑 → (𝐴𝑁) = (𝐵𝑁))       (𝜑𝐴 = 𝐵)
 
Theoremltexp1d 39496 ltmul1d 12460 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝐴 < 𝐵 ↔ (𝐴𝑁) < (𝐵𝑁)))
 
Theoremltexp1dd 39497 Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑𝐴 < 𝐵)       (𝜑 → (𝐴𝑁) < (𝐵𝑁))
 
Theoremzrtelqelz 39498 zsqrtelqelz 16088 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴𝑐(1 / 𝑁)) ∈ ℤ)
 
Theoremzrtdvds 39499 A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℕ) → (𝐴𝑐(1 / 𝑁)) ∥ 𝐴)
 
Theoremrtprmirr 39500 The root of a prime number is irrational. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (ℤ‘2)) → (𝑃𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ))
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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 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