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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevls1pw 21501 Univariate polynomial evaluation for subrings maps the exponentiation of a polynomial to the exponentiation of the evaluated polynomial. (Contributed by SN, 29-Feb-2024.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝐾 = (Base‘𝑆)    &   𝐵 = (Base‘𝑊)    &    = (.g𝐺)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑋𝐵)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆s 𝐾)))(𝑄𝑋)))
 
Theoremevls1varpw 21502 Univariate polynomial evaluation for subrings maps the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑈 = (𝑆s 𝑅)    &   𝑊 = (Poly1𝑈)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑈)    &   𝐵 = (Base‘𝑆)    &    = (.g𝐺)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑆s 𝐵)))(𝑄𝑋)))
 
Theoremevl1fval 21503* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1o eval 𝑅)    &   𝐵 = (Base‘𝑅)       𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)
 
Theoremevl1val 21504* Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑄 = (1o eval 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑀 = (1o mPoly 𝑅)    &   𝐾 = (Base‘𝑀)       ((𝑅 ∈ CRing ∧ 𝐴𝐾) → (𝑂𝐴) = ((𝑄𝐴) ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
 
Theoremevl1fval1lem 21505 Lemma for evl1fval1 21506. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅𝑉𝑄 = (𝑅 evalSub1 𝐵))
 
Theoremevl1fval1 21506 Value of the simple/same ring evaluation map function for univariate polynomials. (Contributed by AV, 11-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐵 = (Base‘𝑅)       𝑄 = (𝑅 evalSub1 𝐵)
 
Theoremevl1rhm 21507 Polynomial evaluation is a homomorphism (into the product ring). (Contributed by Mario Carneiro, 12-Jun-2015.) (Proof shortened by AV, 13-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑅s 𝐵)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom 𝑇))
 
Theoremfveval1fvcl 21508 The function value of the evaluation function of a polynomial is an element of the underlying ring. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑𝑀𝑈)       (𝜑 → ((𝑂𝑀)‘𝑌) ∈ 𝐵)
 
Theoremevl1sca 21509 Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝑂‘(𝐴𝑋)) = (𝐵 × {𝑋}))
 
Theoremevl1scad 21510 Polynomial evaluation builder for scalars. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐴𝑋) ∈ 𝑈 ∧ ((𝑂‘(𝐴𝑋))‘𝑌) = 𝑋))
 
Theoremevl1var 21511 Polynomial evaluation maps the variable to the identity function. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ CRing → (𝑂𝑋) = ( I ↾ 𝐵))
 
Theoremevl1vard 21512 Polynomial evaluation builder for the variable. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝑈 ∧ ((𝑂𝑋)‘𝑌) = 𝑌))
 
Theoremevls1var 21513 Univariate polynomial evaluation for subrings maps the variable to the identity function. (Contributed by AV, 13-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑋 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑋) = ( I ↾ 𝐵))
 
Theoremevls1scasrng 21514 The evaluation of a scalar of a subring yields the same result as evaluated as a scalar over the ring itself. (Contributed by AV, 13-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑂 = (eval1𝑆)    &   𝑊 = (Poly1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝑃 = (Poly1𝑆)    &   𝐵 = (Base‘𝑆)    &   𝐴 = (algSc‘𝑊)    &   𝐶 = (algSc‘𝑃)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))    &   (𝜑𝑋𝑅)       (𝜑 → (𝑄‘(𝐴𝑋)) = (𝑂‘(𝐶𝑋)))
 
Theoremevls1varsrng 21515 The evaluation of the variable of univariate polynomials over subring yields the same result as evaluated as variable of the polynomials over the ring itself. (Contributed by AV, 12-Sep-2019.)
𝑄 = (𝑆 evalSub1 𝑅)    &   𝑂 = (eval1𝑆)    &   𝑉 = (var1𝑈)    &   𝑈 = (𝑆s 𝑅)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝑆 ∈ CRing)    &   (𝜑𝑅 ∈ (SubRing‘𝑆))       (𝜑 → (𝑄𝑉) = (𝑂𝑉))
 
Theoremevl1addd 21516 Polynomial evaluation builder for addition of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (+g𝑃)    &    + = (+g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉 + 𝑊)))
 
Theoremevl1subd 21517 Polynomial evaluation builder for subtraction of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (-g𝑃)    &   𝐷 = (-g𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉𝐷𝑊)))
 
Theoremevl1muld 21518 Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑 → (𝑁𝑈 ∧ ((𝑂𝑁)‘𝑌) = 𝑊))    &    = (.r𝑃)    &    · = (.r𝑅)       (𝜑 → ((𝑀 𝑁) ∈ 𝑈 ∧ ((𝑂‘(𝑀 𝑁))‘𝑌) = (𝑉 · 𝑊)))
 
Theoremevl1vsd 21519 Polynomial evaluation builder for scalar multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &   (𝜑𝑁𝐵)    &    = ( ·𝑠𝑃)    &    · = (.r𝑅)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 · 𝑉)))
 
Theoremevl1expd 21520 Polynomial evaluation builder for an exponential. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → (𝑀𝑈 ∧ ((𝑂𝑀)‘𝑌) = 𝑉))    &    = (.g‘(mulGrp‘𝑃))    &    = (.g‘(mulGrp‘𝑅))    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝑁 𝑀) ∈ 𝑈 ∧ ((𝑂‘(𝑁 𝑀))‘𝑌) = (𝑁 𝑉)))
 
Theorempf1const 21521 Constants are polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵) → (𝐵 × {𝑋}) ∈ 𝑄)
 
Theorempf1id 21522 The identity is a polynomial function. (Contributed by Mario Carneiro, 20-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → ( I ↾ 𝐵) ∈ 𝑄)
 
Theorempf1subrg 21523 Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝑄 = ran (eval1𝑅)       (𝑅 ∈ CRing → 𝑄 ∈ (SubRing‘(𝑅s 𝐵)))
 
Theorempf1rcl 21524 Reverse closure for the set of polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)       (𝑋𝑄𝑅 ∈ CRing)
 
Theorempf1f 21525 Polynomial functions are functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)       (𝐹𝑄𝐹:𝐵𝐵)
 
Theoremmpfpf1 21526* Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝐸 → (𝐹 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))) ∈ 𝑄)
 
Theorempf1mpf 21527* Convert a univariate polynomial function to multivariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝐸 = ran (1o eval 𝑅)       (𝐹𝑄 → (𝐹 ∘ (𝑥 ∈ (𝐵m 1o) ↦ (𝑥‘∅))) ∈ 𝐸)
 
Theorempf1addcl 21528 The sum of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    + = (+g𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹f + 𝐺) ∈ 𝑄)
 
Theorempf1mulcl 21529 The product of multivariate polynomial functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑄 = ran (eval1𝑅)    &    · = (.r𝑅)       ((𝐹𝑄𝐺𝑄) → (𝐹f · 𝐺) ∈ 𝑄)
 
Theorempf1ind 21530* Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)    &   𝑄 = ran (eval1𝑅)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)    &   ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)    &   (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))    &   (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))    &   (𝑥 = 𝑓 → (𝜓𝜏))    &   (𝑥 = 𝑔 → (𝜓𝜂))    &   (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))    &   (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))    &   (𝑥 = 𝐴 → (𝜓𝜌))    &   ((𝜑𝑓𝐵) → 𝜒)    &   (𝜑𝜃)    &   (𝜑𝐴𝑄)       (𝜑𝜌)
 
Theoremevl1gsumdlem 21531* Lemma for evl1gsumd 21532 (induction step). (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)       ((𝑚 ∈ Fin ∧ ¬ 𝑎𝑚𝜑) → ((∀𝑥𝑚 𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥𝑚𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑚 ↦ ((𝑂𝑀)‘𝑌)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀𝑈 → ((𝑂‘(𝑃 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝑌) = (𝑅 Σg (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((𝑂𝑀)‘𝑌))))))
 
Theoremevl1gsumd 21532* Polynomial evaluation builder for a finite group sum of polynomials. (Contributed by AV, 17-Sep-2019.)
𝑂 = (eval1𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Base‘𝑃)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑌𝐵)    &   (𝜑 → ∀𝑥𝑁 𝑀𝑈)    &   (𝜑𝑁 ∈ Fin)       (𝜑 → ((𝑂‘(𝑃 Σg (𝑥𝑁𝑀)))‘𝑌) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑂𝑀)‘𝑌))))
 
Theoremevl1gsumadd 21533* Univariate polynomial evaluation maps (additive) group sums to group sums. Remark: the proof would be shorter if the theorem is proved directly instead of using evls1gsumadd 21499. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    0 = (0g𝑊)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 0 )       (𝜑 → (𝑄‘(𝑊 Σg (𝑥𝑁𝑌))) = (𝑃 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
Theoremevl1gsumaddval 21534* Value of a univariate polynomial evaluation mapping an additive group sum to a group sum of the evaluated variable. (Contributed by AV, 17-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑁𝑌)))‘𝐶) = (𝑅 Σg (𝑥𝑁 ↦ ((𝑄𝑌)‘𝐶))))
 
Theoremevl1gsummul 21535* Univariate polynomial evaluation maps (multiplicative) group sums to group sums. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝑃 = (𝑅s 𝐾)    &   𝐵 = (Base‘𝑊)    &   (𝜑𝑅 ∈ CRing)    &   ((𝜑𝑥𝑁) → 𝑌𝐵)    &   (𝜑𝑁 ⊆ ℕ0)    &    1 = (1r𝑊)    &   𝐺 = (mulGrp‘𝑊)    &   𝐻 = (mulGrp‘𝑃)    &   (𝜑 → (𝑥𝑁𝑌) finSupp 1 )       (𝜑 → (𝑄‘(𝐺 Σg (𝑥𝑁𝑌))) = (𝐻 Σg (𝑥𝑁 ↦ (𝑄𝑌))))
 
Theoremevl1varpw 21536 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 21533, the proof is shorter using evls1varpw 21502 instead of proving it directly. (Contributed by AV, 15-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → (𝑄‘(𝑁 𝑋)) = (𝑁(.g‘(mulGrp‘(𝑅s 𝐵)))(𝑄𝑋)))
 
Theoremevl1varpwval 21537 Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)       (𝜑 → ((𝑄‘(𝑁 𝑋))‘𝐶) = (𝑁𝐸𝐶))
 
Theoremevl1scvarpw 21538 Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   𝑆 = (𝑅s 𝐵)    &    = (.r𝑆)    &   𝑀 = (mulGrp‘𝑆)    &   𝐹 = (.g𝑀)       (𝜑 → (𝑄‘(𝐴 × (𝑁 𝑋))) = ((𝐵 × {𝐴}) (𝑁𝐹(𝑄𝑋))))
 
Theoremevl1scvarpwval 21539 Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐺 = (mulGrp‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐵 = (Base‘𝑅)    &    = (.g𝐺)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑁 ∈ ℕ0)    &    × = ( ·𝑠𝑊)    &   (𝜑𝐴𝐵)    &   (𝜑𝐶𝐵)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &    · = (.r𝑅)       (𝜑 → ((𝑄‘(𝐴 × (𝑁 𝑋)))‘𝐶) = (𝐴 · (𝑁𝐸𝐶)))
 
Theoremevl1gsummon 21540* Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
𝑄 = (eval1𝑅)    &   𝐾 = (Base‘𝑅)    &   𝑊 = (Poly1𝑅)    &   𝐵 = (Base‘𝑊)    &   𝑋 = (var1𝑅)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐺 = (mulGrp‘𝑊)    &    = (.g𝐺)    &    × = ( ·𝑠𝑊)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑 → ∀𝑥𝑀 𝐴𝐾)    &   (𝜑𝑀 ⊆ ℕ0)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑 → ∀𝑥𝑀 𝑁 ∈ ℕ0)    &   (𝜑𝐶𝐾)       (𝜑 → ((𝑄‘(𝑊 Σg (𝑥𝑀 ↦ (𝐴 × (𝑁 𝑋)))))‘𝐶) = (𝑅 Σg (𝑥𝑀 ↦ (𝐴 · (𝑁𝐸𝐶)))))
 
11.4  Matrices

According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 20963) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 20980) and scalar multiplication (see frlmvscafval 20982) for free modules. Actually, there is no definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 21542. Thus, a statement like "Then the set of m x n matrices in R is a module (i.e., an R-module)" as in [Lang] p. 504 follows immediately from frlmlmod 20965.

However, for square matrices there is Definition df-mat 21564, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication.

A "usual" matrix (aij), (i = 1,..., m and j = 1,... n) would be represented as an element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))) and a square matrix (aij), (i = 1,..., n and j = 1,... n) would be represented as an element of (the base set of) ((1...𝑛) Mat 𝑅).

Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which are excluded from the definitions of many authors, e.g., in [Lang] p. 503. It is shown in mat0dimbas0 21624 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 21624 Its determinant is the ring unit, see mdet0fv0 21752.

 
11.4.1  The matrix multiplication

This section is about the multiplication of m x n matrices.

 
Syntaxcmmul 21541 Syntax for the matrix multiplication operator.
class maMul
 
Definitiondf-mamu 21542* The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))
 
Theoremmamufval 21543* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)       (𝜑𝐹 = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))
 
Theoremmamuval 21544* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))
 
Theoremmamufv 21545* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))    &   (𝜑𝐼𝑀)    &   (𝜑𝐾𝑃)       (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))
 
Theoremmamudm 21546 The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   𝐵 = (Base‘𝐸)    &   𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   𝐶 = (Base‘𝐹)    &    × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)       ((𝑅𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))
 
Theoremmamufacex 21547 Every solution of the equation 𝐴𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.)
𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   𝐵 = (Base‘𝐸)    &   𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   𝐶 = (Base‘𝐹)    &    × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 freeLMod (𝑀 × 𝑃))    &   𝐷 = (Base‘𝐺)       (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅𝑉𝑌𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌𝑍𝐶))
 
Theoremmamures 21548 Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 maMul ⟨𝐼, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))       (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌))
 
Theoremmndvcl 21549 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼) ∧ 𝑌 ∈ (𝐵m 𝐼)) → (𝑋f + 𝑌) ∈ (𝐵m 𝐼))
 
Theoremmndvass 21550 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵m 𝐼) ∧ 𝑌 ∈ (𝐵m 𝐼) ∧ 𝑍 ∈ (𝐵m 𝐼))) → ((𝑋f + 𝑌) ∘f + 𝑍) = (𝑋f + (𝑌f + 𝑍)))
 
Theoremmndvlid 21551 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → ((𝐼 × { 0 }) ∘f + 𝑋) = 𝑋)
 
Theoremmndvrid 21552 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵m 𝐼)) → (𝑋f + (𝐼 × { 0 })) = 𝑋)
 
Theoremgrpvlinv 21553 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → ((𝑁𝑋) ∘f + 𝑋) = (𝐼 × { 0 }))
 
Theoremgrpvrinv 21554 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵m 𝐼)) → (𝑋f + (𝑁𝑋)) = (𝐼 × { 0 }))
 
Theoremmhmvlin 21555 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+g𝑁)       ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵m 𝐼) ∧ 𝑌 ∈ (𝐵m 𝐼)) → (𝐹 ∘ (𝑋f + 𝑌)) = ((𝐹𝑋) ∘f (𝐹𝑌)))
 
Theoremringvcl 21556 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵m 𝐼) ∧ 𝑌 ∈ (𝐵m 𝐼)) → (𝑋f · 𝑌) ∈ (𝐵m 𝐼))
 
Theoremmamucl 21557 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵m (𝑀 × 𝑃)))
 
Theoremmamuass 21558 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵m (𝑂 × 𝑃)))    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)       (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))
 
Theoremmamudi 21559 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × 𝑂)))       (𝜑 → ((𝑋f + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘f + (𝑌𝐹𝑍)))
 
Theoremmamudir 21560 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(𝑌f + 𝑍)) = ((𝑋𝐹𝑌) ∘f + (𝑋𝐹𝑍)))
 
Theoremmamuvs1 21561 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × 𝑂)))       (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘f · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘f · (𝑌𝐹𝑍)))
 
Theoremmamuvs2 21562 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(𝜑𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘f · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘f · (𝑋𝐹𝑍)))
 
11.4.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 21587. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.

 
Syntaxcmat 21563 Syntax for the square matrix algebra.
class Mat
 
Definitiondf-mat 21564* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))
 
Theoremmatbas0pc 21565 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
 
Theoremmatbas0 21566 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)
 
Theoremmatval 21567 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))
 
Theoremmatrcl 21568 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
 
Theoremmatbas 21569 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘𝐺) = (Base‘𝐴))
 
Theoremmatplusg 21570 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (+g𝐺) = (+g𝐴))
 
Theoremmatsca 21571 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
 
TheoremmatscaOLD 21572 Obsolete proof of matsca 21571 as of 12-Nov-2024. The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))
 
Theoremmatvsca 21573 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))
 
TheoremmatvscaOLD 21574 Obsolete proof of matvsca 21573 as of 12-Nov-2024. The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))
 
Theoremmat0 21575 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (0g𝐺) = (0g𝐴))
 
Theoremmatinvg 21576 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (invg𝐺) = (invg𝐴))
 
Theoremmat0op 21577* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁0 ))
 
Theoremmatsca2 21578 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 = (Scalar‘𝐴))
 
Theoremmatbas2 21579 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝐾m (𝑁 × 𝑁)) = (Base‘𝐴))
 
Theoremmatbas2i 21580 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 ∈ (𝐾m (𝑁 × 𝑁)))
 
Theoremmatbas2d 21581* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅𝑉)    &   ((𝜑𝑥𝑁𝑦𝑁) → 𝐶𝐾)       (𝜑 → (𝑥𝑁, 𝑦𝑁𝐶) ∈ 𝐵)
 
Theoremeqmat 21582* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))
 
Theoremmatecl 21583 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐼𝑁𝐽𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)
 
Theoremmatecld 21584 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝑀𝐵)       (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)
 
Theoremmatplusg2 21585 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋f + 𝑌))
 
Theoremmatvsca2 21586 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)    &   𝐶 = (𝑁 × 𝑁)       ((𝑋𝐾𝑌𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘f × 𝑌))
 
Theoremmatlmod 21587 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
 
Theoremmatgrp 21588 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)
 
Theoremmatvscl 21589 Closure of the scalar multiplication in the matrix ring. (lmodvscl 20149 analog.) (Contributed by AV, 27-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)
 
Theoremmatsubg 21590 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (-g𝐺) = (-g𝐴))
 
Theoremmatplusgcell 21591 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       (((𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))
 
Theoremmatsubgcell 21592 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (-g𝐴)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) (𝐼𝑌𝐽)))
 
Theoremmatinvgcell 21593 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = (invg𝑅)    &   𝑊 = (invg𝐴)       ((𝑅 ∈ Ring ∧ 𝑋𝐵 ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑊𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))
 
Theoremmatvscacell 21594 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))
 
Theoremmatgsum 21595* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑖𝑁, 𝑗𝑁𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈)) finSupp 0 )       (𝜑 → (𝐴 Σg (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
11.4.3  The matrix algebra

The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 21601) and an associative algebra (see matassa 21602). Additionally, theorems for the identity matrix and transposed matrices are provided.

 
Theoremmatmulr 21596 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → · = (.r𝐴))
 
Theoremmamumat1cl 21597* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       (𝜑𝐼 ∈ (𝐵m (𝑀 × 𝑀)))
 
Theoremmat1comp 21598* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))
 
Theoremmamulid 21599* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))       (𝜑 → (𝐼𝐹𝑋) = 𝑋)
 
Theoremmamurid 21600* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   (𝜑𝑋 ∈ (𝐵m (𝑁 × 𝑀)))       (𝜑 → (𝑋𝐹𝐼) = 𝑋)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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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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