P. 227 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mdettpos 22601	Determinant is invariant under transposition. Proposition 4.8 in [Lang] p. 514. (Contributed by Stefan O'Rear, 9-Jul-2018.)
⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐷‘tpos 𝑀) = (𝐷‘𝑀))
Theorem	mdetunilem1 22602*	Lemma for mdetuni 22612. (Contributed by SO, 14-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → (𝐷‘𝐸) = 0 )
Theorem	mdetunilem2 22603*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺))    &   ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )
Theorem	mdetunilem3 22604*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐵) ∧ (𝐺 ∈ 𝐵 ∧ 𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((𝐹 ↾ ({𝐻} × 𝑁)) ∘f + (𝐺 ↾ ({𝐻} × 𝑁)))) ∧ ((𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐹 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = ((𝐷‘𝐹) + (𝐷‘𝐺)))
Theorem	mdetunilem4 22605*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))
Theorem	mdetunilem5 22606*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → 𝐸 ∈ 𝑁)    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, (𝐹 + 𝐺), 𝐻))) = ((𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, 𝐻))) + (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, 𝐻)))))
Theorem	mdetunilem6 22607*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜓 → 𝜑)    &   ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐹 ∈ 𝑁 ∧ 𝐸 ≠ 𝐹))    &   ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → (𝐺 ∈ 𝐾 ∧ 𝐻 ∈ 𝐾))    &   ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐼 ∈ 𝐾)    ⇒   ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐺, if(𝑎 = 𝐹, 𝐻, 𝐼)))) = ((invg‘𝑅)‘(𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐻, if(𝑎 = 𝐹, 𝐺, 𝐼))))))
Theorem	mdetunilem7 22608*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    ⇒   ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ 𝐹 ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)𝐹𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘𝐹)))
Theorem	mdetunilem8 22609*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )    ⇒   ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )
Theorem	mdetunilem9 22610*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )    &   ⊢ 𝑌 = {𝑥 ∣ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑁 ↑m 𝑁)(∀𝑤 ∈ 𝑥 (𝑦‘𝑤) = if(𝑤 ∈ 𝑧, 1 , 0 ) → (𝐷‘𝑦) = 0 )}    ⇒   ⊢ (𝜑 → 𝐷 = (𝐵 × { 0 }))
Theorem	mdetuni0 22611*	Lemma for mdetuni 22612. (Contributed by SO, 15-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ 𝐸 = (𝑁 maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = ((𝐷‘(1r‘𝐴)) · (𝐸‘𝐹)))
Theorem	mdetuni 22612*	According to the definition in [Weierstrass] p. 272, the determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring. So for any multilinear (mdetuni.li and mdetuni.sc), alternating (mdetuni.al) and normalized (mdetuni.no) function D (mdetuni.ff) from the algebra of square matrices (mdetuni.a) to their underlying commutative ring (mdetuni.cr), the function value of this function D for a matrix F (mdetuni.f) is the determinant of this matrix. (Contributed by Stefan O'Rear, 15-Jul-2018.) (Revised by Alexander van der Vekens, 8-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))    &   ⊢ 𝐸 = (𝑁 maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 1 )    ⇒   ⊢ (𝜑 → (𝐷‘𝐹) = (𝐸‘𝐹))
Theorem	mdetmul 22613	Multiplicativity of the determinant function: the determinant of a matrix product of square matrices equals the product of their determinants. Proposition 4.15 in [Lang] p. 517. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∙ = (.r‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐷‘(𝐹 ∙ 𝐺)) = ((𝐷‘𝐹) · (𝐷‘𝐺)))
11.5.2  Determinants of 2 x 2 -matrices
Theorem	m2detleiblem1 22614	Lemma 1 for m2detleib 22621. (Contributed by AV, 12-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃) → (𝑌‘(𝑆‘𝑄)) = (((pmSgn‘𝑁)‘𝑄)(.g‘𝑅) 1 ))
Theorem	m2detleiblem5 22615	Lemma 5 for m2detleib 22621. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩}) → (𝑌‘(𝑆‘𝑄)) = 1 )
Theorem	m2detleiblem6 22616	Lemma 6 for m2detleib 22621. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩}) → (𝑌‘(𝑆‘𝑄)) = (𝐼‘ 1 ))
Theorem	m2detleiblem7 22617	Lemma 7 for m2detleib 22621. (Contributed by AV, 20-Dec-2018.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅) ∧ 𝑍 ∈ (Base‘𝑅)) → (𝑋(+g‘𝑅)((𝐼‘ 1 ) · 𝑍)) = (𝑋 − 𝑍))
Theorem	m2detleiblem2 22618*	Lemma 2 for m2detleib 22621. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 1-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) ∈ (Base‘𝑅))
Theorem	m2detleiblem3 22619*	Lemma 3 for m2detleib 22621. (Contributed by AV, 16-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (+g‘𝐺)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 1⟩, ⟨2, 2⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((1𝑀1) · (2𝑀2)))
Theorem	m2detleiblem4 22620*	Lemma 4 for m2detleib 22621. (Contributed by AV, 20-Dec-2018.) (Proof shortened by AV, 2-Jan-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ · = (+g‘𝐺)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑄 = {⟨1, 2⟩, ⟨2, 1⟩} ∧ 𝑀 ∈ 𝐵) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ ((𝑄‘𝑛)𝑀𝑛))) = ((2𝑀1) · (1𝑀2)))
Theorem	m2detleib 22621	Leibniz' Formula for 2x2-matrices. (Contributed by AV, 21-Dec-2018.) (Revised by AV, 26-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
⊢ 𝑁 = {1, 2}    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ − = (-g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐷‘𝑀) = (((1𝑀1) · (2𝑀2)) − ((2𝑀1) · (1𝑀2))))
11.5.3  The matrix adjugate/adjunct
Syntax	cmadu 22622	Syntax for the matrix adjugate/adjunct function.
class maAdju
Syntax	cminmar1 22623	Syntax for the minor matrices of a square matrix.
class minMatR1
Definition	df-madu 22624*	Define the adjugate or adjunct (matrix of cofactors) of a square matrix. This definition gives the standard cofactors, however the internal minors are not the standard minors, see definition in [Lang] p. 518. (Contributed by Stefan O'Rear, 7-Sep-2015.) (Revised by SO, 10-Jul-2018.)
⊢ maAdju = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ ((𝑛 maDet 𝑟)‘(𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, (1r‘𝑟), (0g‘𝑟)), (𝑘𝑚𝑙)))))))
Definition	df-minmar1 22625*	Define the matrices whose determinants are the minors of a square matrix. In contrast to the standard definition of minors, a row is replaced by 0's and one 1 instead of deleting the column and row (e.g., definition in [Lang] p. 515). By this, the determinant of such a matrix is equal to the minor determined in the standard way (as determinant of a submatrix, see df-subma 22567- note that the matrix is transposed compared with the submatrix defined in df-subma 22567, but this does not matter because the determinants are the same, see mdettpos 22601). Such matrices are used in the definition of an adjunct of a square matrix, see df-madu 22624. (Contributed by AV, 27-Dec-2018.)
⊢ minMatR1 = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘 ∈ 𝑛, 𝑙 ∈ 𝑛 ↦ (𝑖 ∈ 𝑛, 𝑗 ∈ 𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, (1r‘𝑟), (0g‘𝑟)), (𝑖𝑚𝑗))))))
Theorem	mndifsplit 22626	Lemma for maducoeval2 22630. (Contributed by SO, 16-Jul-2018.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 0 = (0g‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
Theorem	madufval 22627*	First substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 𝐽 = (𝑚 ∈ 𝐵 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑚𝑙))))))
Theorem	maduval 22628*	Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝐽‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑗, if(𝑙 = 𝑖, 1 , 0 ), (𝑘𝑀𝑙))))))
Theorem	maducoeval 22629*	An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
Theorem	maducoeval2 22630*	An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))
Theorem	maduf 22631	Creating the adjunct of matrices is a function from the set of matrices into the set of matrices. (Contributed by Stefan O'Rear, 11-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑅 ∈ CRing → 𝐽:𝐵⟶𝐵)
Theorem	madutpos 22632	The adjuct of a transposed matrix is the transposition of the adjunct of the matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐽‘tpos 𝑀) = tpos (𝐽‘𝑀))
Theorem	madugsum 22633*	The determinant of a matrix with a row 𝐿 consisting of the same element 𝑋 is the sum of the elements of the 𝐿-th column of the adjunct of the matrix multiplied with 𝑋. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑋 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐿 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝑅 Σg (𝑖 ∈ 𝑁 ↦ (𝑋 · (𝑖(𝐽‘𝑀)𝐿)))) = (𝐷‘(𝑗 ∈ 𝑁, 𝑖 ∈ 𝑁 ↦ if(𝑗 = 𝐿, 𝑋, (𝑗𝑀𝑖)))))
Theorem	madurid 22634	Multiplying a matrix with its adjunct results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 16-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → (𝑀 · (𝐽‘𝑀)) = ((𝐷‘𝑀) ∙ 1 ))
Theorem	madulid 22635	Multiplying the adjunct of a matrix with the matrix results in the identity matrix multiplied with the determinant of the matrix. See Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 1 = (1r‘𝐴)    &   ⊢ · = (.r‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝑅 ∈ CRing) → ((𝐽‘𝑀) · 𝑀) = ((𝐷‘𝑀) ∙ 1 ))
Theorem	minmar1fval 22636*	First substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 𝑄 = (𝑚 ∈ 𝐵 ↦ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑚𝑗)))))
Theorem	minmar1val0 22637*	Second substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝑄‘𝑀) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 1 , 0 ), (𝑖𝑀𝑗)))))
Theorem	minmar1val 22638*	Third substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝐾(𝑄‘𝑀)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))))
Theorem	minmar1eval 22639	An entry of a matrix for a minor. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑄 = (𝑁 minMatR1 𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝐾(𝑄‘𝑀)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 1 , 0 ), (𝐼𝑀𝐽)))
Theorem	minmar1marrep 22640	The minor matrix is a special case of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.) (Revised by AV, 4-Jul-2022.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑁 minMatR1 𝑅)‘𝑀) = (𝑀(𝑁 matRRep 𝑅) 1 ))
Theorem	minmar1cl 22641	Closure of the row replacement function for square matrices: The matrix for a minor is a matrix. (Contributed by AV, 13-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐿) ∈ 𝐵)
Theorem	maducoevalmin1 22642	The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝐻((𝑁 minMatR1 𝑅)‘𝑀)𝐼)))
11.5.4  Laplace expansion of determinants (special case) According to Wikipedia ("Laplace expansion", 08-Mar-2019, https://en.wikipedia.org/wiki/Laplace_expansion) "In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression for the determinant det(B) of an n x n -matrix B that is a weighted sum of the determinants of n sub-matrices of B, each of size (n-1) x (n-1)". The expansion is usually performed for a row of matrix B (alternately for a column of matrix B). The mentioned "sub-matrices" are the matrices resultung from deleting the i-th row and the j-th column of matrix B. The mentioned "weights" (factors/coefficients) are the elements at position i and j in matrix B. If the expansion is performed for a row, the coefficients are the elements of the selected row. In the following, only the case where the row for the expansion contains only the zero element of the underlying ring except at the diagonal position. By this, the sum for the Laplace expansion is reduced to one summand, consisting of the element at the diagonal position multiplied with the determinant of the corresponding submatrix, see smadiadetg 22663 or smadiadetr 22665.
Theorem	symgmatr01lem 22643*	Lemma for symgmatr01 22644. (Contributed by AV, 3-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 if(𝑘 = 𝐾, if((𝑄‘𝑘) = 𝐿, 𝐴, 𝐵), (𝑘𝑀(𝑄‘𝑘))) = 𝐵))
Theorem	symgmatr01 22644*	Applying a permutation that does not fix a certain element of a set to a second element to an index of a matrix a row with 0's and a 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑘 ∈ 𝑁 (𝑘(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑘)) = 0 ))
Theorem	gsummatr01lem1 22645*	Lemma A for gsummatr01 22649. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    ⇒   ⊢ ((𝑄 ∈ 𝑅 ∧ 𝑋 ∈ 𝑁) → (𝑄‘𝑋) ∈ 𝑁)
Theorem	gsummatr01lem2 22646*	Lemma B for gsummatr01 22649. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    ⇒   ⊢ ((𝑄 ∈ 𝑅 ∧ 𝑋 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ (Base‘𝐺) → (𝑋𝐴(𝑄‘𝑋)) ∈ (Base‘𝐺)))
Theorem	gsummatr01lem3 22647*	Lemma 1 for gsummatr01 22649. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) → (𝐺 Σg (𝑛 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)))) = ((𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛))))(+g‘𝐺)(𝐾(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝐾))))
Theorem	gsummatr01lem4 22648*	Lemma 2 for gsummatr01 22649. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ ((((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) ∧ 𝑛 ∈ (𝑁 ∖ {𝐾})) → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)) = (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄‘𝑛)))
Theorem	gsummatr01 22649*	Lemma 1 for smadiadetlem4 22659. (Contributed by AV, 8-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑅 = {𝑟 ∈ 𝑃 ∣ (𝑟‘𝐾) = 𝐿}    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐺)    ⇒   ⊢ (((𝐺 ∈ CMnd ∧ 𝑁 ∈ Fin) ∧ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝐴𝑗) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ∧ 𝑄 ∈ 𝑅)) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 0 , 𝐵), (𝑖𝐴𝑗)))(𝑄‘𝑛)))) = (𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝐴𝑗))(𝑄‘𝑛)))))
Theorem	marep01ma 22650*	Replacing a row of a square matrix by a row with 0's and a 1 results in a square matrix of the same dimension. (Contributed by AV, 30-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑀 ∈ 𝐵 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))) ∈ 𝐵)
Theorem	smadiadetlem0 22651*	Lemma 0 for smadiadet 22660: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)))) = 0 ))
Theorem	smadiadetlem1 22652*	Lemma 1 for smadiadet 22660: A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑝 ∈ 𝑃) → (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) ∈ (Base‘𝑅))
Theorem	smadiadetlem1a 22653*	Lemma 1a for smadiadet 22660: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )
Theorem	smadiadetlem2 22654*	Lemma 2 for smadiadet 22660: The summands of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to itself. (Contributed by AV, 31-Dec-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )
Theorem	smadiadetlem3lem0 22655*	Lemma 0 for smadiadetlem3 22658. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) ∧ 𝑄 ∈ 𝑊) → (((𝑌 ∘ 𝑍)‘𝑄)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑄‘𝑛))))) ∈ (Base‘𝑅))
Theorem	smadiadetlem3lem1 22656*	Lemma 1 for smadiadetlem3 22658. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))):𝑊⟶(Base‘𝑅))
Theorem	smadiadetlem3lem2 22657*	Lemma 2 for smadiadetlem3 22658. (Contributed by AV, 12-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → ran (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛)))))) ⊆ ((Cntz‘𝑅)‘ran (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadetlem3 22658*	Lemma 3 for smadiadet 22660. (Contributed by AV, 31-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))) = (𝑅 Σg (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadetlem4 22659*	Lemma 4 for smadiadet 22660. (Contributed by AV, 31-Jan-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝐺 = (mulGrp‘𝑅)    &   ⊢ 𝑌 = (ℤRHom‘𝑅)    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑊 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   ⊢ 𝑍 = (pmSgn‘(𝑁 ∖ {𝐾}))    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝑅 Σg (𝑝 ∈ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐾} ↦ (((𝑌 ∘ 𝑆)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐾, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σg (𝑝 ∈ 𝑊 ↦ (((𝑌 ∘ 𝑍)‘𝑝)(.r‘𝑅)(𝐺 Σg (𝑛 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑛(𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐾}) ↦ (𝑖𝑀𝑗))(𝑝‘𝑛))))))))
Theorem	smadiadet 22660	The determinant of a submatrix of a square matrix obtained by removing a row and a column at the same index equals the determinant of the original matrix with the row replaced with 0's and a 1 at the diagonal position. (Contributed by AV, 31-Jan-2019.) (Proof shortened by AV, 24-Jul-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁) → (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾)) = (𝐷‘(𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾)))
Theorem	smadiadetglem1 22661	Lemma 1 for smadiadetg 22663. (Contributed by AV, 13-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)) = ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ((𝑁 ∖ {𝐾}) × 𝑁)))
Theorem	smadiadetglem2 22662	Lemma 2 for smadiadetg 22663. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾) ↾ ({𝐾} × 𝑁)) = ((({𝐾} × 𝑁) × {𝑆}) ∘f · ((𝐾((𝑁 minMatR1 𝑅)‘𝑀)𝐾) ↾ ({𝐾} × 𝑁))))
Theorem	smadiadetg 22663	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑅 ∈ CRing    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐸 = ((𝑁 ∖ {𝐾}) maDet 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → (𝐷‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆 · (𝐸‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetg0 22664	Lemma for smadiadetr 22665: version of smadiadetg 22663 with all hypotheses defining class variables removed, i.e. all class variables defined in the hypotheses replaced in the theorem by their definition. (Contributed by AV, 15-Feb-2019.)
⊢ 𝑅 ∈ CRing    ⇒   ⊢ ((𝑀 ∈ (Base‘(𝑁 Mat 𝑅)) ∧ 𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅)) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
Theorem	smadiadetr 22665	The determinant of a square matrix with one row replaced with 0's and an arbitrary element of the underlying ring at the diagonal position equals the ring element multiplied with the determinant of a submatrix of the square matrix obtained by removing the row and the column at the same index. Closed form of smadiadetg 22663. Special case of the "Laplace expansion", see definition in [Lang] p. 515. (Contributed by AV, 15-Feb-2019.)
⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ (Base‘(𝑁 Mat 𝑅))) ∧ (𝐾 ∈ 𝑁 ∧ 𝑆 ∈ (Base‘𝑅))) → ((𝑁 maDet 𝑅)‘(𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐾)) = (𝑆(.r‘𝑅)(((𝑁 ∖ {𝐾}) maDet 𝑅)‘(𝐾((𝑁 subMat 𝑅)‘𝑀)𝐾))))
11.5.5  Inverse matrix
Theorem	invrvald 22666	If a matrix multiplied with a given matrix (from the left as well as from the right) results in the identity matrix, this matrix is the inverse (matrix) of the given matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 · 𝑌) = 1 )    &   ⊢ (𝜑 → (𝑌 · 𝑋) = 1 )    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∧ (𝐼‘𝑋) = 𝑌))
Theorem	matinv 22667	The inverse of a matrix is the adjunct of the matrix multiplied with the inverse of the determinant of the matrix if the determinant is a unit in the underlying ring. Proposition 4.16 in [Lang] p. 518. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐽 = (𝑁 maAdju 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (Unit‘𝐴)    &   ⊢ 𝑉 = (Unit‘𝑅)    &   ⊢ 𝐻 = (invr‘𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    &   ⊢ ∙ = ( ·𝑠 ‘𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ (𝐷‘𝑀) ∈ 𝑉) → (𝑀 ∈ 𝑈 ∧ (𝐼‘𝑀) = ((𝐻‘(𝐷‘𝑀)) ∙ (𝐽‘𝑀))))
Theorem	matunit 22668	A matrix is a unit in the ring of matrices iff its determinant is a unit in the underlying ring. (Contributed by Stefan O'Rear, 17-Jul-2018.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑈 = (Unit‘𝐴)    &   ⊢ 𝑉 = (Unit‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑈 ↔ (𝐷‘𝑀) ∈ 𝑉))
11.5.6  Cramer's rule In the following, Cramer's rule cramer 22681 is proven. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule 22681: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations." The outline of the proof for systems of linear equations with coefficients from a commutative ring, according to the proof in Wikipedia (https://en.wikipedia.org/wiki/Cramer's_rule#A_short_proof), 22681 is as follows: The system of linear equations 𝐴 × 𝑋 = 𝐵 to be solved shall be given by the N x N coefficient matrix 𝐴 and the N-dimensional vector 𝐵. Let (𝐴‘𝑖) be the matrix obtained by replacing the i-th column of the coefficient matrix 𝐴 by the right-hand side vector 𝐵. Additionally, let (𝑋‘𝑖) be the matrix obtained by replacing the i-th column of the identity matrix by the solution vector 𝑋, with 𝑋 = (𝑥‘𝑖). Finally, it is assumed that det 𝐴 is a unit in the underlying ring. With these definitions, it follows that 𝐴 × (𝑋‘𝑖) = (𝐴‘𝑖) (cramerimplem2 22674), using matrix multiplication (mamuval 22383) and multiplication of a vector with a matrix (mulmarep1gsum2 22564). By using the multiplicativity of the determinant (mdetmul 22613) it follows that det (𝐴‘𝑖) = det (𝐴 × (𝑋‘𝑖)) = det 𝐴 · det (𝑋‘𝑖) (cramerimplem3 22675). Furthermore, it follows that det (𝑋‘𝑖) = (𝑥‘𝑖) (cramerimplem1 22673). To show this, a special case of the Laplace expansion is used (smadiadetg 22663). From these equations and the cancellation law for division in a ring (dvrcan3 20388) it follows that (𝑥‘𝑖) = det (𝑋‘𝑖) = det (𝐴‘𝑖) / det 𝐴. This is the right to left implication (cramerimp 22676, cramerlem1 22677, cramerlem2 22678) of Cramer's rule (cramer 22681). The left to right implication is shown by cramerlem3 22679, using the fact that a solution of the system of linear equations exists (slesolex 22672). Notice that for the special case of 0-dimensional matrices/vectors only the left to right implication is valid (see cramer0 22680), because assuming the right-hand side of the implication ((𝑋 · 𝑍) = 𝑌), 𝑍 could be anything (see mavmul0g 22543).
Theorem	slesolvec 22669	Every solution of a system of linear equations represented by a matrix and a vector is a vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 27-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
Theorem	slesolinv 22670	The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = ((𝐼‘𝑋) · 𝑌))
Theorem	slesolinvbi 22671	The solution of a system of linear equations represented by a matrix with a unit as determinant is the multiplication of the inverse of the matrix with the right-hand side vector. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐼 = (invr‘𝐴)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ((𝑋 · 𝑍) = 𝑌 ↔ 𝑍 = ((𝐼‘𝑋) · 𝑌)))
Theorem	slesolex 22672*	Every system of linear equations represented by a matrix with a unit as determinant has a solution. (Contributed by AV, 11-Feb-2019.) (Revised by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∃𝑧 ∈ 𝑉 (𝑋 · 𝑧) = 𝑌)
Theorem	cramerimplem1 22673	Lemma 1 for cramerimp 22676: The determinant of the identity matrix with the ith column replaced by a (column) vector equals the ith component of the vector. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 5-Jul-2022.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ 𝑍 ∈ 𝑉) → (𝐷‘𝐸) = (𝑍‘𝐼))
Theorem	cramerimplem2 22674	Lemma 2 for cramerimp 22676: The matrix of a system of linear equations multiplied with the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋 × 𝐸) = 𝐻)
Theorem	cramerimplem3 22675	Lemma 3 for cramerimp 22676: The determinant of the matrix of a system of linear equations multiplied with the determinant of the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ ⊗ = (.r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)) = (𝐷‘𝐻))
Theorem	cramerimp 22676	One direction of Cramer's rule (according to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations."): The ith component of the solution vector of a system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the righthand side vector of the system of linear equations divided by the determinant of the matrix of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐸 = (((1r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)    &   ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝐼) = ((𝐷‘𝐻) / (𝐷‘𝑋)))
Theorem	cramerlem1 22677*	Lemma 1 for cramer 22681. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))
Theorem	cramerlem2 22678*	Lemma 2 for cramer 22681. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))))
Theorem	cramerlem3 22679*	Lemma 3 for cramer 22681. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))
Theorem	cramer0 22680*	Special case of Cramer's rule for 0-dimensional matrices/vectors. (Contributed by AV, 28-Feb-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑁 = ∅ ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) → (𝑋 · 𝑍) = 𝑌))
Theorem	cramer 22681*	Cramer's rule. According to Wikipedia "Cramer's rule", 21-Feb-2019, https://en.wikipedia.org/wiki/Cramer%27s_rule: "[Cramer's rule] ... expresses the [unique] solution [of a system of linear equations] in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand sides of the equations." If it is assumed that a (unique) solution exists, it can be obtained by Cramer's rule (see also cramerimp 22676). On the other hand, if a vector can be constructed by Cramer's rule, it is a solution of the system of linear equations, so at least one solution exists. The uniqueness is ensured by considering only systems of linear equations whose matrix has a unit (of the underlying ring) as determinant, see matunit 22668 or slesolinv 22670. For fields as underlying rings, this requirement is equivalent to the determinant not being 0. Theorem 4.4 in [Lang] p. 513. This is Metamath 100 proof #97. (Contributed by Alexander van der Vekens, 21-Feb-2019.) (Revised by Alexander van der Vekens, 1-Mar-2019.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝑉 = ((Base‘𝑅) ↑m 𝑁)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ≠ ∅) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → (𝑍 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))) ↔ (𝑋 · 𝑍) = 𝑌))
11.6  Polynomial matrices A polynomial matrix or matrix of polynomials is a matrix whose elements are univariate (or multivariate) polynomials. See Wikipedia "Polynomial matrix" https://en.wikipedia.org/wiki/Polynomial_matrix (18-Nov-2019). In this section, only square matrices whose elements are univariate polynomials are considered. Usually, the ring of such matrices, the ring of n x n matrices over the polynomial ring over a ring 𝑅, is denoted by M(n, R[t]). The elements of this ring are called "polynomial matrices (over the ring 𝑅)" in the following. In Metamath notation, this ring is defined by (𝑁 Mat (Poly1‘𝑅)), usually represented by the class variable 𝐶 (or 𝑌, if 𝐶 is already occupied): 𝐶 = (𝑁 Mat 𝑃) with 𝑃 = (Poly1‘𝑅).
11.6.1  Basic properties
Theorem	pmatring 22682	The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
Theorem	pmatlmod 22683	The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod)
Theorem	pmatassa 22684	The set of polynomial matrices over a commutative ring is an associative algebra. (Contributed by AV, 16-Jun-2024.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ AssAlg)
Theorem	pmat0op 22685*	The zero polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 ))
Theorem	pmat1op 22686*	The identity polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 1 = (1r‘𝑃)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))
Theorem	pmat1ovd 22687	Entries of the identity polynomial matrix over a ring, deduction form. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 0 = (0g‘𝑃)    &   ⊢ 1 = (1r‘𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ 𝑈 = (1r‘𝐶)    ⇒   ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))
Theorem	pmat0opsc 22688*	The zero polynomial matrix over a ring represented as operation with "lifted scalars" (i.e. elements of the ring underlying the polynomial ring embedded into the polynomial ring by the scalar injection/algebra scalar lifting function algSc). (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐴‘ 0 )))
Theorem	pmat1opsc 22689*	The identity polynomial matrix over a ring represented as operation with "lifted scalars". (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝐴‘ 1 ), (𝐴‘ 0 ))))
Theorem	pmat1ovscd 22690	Entries of the identity polynomial matrix over a ring represented with "lifted scalars", deduction form. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐴 = (algSc‘𝑃)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑁)    &   ⊢ 𝑈 = (1r‘𝐶)    ⇒   ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 )))
Theorem	pmatcoe1fsupp 22691*	For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 ))
Theorem	1pmatscmul 22692	The scalar product of the identity polynomial matrix with a polynomial is a polynomial matrix. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 𝐸 = (Base‘𝑃)    &   ⊢ ∗ = ( ·𝑠 ‘𝐶)    &   ⊢ 1 = (1r‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵)
11.6.2  Constant polynomial matrices A constant polynomial matrix is a polynomial matrix whose elements are constant polynomials, i.e., polynomials with no indeterminates. Constant polynomials are obtained by "lifting" a "scalar" (i.e. an element of the underlying ring) into the polynomial ring/algebra by a "scalar injection", i.e., applying the "algebra scalar injection function" algSc (see df-ascl 21837) to a scalar 𝐴 ∈ 𝑅: ((algSc‘𝑃)‘𝐴). Analogously, constant polynomial matrices (over the ring 𝑅) are obtained by "lifting" matrices over the ring 𝑅 by the function matToPolyMat (see df-mat2pmat 22697), called "matrix transformation" in the following. In this section it is shown that the set 𝑆 = (𝑁 ConstPolyMat 𝑅) of constant polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 is a subring of the ring of polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 (cpmatsrgpmat 22711) and that 𝑇 = (𝑁 matToPolyMat 𝑅) is a ring isomorphism from the ring of matrices over a ring 𝑅 onto the ring of constant polynomial matrices over the ring 𝑅 (see m2cpmrngiso 22748). Thus, the ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring, see matcpmric 22749. Finally, 𝐼 = (𝑁 cPolyMatToMat 𝑅), the transformation of a constant polynomial matrix into a matrix, is the inverse function of the matrix transformation 𝑇 = (𝑁 matToPolyMat 𝑅), see m2cpminv 22750.
Syntax	ccpmat 22693	Extend class notation with the set of all constant polynomial matrices.
class ConstPolyMat
Syntax	cmat2pmat 22694	Extend class notation with the transformation of a matrix into a matrix of polynomials.
class matToPolyMat
Syntax	ccpmat2mat 22695	Extend class notation with the transformation of a constant polynomial matrix into a matrix.
class cPolyMatToMat
Definition	df-cpmat 22696*	The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 22278). (Contributed by AV, 15-Nov-2019.)
⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)})
Definition	df-mat2pmat 22697*	Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.)
⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦)))))
Definition	df-cpmat2mat 22698*	Transformation of a constant polynomial matrix (over a ring) into a matrix over the corresponding ring. Since this function is the inverse function of matToPolyMat, see m2cpminv 22750, it is also called "inverse matrix transformation" in the following. (Contributed by AV, 14-Dec-2019.)
⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
Theorem	cpmat 22699*	Value of the constructor of the set of all constant polynomial matrices, i.e. the set of all 𝑁 x 𝑁 matrices of polynomials over a ring 𝑅. (Contributed by AV, 15-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)})
Theorem	cpmatpmat 22700	A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    &   ⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐶 = (𝑁 Mat 𝑃)    &   ⊢ 𝐵 = (Base‘𝐶)    ⇒   ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵)