P. 338 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	constrrtcc 33701	In the construction of constructible numbers, circle-circle intersections are roots of a quadratic equation. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    &   ⊢ 𝑃 = ((𝐵 − 𝐶) · (∗‘(𝐵 − 𝐶)))    &   ⊢ 𝑄 = ((𝐸 − 𝐹) · (∗‘(𝐸 − 𝐹)))    &   ⊢ 𝑀 = (((𝑄 − ((∗‘𝐷) · (𝐷 + 𝐴))) − (𝑃 − ((∗‘𝐴) · (𝐷 + 𝐴)))) / ((∗‘𝐷) − (∗‘𝐴)))    &   ⊢ 𝑁 = -(((((∗‘𝐴) · (𝐷 · 𝐴)) − (𝑃 · 𝐷)) − (((∗‘𝐷) · (𝐷 · 𝐴)) − (𝑄 · 𝐴))) / ((∗‘𝐷) − (∗‘𝐴)))    ⇒   ⊢ (𝜑 → ((𝑋↑2) + ((𝑀 · 𝑋) + 𝑁)) = 0)
Theorem	isconstr 33702*	Property of being a constructible number. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    ⇒   ⊢ (𝐴 ∈ Constr ↔ ∃𝑚 ∈ ω 𝐴 ∈ (𝐶‘𝑚))
Theorem	constr0 33703	The first step of the construction of constructible numbers is the pair {0, 1}. In this theorem and the following, we use (𝐶‘𝑁) for the 𝑁-th intermediate iteration of the constructible number. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    ⇒   ⊢ (𝐶‘∅) = {0, 1}
Theorem	constrsuc 33704*	Membership in the successor step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ 𝑆 = (𝐶‘𝑁)    ⇒   ⊢ (𝜑 → (𝑋 ∈ (𝐶‘suc 𝑁) ↔ (𝑋 ∈ ℂ ∧ (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑋 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 ∃𝑡 ∈ ℝ (𝑋 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑋 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 ∃𝑐 ∈ 𝑆 ∃𝑑 ∈ 𝑆 ∃𝑒 ∈ 𝑆 ∃𝑓 ∈ 𝑆 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑋 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑋 − 𝑑)) = (abs‘(𝑒 − 𝑓)))))))
Theorem	constrlim 33705*	Limit step of the construction of constructible numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ 𝑉)    &   ⊢ (𝜑 → Lim 𝑁)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) = ∪ 𝑛 ∈ 𝑁 (𝐶‘𝑛))
Theorem	constrsscn 33706*	Closure of the constructible points in the complex numbers. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ ℂ)
Theorem	constrsslem 33707*	Lemma for constrss 33709. This lemma requires the additional condition that 0 is a constructible number; that condition is removed in constrss 33709. (Proposed by Saveliy Skresanov, 23-JUn-2025.) (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 0 ∈ (𝐶‘𝑁))    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
Theorem	constr01 33708*	0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
Theorem	constrss 33709*	Constructed points are in the next generation constructed points. Lemma 7.3 of [Stewart] p. 91 (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (𝐶‘suc 𝑁))
Theorem	constrmon 33710*	The construction of constructible numbers is monotonous, i.e. if the ordinal 𝑀 is less than the ordinal 𝑁, which is denoted by 𝑀 ∈ 𝑁, then the 𝑀-th step of the constructible numbers is included in the 𝑁-th step. (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑁)    ⇒   ⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))
Theorem	constrconj 33711*	If a point 𝑋 of the complex plane is constructible, so is its conjugate (∗‘𝑋). (Proposed by Saveliy Skresanov, 25-Jun-2025.) (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶‘𝑁))    ⇒   ⊢ (𝜑 → (∗‘𝑋) ∈ (𝐶‘𝑁))
Theorem	constrfin 33712*	Each step of the construction of constructible numbers is finite. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝑁 ∈ ω)    ⇒   ⊢ (𝜑 → (𝐶‘𝑁) ∈ Fin)
Theorem	constrelextdg2 33713*	If the 𝑁-th step (𝐶‘𝑁) of the construction of constuctible numbers is included in a subfield 𝐹 of the complex numbers, then any element 𝑋 of the next step (𝐶‘suc 𝑁) is either in 𝐹 or in a quadratic extension of 𝐹. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐾 = (ℂfld ↾s 𝐹)    &   ⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (𝐹 ∪ {𝑋})))    &   ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘ℂfld))    &   ⊢ (𝜑 → 𝑁 ∈ On)    &   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐶‘suc 𝑁))    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐹 ∨ (𝐿[:]𝐾) = 2))
Theorem	constrextdg2lem 33714*	Lemma for constrextdg2 33715 (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ (𝜑 → 𝑅 ∈ ( < Chain(SubDRing‘ℂfld)))    &   ⊢ (𝜑 → (𝑅‘0) = ℚ)    &   ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain(SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))
Theorem	constrextdg2 33715*	Any step (𝐶‘𝑁) of the construction of constructible numbers is contained in the last field of a tower of quadratic field extensions starting with ℚ. See Theorem 7.11 of [Stewart] p. 97. (Contributed by Thierry Arnoux, 19-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    ⇒   ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain(SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘𝑁) ⊆ (lastS‘𝑣)))
Theorem	constrext2chnlem 33716*	Lemma for constrext2chn 33725. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ 𝐸 = (ℂfld ↾s 𝑒)    &   ⊢ 𝐹 = (ℂfld ↾s 𝑓)    &   ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}    &   ⊢ (𝜑 → 𝑁 ∈ ω)    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝐿 = (ℂfld ↾s (ℂfld fldGen (ℚ ∪ {𝐴})))    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
Theorem	constrfiss 33717*	For any finite set 𝐴 of constructible numbers, there is a 𝑛 -th step (𝐶‘𝑛) containing all numbers in 𝐴. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ⊆ Constr)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ω 𝐴 ⊆ (𝐶‘𝑛))
Theorem	constrllcllem 33718*	Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrlccllem 33719*	Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcccllem 33720*	Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐺)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcbvlem 33721*	Technical lemma for eliminating the hypothesis of constr0 33703 and co. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ rec((𝑧 ∈ V ↦ {𝑦 ∈ ℂ ∣ (∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑜 ∈ ℝ ∃𝑝 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ 𝑦 = (𝑘 + (𝑝 · (𝑙 − 𝑘))) ∧ (ℑ‘((∗‘(𝑗 − 𝑖)) · (𝑙 − 𝑘))) ≠ 0) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 ∃𝑜 ∈ ℝ (𝑦 = (𝑖 + (𝑜 · (𝑗 − 𝑖))) ∧ (abs‘(𝑦 − 𝑘)) = (abs‘(𝑚 − 𝑞))) ∨ ∃𝑖 ∈ 𝑧 ∃𝑗 ∈ 𝑧 ∃𝑘 ∈ 𝑧 ∃𝑙 ∈ 𝑧 ∃𝑚 ∈ 𝑧 ∃𝑞 ∈ 𝑧 (𝑖 ≠ 𝑙 ∧ (abs‘(𝑦 − 𝑖)) = (abs‘(𝑗 − 𝑘)) ∧ (abs‘(𝑦 − 𝑙)) = (abs‘(𝑚 − 𝑞))))}), {0, 1}) = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
Theorem	constrllcl 33722	Constructible numbers are closed under line-line intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → 𝑋 = (𝐺 + (𝑅 · (𝐷 − 𝐺))))    &   ⊢ (𝜑 → (ℑ‘((∗‘(𝐵 − 𝐴)) · (𝐷 − 𝐺))) ≠ 0)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrlccl 33723	Constructible numbers are closed under line-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐺 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑇 ∈ ℝ)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝑋 = (𝐴 + (𝑇 · (𝐵 − 𝐴))))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐺)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrcccl 33724	Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ Constr)    &   ⊢ (𝜑 → 𝐵 ∈ Constr)    &   ⊢ (𝜑 → 𝐶 ∈ Constr)    &   ⊢ (𝜑 → 𝐷 ∈ Constr)    &   ⊢ (𝜑 → 𝐸 ∈ Constr)    &   ⊢ (𝜑 → 𝐹 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐴)) = (abs‘(𝐵 − 𝐶)))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝐷)) = (abs‘(𝐸 − 𝐹)))    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrext2chn 33725*	If a constructible number generates some subfield 𝐿 of ℂ, then the degree of the extension of 𝐿 over ℚ is a power of two. Theorem 7.12 of [Stewart] p. 98. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ 𝐿 = (ℂfld ↾s 𝑆)    &   ⊢ 𝑆 = (ℂfld fldGen (ℚ ∪ {𝐴}))    &   ⊢ (𝜑 → 𝐴 ∈ Constr)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ0 (𝐿[:]𝑄) = (2↑𝑛))
Theorem	constrcn 33726	Constructible numbers are complex numbers. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → 𝑋 ∈ ℂ)
Theorem	nn0constr 33727	Nonnegative integers are constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    ⇒   ⊢ (𝜑 → 𝑁 ∈ Constr)
Theorem	constraddcl 33728	Constructive numbers are closed under complex addition. Item (1) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ Constr)
Theorem	constrnegcl 33729	Constructible numbers are closed under additive inverse. Item (2) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → -𝑋 ∈ Constr)
Theorem	zconstr 33730	Integers are constructible. (Contributed by Thierry Arnoux, 3-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → 𝑋 ∈ Constr)
Theorem	constrdircl 33731	Constructible numbers are closed under taking the point on the unit circle having the same argument. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (𝑋 / (abs‘𝑋)) ∈ Constr)
Theorem	iconstr 33732	The imaginary unit i is constructible. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ i ∈ Constr
Theorem	constrremulcl 33733	If two real numbers 𝑋 and 𝑌 are constructible, then, so is their product. (Contributed by Thierry Arnoux, 2-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 𝑌 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ Constr)
Theorem	constrcjcl 33734	Constructible numbers are closed under complex conjugate. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (∗‘𝑋) ∈ Constr)
Theorem	constrrecl 33735	Constructible numbers are closed under taking the real part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (ℜ‘𝑋) ∈ Constr)
Theorem	constrimcl 33736	Constructible numbers are closed under taking the imaginary part. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (ℑ‘𝑋) ∈ Constr)
Theorem	constrmulcl 33737	Constructible numbers are closed under complex multiplication. Item (3) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑌 ∈ Constr)    ⇒   ⊢ (𝜑 → (𝑋 · 𝑌) ∈ Constr)
Theorem	constrreinvcl 33738	If a real number 𝑋 is constructible, then, so is its inverse. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    ⇒   ⊢ (𝜑 → (1 / 𝑋) ∈ Constr)
Theorem	constrinvcl 33739	Constructible numbers are closed under complex inverse. Item (4) of Theorem 7.10 of [Stewart] p. 96 (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ≠ 0)    ⇒   ⊢ (𝜑 → (1 / 𝑋) ∈ Constr)
Theorem	constrcon 33740*	Contradiction of constructibility: If a complex number 𝐴 has minimal polynomial 𝐹 over ℚ of a degree that is not a power of 2, then 𝐴 is not constructible. (Contributed by Thierry Arnoux, 26-Oct-2025.)
⊢ 𝐷 = (deg1‘(ℂfld ↾s ℚ))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐹 = (𝑀‘𝐴))    &   ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐷‘𝐹) ≠ (2↑𝑛))    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ Constr)
Theorem	constrsdrg 33741	Constructible numbers form a subfield of the complex numbers. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ Constr ∈ (SubDRing‘ℂfld)
Theorem	constrfld 33742	The constructible numbers form a field. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (ℂfld ↾s Constr) ∈ Field
Theorem	constrresqrtcl 33743	If a positive real number 𝑋 is constructible, then, so is its square root. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    &   ⊢ (𝜑 → 𝑋 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝑋)    ⇒   ⊢ (𝜑 → (√‘𝑋) ∈ Constr)
Theorem	constrabscl 33744	Constructible numbers are closed under absolute value (modulus). (Contributed by Thierry Arnoux, 6-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (abs‘𝑋) ∈ Constr)
Theorem	constrsqrtcl 33745	Constructible numbers are closed under taking the square root. This is not generally the case for the cubic root operation, see 2sqr3nconstr 33747. Item (5) of Theorem 7.10 of [Stewart] p. 96 (Proposed by Saveliy Skresanov, 3-Nov-2025.) (Contributed by Thierry Arnoux, 6-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ Constr)    ⇒   ⊢ (𝜑 → (√‘𝑋) ∈ Constr)
21.3.12.1  Impossible constructions
Theorem	2sqr3minply 33746	The polynomial ((𝑋↑3) − 2) is the minimal polynomial for (2↑𝑐(1 / 3)) over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Jun-2025.)
⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ − = (-g‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) − (𝐾‘2))    &   ⊢ 𝐴 = (2↑𝑐(1 / 3))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    ⇒   ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
Theorem	2sqr3nconstr 33747	Doubling the cube is an impossible construction, i.e. the cube root of 2 is not constructible with straightedge and compass. Given a cube of edge of length one, a cube of double volume would have an edge of length (2↑𝑐(1 / 3)), however that number is not constructible. This is the first part of Metamath 100 proof #8. Theorem 7.13 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.)
⊢ (2↑𝑐(1 / 3)) ∉ Constr
Theorem	cos9thpiminplylem1 33748	The polynomial ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) has no integer roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℤ)    ⇒   ⊢ (𝜑 → ((𝑋↑3) + ((-3 · (𝑋↑2)) + 1)) ≠ 0)
Theorem	cos9thpiminplylem2 33749	The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) has no rational roots. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ ℚ)    ⇒   ⊢ (𝜑 → ((𝑋↑3) + ((-3 · 𝑋) + 1)) ≠ 0)
Theorem	cos9thpiminplylem3 33750	Lemma for cos9thpiminply 33754. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ⊢ ((𝑂↑2) + (𝑂 + 1)) = 0
Theorem	cos9thpiminplylem4 33751	Lemma for cos9thpiminply 33754. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    ⇒   ⊢ ((𝑍↑6) + (𝑍↑3)) = -1
Theorem	cos9thpiminplylem5 33752	The constructed complex number 𝐴 is a root of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    ⇒   ⊢ ((𝐴↑3) + ((-3 · 𝐴) + 1)) = 0
Theorem	cos9thpiminplylem6 33753	Evaluation of the polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)). (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ + = (+g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   ⊢ (𝜑 → 𝑌 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((ℂfld evalSub1 ℚ)‘𝐹)‘𝑌) = ((𝑌↑3) + ((-3 · 𝑌) + 1)))
Theorem	cos9thpiminply 33754	The polynomial ((𝑋↑3) + ((-3 · 𝑋) + 1)) is the minimal polynomial for 𝐴 over ℚ, and its degree is 3. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    &   ⊢ 𝑄 = (ℂfld ↾s ℚ)    &   ⊢ + = (+g‘𝑃)    &   ⊢ · = (.r‘𝑃)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑃))    &   ⊢ 𝑃 = (Poly1‘𝑄)    &   ⊢ 𝐾 = (algSc‘𝑃)    &   ⊢ 𝑋 = (var1‘𝑄)    &   ⊢ 𝐷 = (deg1‘𝑄)    &   ⊢ 𝐹 = ((3 ↑ 𝑋) + (((𝐾‘-3) · 𝑋) + (𝐾‘1)))    &   ⊢ 𝑀 = (ℂfld minPoly ℚ)    ⇒   ⊢ (𝐹 = (𝑀‘𝐴) ∧ (𝐷‘𝐹) = 3)
Theorem	cos9thpinconstrlem1 33755	The complex number 𝑂, representing an angle of (2 · π) / 3, is constructible. (Contributed by Thierry Arnoux, 14-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    ⇒   ⊢ 𝑂 ∈ Constr
Theorem	cos9thpinconstrlem2 33756	The complex number 𝐴 is not constructible. (Contributed by Thierry Arnoux, 15-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    &   ⊢ 𝐴 = (𝑍 + (1 / 𝑍))    ⇒   ⊢ ¬ 𝐴 ∈ Constr
Theorem	cos9thpinconstr 33757	Trisecting an angle is an impossible construction. Given for example 𝑂 = (exp‘((i · (2 · π)) / 3)), which represents an angle of ((2 · π) / 3), the cube root of 𝑂 is not constructible with straightedge and compass, while 𝑂 itself is constructible. This is the second part of Metamath 100 proof #8. Theorem 7.14 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 15-Nov-2025.)
⊢ 𝑂 = (exp‘((i · (2 · π)) / 3))    &   ⊢ 𝑍 = (𝑂↑𝑐(1 / 3))    ⇒   ⊢ (𝑂 ∈ Constr ∧ 𝑍 ∉ Constr)
Theorem	trisecnconstr 33758	Not all angles can be trisected. (Contributed by Thierry Arnoux, 15-Nov-2025.)
⊢ ¬ ∀𝑜 ∈ Constr (𝑜↑𝑐(1 / 3)) ∈ Constr
21.3.13  Matrices
21.3.13.1  Submatrices
Syntax	csmat 33759	Syntax for a function generating submatrices.
class subMat1
Definition	df-smat 33760*	Define a function generating submatrices of an integer-indexed matrix. The function maps an index in ((1...𝑀) × (1...𝑁)) into a new index in ((1...(𝑀 − 1)) × (1...(𝑁 − 1))). A submatrix is obtained by deleting a row and a column of the original matrix. Because this function re-indexes the matrix, the resulting submatrix still has the same index set for rows and columns, and its determinent is defined, unlike the current df-subma 22480. (Contributed by Thierry Arnoux, 18-Aug-2020.)
⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
Theorem	smatfval 33761*	Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
Theorem	smatrcl 33762	Closure of the rectangular submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    ⇒   ⊢ (𝜑 → 𝑆 ∈ (𝐵 ↑m ((1...(𝑀 − 1)) × (1...(𝑁 − 1)))))
Theorem	smatlem 33763	Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ ℕ)    &   ⊢ (𝜑 → 𝐽 ∈ ℕ)    &   ⊢ (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)    &   ⊢ (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))
Theorem	smattl 33764	Entries of a submatrix, top left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴𝐽))
Theorem	smattr 33765	Entries of a submatrix, top right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (1..^𝐿))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴𝐽))
Theorem	smatbl 33766	Entries of a submatrix, bottom left. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (1..^𝐾))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝐼𝐴(𝐽 + 1)))
Theorem	smatbr 33767	Entries of a submatrix, bottom right. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))    &   ⊢ (𝜑 → 𝐼 ∈ (𝐾...𝑀))    &   ⊢ (𝜑 → 𝐽 ∈ (𝐿...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑆𝐽) = ((𝐼 + 1)𝐴(𝐽 + 1)))
Theorem	smatcl 33768	Closure of the square submatrix: if 𝑀 is a square matrix of dimension 𝑁 with indices in (1...𝑁), then a submatrix of 𝑀 is of dimension (𝑁 − 1). (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐶 = (Base‘((1...(𝑁 − 1)) Mat 𝑅))    &   ⊢ 𝑆 = (𝐾(subMat1‘𝑀)𝐿)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐶)
Theorem	matmpo 33769*	Write a square matrix as a mapping operation. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ (𝑀 ∈ 𝐵 → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀𝑗)))
Theorem	1smat1 33770	The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 22486. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 1 = (1r‘((1...𝑁) Mat 𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘ 1 )𝐼) = (1r‘((1...(𝑁 − 1)) Mat 𝑅)))
Theorem	submat1n 33771	One case where the submatrix with integer indices, subMat1, and the general submatrix subMat, agree. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑁(((1...𝑁) subMat 𝑅)‘𝑀)𝑁))
Theorem	submatres 33772	Special case where the submatrix is a restriction of the initial matrix, and no renumbering occurs. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ 𝐵) → (𝑁(subMat1‘𝑀)𝑁) = (𝑀 ↾ ((1...(𝑁 − 1)) × (1...(𝑁 − 1)))))
Theorem	submateqlem1 33773	Lemma for submateq 33775. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝐾 ≤ 𝑀)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateqlem2 33774	Lemma for submateq 33775. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 < 𝐾)    ⇒   ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾})))
Theorem	submateq 33775*	Sufficient condition for two submatrices to be equal. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐸 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ ((1...𝑁) ∖ {𝐼}) ∧ 𝑗 ∈ ((1...𝑁) ∖ {𝐽})) → (𝑖𝐸𝑗) = (𝑖𝐹𝑗))    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝐸)𝐽) = (𝐼(subMat1‘𝐹)𝐽))
Theorem	submatminr1 33776	If we take a submatrix by removing the row 𝐼 and column 𝐽, then the result is the same on the matrix with row 𝐼 and column 𝐽 modified by the minMatR1 operator. (Contributed by Thierry Arnoux, 25-Aug-2020.)
⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐸 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑀)𝐽) = (𝐼(subMat1‘𝐸)𝐽))
21.3.13.2  Matrix literals
Syntax	clmat 33777	Extend class notation with the literal matrix conversion function.
class litMat
Definition	df-lmat 33778*	Define a function converting words of words into matrices. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ litMat = (𝑚 ∈ V ↦ (𝑖 ∈ (1...(♯‘𝑚)), 𝑗 ∈ (1...(♯‘(𝑚‘0))) ↦ ((𝑚‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatval 33779*	Value of the literal matrix conversion function. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ (𝑀 ∈ 𝑉 → (litMat‘𝑀) = (𝑖 ∈ (1...(♯‘𝑀)), 𝑗 ∈ (1...(♯‘(𝑀‘0))) ↦ ((𝑀‘(𝑖 − 1))‘(𝑗 − 1))))
Theorem	lmatfval 33780*	Entries of a literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = ((𝑊‘(𝐼 − 1))‘(𝐽 − 1)))
Theorem	lmatfvlem 33781*	Useful lemma to extract literal matrix entries. Suggested by Mario Carneiro. (Contributed by Thierry Arnoux, 3-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝐾 ∈ ℕ0    &   ⊢ 𝐿 ∈ ℕ0    &   ⊢ 𝐼 ≤ 𝑁    &   ⊢ 𝐽 ≤ 𝑁    &   ⊢ (𝐾 + 1) = 𝐼    &   ⊢ (𝐿 + 1) = 𝐽    &   ⊢ (𝑊‘𝐾) = 𝑋    &   ⊢ (𝜑 → (𝑋‘𝐿) = 𝑌)    ⇒   ⊢ (𝜑 → (𝐼𝑀𝐽) = 𝑌)
Theorem	lmatcl 33782*	Closure of the literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘𝑊)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑊 ∈ Word Word 𝑉)    &   ⊢ (𝜑 → (♯‘𝑊) = 𝑁)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑁)) → (♯‘(𝑊‘𝑖)) = 𝑁)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝑂 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝑃 = (Base‘𝑂)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑀 ∈ 𝑃)
Theorem	lmat22lem 33783*	Lemma for lmat22e11 33784 and co. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^2)) → (♯‘(⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩‘𝑖)) = 2)
Theorem	lmat22e11 33784	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 28-Aug-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀1) = 𝐴)
Theorem	lmat22e12 33785	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (1𝑀2) = 𝐵)
Theorem	lmat22e21 33786	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀1) = 𝐶)
Theorem	lmat22e22 33787	Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (2𝑀2) = 𝐷)
Theorem	lmat22det 33788	The determinant of a literal 2x2 complex matrix. (Contributed by Thierry Arnoux, 1-Sep-2020.)
⊢ 𝑀 = (litMat‘⟨“⟨“𝐴𝐵”⟩⟨“𝐶𝐷”⟩”⟩)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ · = (.r‘𝑅)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝐽 = ((1...2) maDet 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    ⇒   ⊢ (𝜑 → (𝐽‘𝑀) = ((𝐴 · 𝐷) − (𝐶 · 𝐵)))
21.3.13.3  Laplace expansion of determinants
Theorem	mdetpmtr1 33789*	The determinant of a matrix with permuted rows is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀𝑗))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
Theorem	mdetpmtr2 33790*	The determinant of a matrix with permuted columns is the determinant of the original matrix multiplied by the sign of the permutation. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑖𝑀(𝑃‘𝑗)))    ⇒   ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) ∧ (𝑀 ∈ 𝐵 ∧ 𝑃 ∈ 𝐺)) → (𝐷‘𝑀) = (((𝑍 ∘ 𝑆)‘𝑃) · (𝐷‘𝐸)))
Theorem	mdetpmtr12 33791*	The determinant of a matrix with permuted rows and columns is the determinant of the original matrix multiplied by the product of the signs of the permutations. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐺 = (Base‘(SymGrp‘𝑁))    &   ⊢ 𝑆 = (pmSgn‘𝑁)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐸 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑃‘𝑖)𝑀(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐷‘𝐸)))
Theorem	mdetlap1 33792*	A Laplace expansion of the determinant of a matrix, using the adjunct (cofactor) matrix. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ 𝐴 = (𝑁 Mat 𝑅)    &   ⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐷 = (𝑁 maDet 𝑅)    &   ⊢ 𝐾 = (𝑁 maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁) → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ 𝑁 ↦ ((𝐼𝑀𝑗) · (𝑗(𝐾‘𝑀)𝐼)))))
Theorem	madjusmdetlem1 33793*	Lemma for madjusmdet 33797. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝐺 = (Base‘(SymGrp‘(1...𝑁)))    &   ⊢ 𝑆 = (pmSgn‘(1...𝑁))    &   ⊢ 𝑈 = (𝐼(((1...𝑁) minMatR1 𝑅)‘𝑀)𝐽)    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ ((𝑃‘𝑖)𝑈(𝑄‘𝑗)))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐺)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐺)    &   ⊢ (𝜑 → (𝑃‘𝑁) = 𝐼)    &   ⊢ (𝜑 → (𝑄‘𝑁) = 𝐽)    &   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘((𝑆‘𝑃) · (𝑆‘𝑄))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdetlem2 33794*	Lemma for madjusmdet 33797. (Contributed by Thierry Arnoux, 26-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (1...(𝑁 − 1))) → if(𝑋 < 𝐼, 𝑋, (𝑋 + 1)) = ((𝑃 ∘ ◡𝑆)‘𝑋))
Theorem	madjusmdetlem3 33795*	Lemma for madjusmdet 33797. (Contributed by Thierry Arnoux, 27-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑊 = (𝑖 ∈ (1...𝑁), 𝑗 ∈ (1...𝑁) ↦ (((𝑃 ∘ ◡𝑆)‘𝑖)𝑈((𝑄 ∘ ◡𝑇)‘𝑗)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼(subMat1‘𝑈)𝐽) = (𝑁(subMat1‘𝑊)𝑁))
Theorem	madjusmdetlem4 33796*	Lemma for madjusmdet 33797. (Contributed by Thierry Arnoux, 22-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ 𝑃 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝐼, if(𝑖 ≤ 𝐼, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑆 = (𝑖 ∈ (1...𝑁) ↦ if(𝑖 = 1, 𝑁, if(𝑖 ≤ 𝑁, (𝑖 − 1), 𝑖)))    &   ⊢ 𝑄 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝐽, if(𝑗 ≤ 𝐽, (𝑗 − 1), 𝑗)))    &   ⊢ 𝑇 = (𝑗 ∈ (1...𝑁) ↦ if(𝑗 = 1, 𝑁, if(𝑗 ≤ 𝑁, (𝑗 − 1), 𝑗)))    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	madjusmdet 33797	Express the cofactor of the matrix, i.e. the entries of its adjunct matrix, using determinant of submatrices. (Contributed by Thierry Arnoux, 23-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐽(𝐾‘𝑀)𝐼) = ((𝑍‘(-1↑(𝐼 + 𝐽))) · (𝐸‘(𝐼(subMat1‘𝑀)𝐽))))
Theorem	mdetlap 33798*	Laplace expansion of the determinant of a square matrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
⊢ 𝐵 = (Base‘𝐴)    &   ⊢ 𝐴 = ((1...𝑁) Mat 𝑅)    &   ⊢ 𝐷 = ((1...𝑁) maDet 𝑅)    &   ⊢ 𝐾 = ((1...𝑁) maAdju 𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑍 = (ℤRHom‘𝑅)    &   ⊢ 𝐸 = ((1...(𝑁 − 1)) maDet 𝑅)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝐼 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝐽 ∈ (1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐷‘𝑀) = (𝑅 Σg (𝑗 ∈ (1...𝑁) ↦ ((𝑍‘(-1↑(𝐼 + 𝑗))) · ((𝐼𝑀𝑗) · (𝐸‘(𝐼(subMat1‘𝑀)𝑗)))))))
21.3.14  Topology
Theorem	ist0cld 33799*	The predicate "is a T0 space", using closed sets. (Contributed by Thierry Arnoux, 16-Aug-2020.)
⊢ (𝜑 → 𝐵 = ∪ 𝐽)    &   ⊢ (𝜑 → 𝐷 = (Clsd‘𝐽))    ⇒   ⊢ (𝜑 → (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∀𝑑 ∈ 𝐷 (𝑥 ∈ 𝑑 ↔ 𝑦 ∈ 𝑑) → 𝑥 = 𝑦))))
21.3.14.1  Open maps
Theorem	txomap 33800*	Given two open maps 𝐹 and 𝐺, 𝐻 mapping pairs of sets, is also an open map for the product topology. (Contributed by Thierry Arnoux, 29-Dec-2019.)
⊢ (𝜑 → 𝐹:𝑋⟶𝑍)    &   ⊢ (𝜑 → 𝐺:𝑌⟶𝑇)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))    &   ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))    &   ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑇))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐾) → (𝐺 “ 𝑦) ∈ 𝑀)    &   ⊢ (𝜑 → 𝐴 ∈ (𝐽 ×t 𝐾))    &   ⊢ 𝐻 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    ⇒   ⊢ (𝜑 → (𝐻 “ 𝐴) ∈ (𝐿 ×t 𝑀))