P. 377 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lindsenlbs 37601	A maximal linearly independent set in a free module of finite dimension over a division ring is a basis. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (((𝑅 ∈ DivRing ∧ 𝐼 ∈ Fin ∧ 𝑋 ∈ (LIndS‘(𝑅 freeLMod 𝐼))) ∧ 𝑋 ≈ 𝐼) → 𝑋 ∈ (LBasis‘(𝑅 freeLMod 𝐼)))
Theorem	matunitlindflem1 37602	One direction of matunitlindf 37604. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ (((𝑅 ∈ Field ∧ 𝑀:(𝐼 × 𝐼)⟶(Base‘𝑅)) ∧ 𝐼 ∈ (Fin ∖ {∅})) → (¬ curry 𝑀 LIndF (𝑅 freeLMod 𝐼) → ((𝐼 maDet 𝑅)‘𝑀) = (0g‘𝑅)))
Theorem	matunitlindflem2 37603	One direction of matunitlindf 37604. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) ∧ 𝐼 ≠ ∅) ∧ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)) → ((𝐼 maDet 𝑅)‘𝑀) ∈ (Unit‘𝑅))
Theorem	matunitlindf 37604	A matrix over a field is invertible iff the rows are linearly independent. (Contributed by Brendan Leahy, 2-Jun-2021.)
⊢ ((𝑅 ∈ Field ∧ 𝑀 ∈ (Base‘(𝐼 Mat 𝑅))) → (𝑀 ∈ (Unit‘(𝐼 Mat 𝑅)) ↔ curry 𝑀 LIndF (𝑅 freeLMod 𝐼)))
Theorem	ptrest 37605*	Expressing a restriction of a product topology as a product topology. (Contributed by Brendan Leahy, 24-Mar-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶Top)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑆 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((∏t‘𝐹) ↾t X𝑘 ∈ 𝐴 𝑆) = (∏t‘(𝑘 ∈ 𝐴 ↦ ((𝐹‘𝑘) ↾t 𝑆))))
Theorem	ptrecube 37606*	Any point in an open set of N-space is surrounded by an open cube within that set. (Contributed by Brendan Leahy, 21-Aug-2020.) (Proof shortened by AV, 28-Sep-2020.)
⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ 𝐷 = ((abs ∘ − ) ↾ (ℝ × ℝ))    ⇒   ⊢ ((𝑆 ∈ 𝑅 ∧ 𝑃 ∈ 𝑆) → ∃𝑑 ∈ ℝ+ X𝑛 ∈ (1...𝑁)((𝑃‘𝑛)(ball‘𝐷)𝑑) ⊆ 𝑆)
Theorem	poimirlem1 37607*	Lemma for poimir 37639- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
Theorem	poimirlem2 37608*	Lemma for poimir 37639- consecutive vertices differ in at most one dimension. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑉 ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ((0...𝑁) ∖ {𝑉}))    ⇒   ⊢ (𝜑 → ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑉 − 1))‘𝑛) ≠ ((𝐹‘𝑉)‘𝑛))
Theorem	poimirlem3 37609*	Lemma for poimir 37639 to add an interior point to an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    &   ⊢ (𝜑 → 𝑇:(1...𝑀)⟶(0..^𝐾))    &   ⊢ (𝜑 → 𝑈:(1...𝑀)–1-1-onto→(1...𝑀))    ⇒   ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋((𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 → (⟨(𝑇 ∪ {⟨(𝑀 + 1), 0⟩}), (𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((𝑇 ∪ {⟨(𝑀 + 1), 0⟩}) ∘f + ((((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ (((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((𝑇 ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ ((𝑈 ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
Theorem	poimirlem4 37610*	Lemma for poimir 37639 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑀 < 𝑁)    ⇒   ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑m (1...𝑀)) × {𝑓 ∣ 𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑m (1...(𝑀 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ⦋(((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
Theorem	poimirlem5 37611*	Lemma for poimir 37639 to establish that, for the simplices defined by a walk along the edges of an 𝑁-cube, if the starting vertex is not opposite a given face, it is the earliest vertex of the face on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 0 < (2nd ‘𝑇))    ⇒   ⊢ (𝜑 → (𝐹‘0) = (1st ‘(1st ‘𝑇)))
Theorem	poimirlem6 37612*	Lemma for poimir 37639 establishing, for a face of a simplex defined by a walk along the edges of an 𝑁-cube, the single dimension in which successive vertices before the opposite vertex differ. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ (1...((2nd ‘𝑇) − 1)))    ⇒   ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀))
Theorem	poimirlem7 37613*	Lemma for poimir 37639, similar to poimirlem6 37612, but for vertices after the opposite vertex. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑀 ∈ ((((2nd ‘𝑇) + 1) + 1)...𝑁))    ⇒   ⊢ (𝜑 → (℩𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 2))‘𝑛) ≠ ((𝐹‘(𝑀 − 1))‘𝑛)) = ((2nd ‘(1st ‘𝑇))‘𝑀))
Theorem	poimirlem8 37614*	Lemma for poimir 37639, establishing that away from the opposite vertex the walks in poimirlem9 37615 yield the same vertices. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})) = ((2nd ‘(1st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))
Theorem	poimirlem9 37615*	Lemma for poimir 37639, establishing the two walks that yield a given face when the opposite vertex is neither first nor last. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) ≠ (2nd ‘(1st ‘𝑇)))    ⇒   ⊢ (𝜑 → (2nd ‘(1st ‘𝑈)) = ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)})))))
Theorem	poimirlem10 37616*	Lemma for poimir 37639 establishing the cube that yields the simplex that yields a face if the opposite vertex was first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ((𝐹‘(𝑁 − 1)) ∘f − ((1...𝑁) × {1})) = (1st ‘(1st ‘𝑇)))
Theorem	poimirlem11 37617*	Lemma for poimir 37639 connecting walks that could yield from a given cube a given face opposite the first vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑈) = 0)    &   ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀)))
Theorem	poimirlem12 37618*	Lemma for poimir 37639 connecting walks that could yield from a given cube a given face opposite the final vertex of the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑈) = 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ((2nd ‘(1st ‘𝑇)) “ (1...𝑀)) ⊆ ((2nd ‘(1st ‘𝑈)) “ (1...𝑀)))
Theorem	poimirlem13 37619*	Lemma for poimir 37639- for at most one simplex associated with a shared face is the opposite vertex first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    ⇒   ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
Theorem	poimirlem14 37620*	Lemma for poimir 37639- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    ⇒   ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
Theorem	poimirlem15 37621*	Lemma for poimir 37639, that the face in poimirlem22 37628 is a face. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → (2nd ‘𝑇) ∈ (1...(𝑁 − 1)))    ⇒   ⊢ (𝜑 → ⟨⟨(1st ‘(1st ‘𝑇)), ((2nd ‘(1st ‘𝑇)) ∘ ({⟨(2nd ‘𝑇), ((2nd ‘𝑇) + 1)⟩, ⟨((2nd ‘𝑇) + 1), (2nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd ‘𝑇), ((2nd ‘𝑇) + 1)}))))⟩, (2nd ‘𝑇)⟩ ∈ 𝑆)
Theorem	poimirlem16 37622*	Lemma for poimir 37639 establishing the vertices of the simplex of poimirlem17 37623. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st ‘𝑇))‘1), 1, 0))) ∘f + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
Theorem	poimirlem17 37623*	Lemma for poimir 37639 establishing existence for poimirlem18 37624. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem18 37624*	Lemma for poimir 37639 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 0)    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem19 37625*	Lemma for poimir 37639 establishing the vertices of the simplex in poimirlem20 37626. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st ‘𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st ‘𝑇))‘𝑁), 1, 0))) ∘f + (((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
Theorem	poimirlem20 37626*	Lemma for poimir 37639 establishing existence for poimirlem21 37627. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem21 37627*	Lemma for poimir 37639 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of the walk, there exists exactly one other simplex containing it. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem22 37628*	Lemma for poimir 37639, that a given face belongs to exactly two simplices, provided it's not on the boundary of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}    &   ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)    ⇒   ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
Theorem	poimirlem23 37629*	Lemma for poimir 37639, two ways of expressing the property that a face is not on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑉 ∈ (0...𝑁))    ⇒   ⊢ (𝜑 → (∃𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))(𝑝‘𝑁) ≠ 0 ↔ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁))))
Theorem	poimirlem24 37630*	Lemma for poimir 37639, two ways of expressing that a simplex has an admissible face on the back face of the cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ (𝜑 → 𝑉 ∈ (0...𝑁))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑m (1...𝑁)) ↑m (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑉, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘f + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑉})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶 ∧ ¬ (𝑉 = 𝑁 ∧ ((𝑇‘𝑁) = 0 ∧ (𝑈‘𝑁) = 𝑁)))))
Theorem	poimirlem25 37631*	Lemma for poimir 37639 stating that for a given simplex such that no vertex maps to 𝑁, the number of admissible faces is even. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   ⊢ (𝜑 → 𝑇:(1...𝑁)⟶(0..^𝐾))    &   ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))    &   ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑁 ≠ ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶)    ⇒   ⊢ (𝜑 → 2 ∥ (♯‘{𝑦 ∈ (0...𝑁) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑦})𝑖 = ⦋⟨𝑇, 𝑈⟩ / 𝑠⦌𝐶}))
Theorem	poimirlem26 37632*	Lemma for poimir 37639 showing an even difference between the number of admissible faces and the number of admissible simplices. Equation (6) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    ⇒   ⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
Theorem	poimirlem27 37633*	Lemma for poimir 37639 showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))    ⇒   ⊢ (𝜑 → 2 ∥ ((♯‘{𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (♯‘{𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})))
Theorem	poimirlem28 37634*	Lemma for poimir 37639, a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝑝 = ((1st ‘𝑠) ∘f + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    ⇒   ⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
Theorem	poimirlem29 37635*	Lemma for poimir 37639 connecting cubes of the tessellation to neighborhoods. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))    &   ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘f / ((1...𝑁) × {𝑘})))‘𝑛)    &   ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))    &   ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)    ⇒   ⊢ (𝜑 → (∀𝑖 ∈ ℕ ∃𝑘 ∈ (ℤ≥‘𝑖)∀𝑚 ∈ (1...𝑁)(((1st ‘(𝐺‘𝑘)) ∘f / ((1...𝑁) × {𝑘}))‘𝑚) ∈ ((𝐶‘𝑚)(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))(1 / 𝑖)) → ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝐶 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛))))
Theorem	poimirlem30 37636*	Lemma for poimir 37639 combining poimirlem29 37635 with bwth 23433. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))    &   ⊢ 𝑋 = ((𝐹‘(((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∘f / ((1...𝑁) × {𝑘})))‘𝑛)    &   ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))    &   ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟𝑋)    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
Theorem	poimirlem31 37637*	Lemma for poimir 37639, assigning values to the vertices of the tessellation that meet the hypotheses of both poimirlem30 37636 and poimirlem28 37634. Equation (2) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)    &   ⊢ 𝑃 = ((1st ‘(𝐺‘𝑘)) ∘f + ((((2nd ‘(𝐺‘𝑘)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(𝐺‘𝑘)) “ ((𝑗 + 1)...𝑁)) × {0})))    &   ⊢ (𝜑 → 𝐺:ℕ⟶((ℕ0 ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran (1st ‘(𝐺‘𝑘)) ⊆ (0..^𝑘))    &   ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑖 ∈ (0...𝑁))) → ∃𝑗 ∈ (0...𝑁)𝑖 = sup(({0} ∪ {𝑎 ∈ (1...𝑁) ∣ ∀𝑏 ∈ (1...𝑎)(0 ≤ ((𝐹‘(𝑃 ∘f / ((1...𝑁) × {𝑘})))‘𝑏) ∧ (𝑃‘𝑏) ≠ 0)}), ℝ, < ))    ⇒   ⊢ ((𝜑 ∧ (𝑘 ∈ ℕ ∧ 𝑛 ∈ (1...𝑁) ∧ 𝑟 ∈ { ≤ , ◡ ≤ })) → ∃𝑗 ∈ (0...𝑁)0𝑟((𝐹‘(𝑃 ∘f / ((1...𝑁) × {𝑘})))‘𝑛))
Theorem	poimirlem32 37638*	Lemma for poimir 37639, combining poimirlem28 37634, poimirlem30 37636, and poimirlem31 37637 to get Equation (1) of [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 ∀𝑛 ∈ (1...𝑁)∀𝑣 ∈ (𝑅 ↾t 𝐼)(𝑐 ∈ 𝑣 → ∀𝑟 ∈ { ≤ , ◡ ≤ }∃𝑧 ∈ 𝑣 0𝑟((𝐹‘𝑧)‘𝑛)))
Theorem	poimir 37639*	Poincare-Miranda theorem. Theorem on [Kulpa] p. 547. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn 𝑅))    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 0)) → ((𝐹‘𝑧)‘𝑛) ≤ 0)    &   ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑧 ∈ 𝐼 ∧ (𝑧‘𝑛) = 1)) → 0 ≤ ((𝐹‘𝑧)‘𝑛))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 (𝐹‘𝑐) = ((1...𝑁) × {0}))
Theorem	broucube 37640*	Brouwer - or as Kulpa calls it, "Bohl-Brouwer" - fixed point theorem for the unit cube. Theorem on [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ 𝐼 = ((0[,]1) ↑m (1...𝑁))    &   ⊢ 𝑅 = (∏t‘((1...𝑁) × {(topGen‘ran (,))}))    &   ⊢ (𝜑 → 𝐹 ∈ ((𝑅 ↾t 𝐼) Cn (𝑅 ↾t 𝐼)))    ⇒   ⊢ (𝜑 → ∃𝑐 ∈ 𝐼 𝑐 = (𝐹‘𝑐))
Theorem	heicant 37641	Heine-Cantor theorem: a continuous mapping between metric spaces whose domain is compact is uniformly continuous. Theorem on [Rosenlicht] p. 80. (Contributed by Brendan Leahy, 13-Aug-2018.) (Proof shortened by AV, 27-Sep-2020.)
⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌))    &   ⊢ (𝜑 → (MetOpen‘𝐶) ∈ Comp)    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))
Theorem	opnmbllem0 37642*	Lemma for ismblfin 37647; could also be used to shorten proof of opnmbllem 25649. (Contributed by Brendan Leahy, 13-Jul-2018.)
⊢ (𝐴 ∈ (topGen‘ran (,)) → ∪ ([,] “ {𝑧 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑧) ⊆ 𝐴}) = 𝐴)
Theorem	mblfinlem1 37643*	Lemma for ismblfin 37647, ordering the sets of dyadic intervals that are antichains under subset and whose unions are contained entirely in 𝐴. (Contributed by Brendan Leahy, 13-Jul-2018.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝐴 ≠ ∅) → ∃𝑓 𝑓:ℕ–1-1-onto→{𝑎 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} ∣ ∀𝑐 ∈ {𝑏 ∈ ran (𝑥 ∈ ℤ, 𝑦 ∈ ℕ0 ↦ ⟨(𝑥 / (2↑𝑦)), ((𝑥 + 1) / (2↑𝑦))⟩) ∣ ([,]‘𝑏) ⊆ 𝐴} (([,]‘𝑎) ⊆ ([,]‘𝑐) → 𝑎 = 𝑐)})
Theorem	mblfinlem2 37644*	Lemma for ismblfin 37647, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different definition of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ ((𝐴 ∈ (topGen‘ran (,)) ∧ 𝑀 ∈ ℝ ∧ 𝑀 < (vol*‘𝐴)) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ 𝑀 < (vol*‘𝑠)))
Theorem	mblfinlem3 37645*	The difference between two sets measurable by the criterion in ismblfin 37647 is itself measurable by the same. Corollary 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘(𝐴 ∖ 𝐵)))
Theorem	mblfinlem4 37646*	Backward direction of ismblfin 37647. (Contributed by Brendan Leahy, 28-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))
Theorem	ismblfin 37647*	Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)
⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∈ dom vol ↔ (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )))
Theorem	ovoliunnfl 37648*	ovoliun 25553 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
⊢ ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓‘𝑛) ⊆ ℝ ∧ (vol*‘(𝑓‘𝑛)) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ ℕ (𝑓‘𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓‘𝑚)))), ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	ex-ovoliunnfl 37649*	Demonstration of ovoliunnfl 37648. (Contributed by Brendan Leahy, 21-Nov-2017.)
⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	voliunnfl 37650*	voliun 25602 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
⊢ 𝑆 = seq1( + , 𝐺)    &   ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝑓‘𝑛)))    &   ⊢ ((∀𝑛 ∈ ℕ ((𝑓‘𝑛) ∈ dom vol ∧ (vol‘(𝑓‘𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝑓‘𝑛)) → (vol‘∪ 𝑛 ∈ ℕ (𝑓‘𝑛)) = sup(ran 𝑆, ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	volsupnfl 37651*	volsup 25604 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)
⊢ ((𝑓:ℕ⟶dom vol ∧ ∀𝑛 ∈ ℕ (𝑓‘𝑛) ⊆ (𝑓‘(𝑛 + 1))) → (vol‘∪ ran 𝑓) = sup((vol “ ran 𝑓), ℝ*, < ))    ⇒   ⊢ ((𝐴 ≼ ℕ ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ℕ) → ∪ 𝐴 ≠ ℝ)
Theorem	mbfresfi 37652*	Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝑆 ∈ Fin)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 (𝐹 ↾ 𝑠) ∈ MblFn)    &   ⊢ (𝜑 → ∪ 𝑆 = 𝐴)    ⇒   ⊢ (𝜑 → 𝐹 ∈ MblFn)
Theorem	mbfposadd 37653*	If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn)
Theorem	cnambfre 37654	A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ∈ dom vol ∧ (vol*‘(𝐴 ∖ ((◡(((topGen‘ran (,)) ↾t 𝐴) CnP (topGen‘ran (,))) ∘ E ) “ {𝐹}))) = 0) → 𝐹 ∈ MblFn)
Theorem	dvtanlem 37655	Lemma for dvtan 37656- the domain of the tangent is open. (Contributed by Brendan Leahy, 8-Aug-2018.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (◡cos “ (ℂ ∖ {0})) ∈ (TopOpen‘ℂfld)
Theorem	dvtan 37656	Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.)
⊢ (ℂ D tan) = (𝑥 ∈ dom tan ↦ ((cos‘𝑥)↑-2))
Theorem	itg2addnclem 37657*	An alternate expression for the ∫2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)
⊢ 𝐿 = {𝑥 ∣ ∃𝑔 ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑦))) ∘r ≤ 𝐹 ∧ 𝑥 = (∫1‘𝑔))}    ⇒   ⊢ (𝐹:ℝ⟶(0[,]+∞) → (∫2‘𝐹) = sup(𝐿, ℝ*, < ))
Theorem	itg2addnclem2 37658*	Lemma for itg2addnc 37660. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    ⇒   ⊢ (((𝜑 ∧ ℎ ∈ dom ∫1) ∧ 𝑣 ∈ ℝ+) → (𝑥 ∈ ℝ ↦ if(((((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)) ≤ (ℎ‘𝑥) ∧ (ℎ‘𝑥) ≠ 0), (((⌊‘((𝐹‘𝑥) / (𝑣 / 3))) − 1) · (𝑣 / 3)), (ℎ‘𝑥))) ∈ dom ∫1)
Theorem	itg2addnclem3 37659*	Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 37660. (Contributed by Brendan Leahy, 11-Mar-2018.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∃ℎ ∈ dom ∫1(∃𝑦 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((ℎ‘𝑧) = 0, 0, ((ℎ‘𝑧) + 𝑦))) ∘r ≤ (𝐹 ∘f + 𝐺) ∧ 𝑠 = (∫1‘ℎ)) → ∃𝑡∃𝑢(∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1((∃𝑐 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑓‘𝑧) = 0, 0, ((𝑓‘𝑧) + 𝑐))) ∘r ≤ 𝐹 ∧ 𝑡 = (∫1‘𝑓)) ∧ (∃𝑑 ∈ ℝ+ (𝑧 ∈ ℝ ↦ if((𝑔‘𝑧) = 0, 0, ((𝑔‘𝑧) + 𝑑))) ∘r ≤ 𝐺 ∧ 𝑢 = (∫1‘𝑔))) ∧ 𝑠 = (𝑡 + 𝑢))))
Theorem	itg2addnc 37660	Alternate proof of itg2add 25808 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 25757, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 10472, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
⊢ (𝜑 → 𝐹 ∈ MblFn)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐹) ∈ ℝ)    &   ⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞))    &   ⊢ (𝜑 → (∫2‘𝐺) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘(𝐹 ∘f + 𝐺)) = ((∫2‘𝐹) + (∫2‘𝐺)))
Theorem	itg2gt0cn 37661*	itg2gt0 25809 holds on functions continuous on an open interval in the absence of ax-cc 10472. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)
⊢ (𝜑 → 𝑋 < 𝑌)    &   ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 0 < (𝐹‘𝑥))    &   ⊢ (𝜑 → (𝐹 ↾ (𝑋(,)𝑌)) ∈ ((𝑋(,)𝑌)–cn→ℂ))    ⇒   ⊢ (𝜑 → 0 < (∫2‘𝐹))
Theorem	ibladdnclem 37662*	Lemma for ibladdnc 37663; cf ibladdlem 25869, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 37660. (Contributed by Brendan Leahy, 31-Oct-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)    &   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)    &   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)    ⇒   ⊢ (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
Theorem	ibladdnc 37663*	Choice-free analogue of itgadd 25874. A measurability hypothesis is necessitated by the loss of mbfadd 25709; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ 𝐿1)
Theorem	itgaddnclem1 37664*	Lemma for itgaddnc 37666; cf. itgaddlem1 25872. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐶)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	itgaddnclem2 37665*	Lemma for itgaddnc 37666; cf. itgaddlem2 25873. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	itgaddnc 37666*	Choice-free analogue of itgadd 25874. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 + 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 + ∫𝐴𝐶 d𝑥))
Theorem	iblsubnc 37667*	Choice-free analogue of iblsub 25871. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ 𝐿1)
Theorem	itgsubnc 37668*	Choice-free analogue of itgsub 25875. (Contributed by Brendan Leahy, 11-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 − 𝐶)) ∈ MblFn)    ⇒   ⊢ (𝜑 → ∫𝐴(𝐵 − 𝐶) d𝑥 = (∫𝐴𝐵 d𝑥 − ∫𝐴𝐶 d𝑥))
Theorem	iblabsnclem 37669*	Lemma for iblabsnc 37670; cf. iblabslem 25877. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ 𝐺 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (abs‘(𝐹‘𝐵)), 0))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐹‘𝐵)) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝐵) ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐺 ∈ MblFn ∧ (∫2‘𝐺) ∈ ℝ))
Theorem	iblabsnc 37670*	Choice-free analogue of iblabs 25878. As with ibladdnc 37663, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ 𝐿1)
Theorem	iblmulc2nc 37671*	Choice-free analogue of iblmulc2 25880. (Contributed by Brendan Leahy, 17-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ 𝐿1)
Theorem	itgmulc2nclem1 37672*	Lemma for itgmulc2nc 37674; cf. itgmulc2lem1 25881. (Contributed by Brendan Leahy, 17-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgmulc2nclem2 37673*	Lemma for itgmulc2nc 37674; cf. itgmulc2lem2 25882. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgmulc2nc 37674*	Choice-free analogue of itgmulc2 25883. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐶 · 𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (𝐶 · ∫𝐴𝐵 d𝑥) = ∫𝐴(𝐶 · 𝐵) d𝑥)
Theorem	itgabsnc 37675*	Choice-free analogue of itgabs 25884. (Contributed by Brendan Leahy, 19-Nov-2017.) (Revised by Brendan Leahy, 19-Jun-2018.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿1)    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ MblFn)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((∗‘∫𝐴𝐵 d𝑥) · ⦋𝑦 / 𝑥⦌𝐵)) ∈ MblFn)    ⇒   ⊢ (𝜑 → (abs‘∫𝐴𝐵 d𝑥) ≤ ∫𝐴(abs‘𝐵) d𝑥)
Theorem	itggt0cn 37676*	itggt0 25893 holds for continuous functions in the absence of ax-cc 10472. (Contributed by Brendan Leahy, 16-Nov-2017.)
⊢ (𝜑 → 𝑋 < 𝑌)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ 𝐿1)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋(,)𝑌)) → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → (𝑥 ∈ (𝑋(,)𝑌) ↦ 𝐵) ∈ ((𝑋(,)𝑌)–cn→ℂ))    ⇒   ⊢ (𝜑 → 0 < ∫(𝑋(,)𝑌)𝐵 d𝑥)
Theorem	ftc1cnnclem 37677*	Lemma for ftc1cnnc 37678; cf. ftc1lem4 26094. The stronger assumptions of ftc1cn 26098 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝑐 ∈ (𝐴(,)𝐵))    &   ⊢ 𝐻 = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝑐}) ↦ (((𝐺‘𝑧) − (𝐺‘𝑐)) / (𝑧 − 𝑐)))    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((abs‘(𝑦 − 𝑐)) < 𝑅 → (abs‘((𝐹‘𝑦) − (𝐹‘𝑐))) < 𝐸))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑋 − 𝑐)) < 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐴[,]𝐵))    &   ⊢ (𝜑 → (abs‘(𝑌 − 𝑐)) < 𝑅)    ⇒   ⊢ ((𝜑 ∧ 𝑋 < 𝑌) → (abs‘((((𝐺‘𝑌) − (𝐺‘𝑋)) / (𝑌 − 𝑋)) − (𝐹‘𝑐))) < 𝐸)
Theorem	ftc1cnnc 37678*	Choice-free proof of ftc1cn 26098. (Contributed by Brendan Leahy, 20-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    ⇒   ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)
Theorem	ftc1anclem1 37679	Lemma for ftc1anc 37687- the absolute value of a real-valued measurable function is measurable. Would be trivial with cncombf 25706, but this proof avoids ax-cc 10472. (Contributed by Brendan Leahy, 18-Jun-2018.)
⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐹 ∈ MblFn) → (abs ∘ 𝐹) ∈ MblFn)
Theorem	ftc1anclem2 37680*	Lemma for ftc1anc 37687- restriction of an integrable function to the absolute value of its real or imaginary part. (Contributed by Brendan Leahy, 19-Jun-2018.) (Revised by Brendan Leahy, 8-Aug-2018.)
⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐹 ∈ 𝐿1 ∧ 𝐺 ∈ {ℜ, ℑ}) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝐴, (abs‘(𝐺‘(𝐹‘𝑡))), 0))) ∈ ℝ)
Theorem	ftc1anclem3 37681	Lemma for ftc1anc 37687- the absolute value of the sum of a simple function and i times another simple function is itself a simple function. (Contributed by Brendan Leahy, 27-May-2018.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ dom ∫1) → (abs ∘ (𝐹 ∘f + ((ℝ × {i}) ∘f · 𝐺))) ∈ dom ∫1)
Theorem	ftc1anclem4 37682*	Lemma for ftc1anc 37687. (Contributed by Brendan Leahy, 17-Jun-2018.)
⊢ ((𝐹 ∈ dom ∫1 ∧ 𝐺 ∈ 𝐿1 ∧ 𝐺:ℝ⟶ℝ) → (∫2‘(𝑡 ∈ ℝ ↦ (abs‘((𝐺‘𝑡) − (𝐹‘𝑡))))) ∈ ℝ)
Theorem	ftc1anclem5 37683*	Lemma for ftc1anc 37687, the existence of a simple function the integral of whose pointwise difference from the function is less than a given positive real. (Contributed by Brendan Leahy, 17-Jun-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘((ℜ‘if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0)) − (𝑓‘𝑡))))) < 𝑌)
Theorem	ftc1anclem6 37684*	Lemma for ftc1anc 37687- construction of simple functions within an arbitrary absolute distance of the given function. Similar to Lemma 565Ib of [Fremlin5] p. 218, but without Fremlin's additional step of converting the simple function into a continuous one, which is unnecessary to this lemma's use; also, two simple functions are used to allow for complex-valued 𝐹. (Contributed by Brendan Leahy, 31-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ ℝ+) → ∃𝑓 ∈ dom ∫1∃𝑔 ∈ dom ∫1(∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < 𝑌)
Theorem	ftc1anclem7 37685*	Lemma for ftc1anc 37687. (Contributed by Brendan Leahy, 13-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → ((∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡)))), 0))) + (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), (abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0)))) < ((𝑦 / 2) + (𝑦 / 2)))
Theorem	ftc1anclem8 37686*	Lemma for ftc1anc 37687. (Contributed by Brendan Leahy, 29-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    ⇒   ⊢ (((((((𝜑 ∧ (𝑓 ∈ dom ∫1 ∧ 𝑔 ∈ dom ∫1)) ∧ (∫2‘(𝑡 ∈ ℝ ↦ (abs‘(if(𝑡 ∈ 𝐷, (𝐹‘𝑡), 0) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))))) < (𝑦 / 2)) ∧ ∃𝑟 ∈ (ran 𝑓 ∪ ran 𝑔)𝑟 ≠ 0) ∧ 𝑦 ∈ ℝ+) ∧ (𝑢 ∈ (𝐴[,]𝐵) ∧ 𝑤 ∈ (𝐴[,]𝐵) ∧ 𝑢 ≤ 𝑤)) ∧ (abs‘(𝑤 − 𝑢)) < ((𝑦 / 2) / (2 · sup((abs “ (ran 𝑓 ∪ ran 𝑔)), ℝ, < )))) → (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ (𝑢(,)𝑤), ((abs‘((𝐹‘𝑡) − ((𝑓‘𝑡) + (i · (𝑔‘𝑡))))) + (abs‘((𝑓‘𝑡) + (i · (𝑔‘𝑡))))), 0))) < 𝑦)
Theorem	ftc1anc 37687*	ftc1a 26092 holds for functions that obey the triangle inequality in the absence of ax-cc 10472. Theorem 565Ma of [Fremlin5] p. 220. (Contributed by Brendan Leahy, 11-May-2018.)
⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐷 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)    &   ⊢ (𝜑 → ∀𝑠 ∈ ((,) “ ((𝐴[,]𝐵) × (𝐴[,]𝐵)))(abs‘∫𝑠(𝐹‘𝑡) d𝑡) ≤ (∫2‘(𝑡 ∈ ℝ ↦ if(𝑡 ∈ 𝑠, (abs‘(𝐹‘𝑡)), 0))))    ⇒   ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
Theorem	ftc2nc 37688*	Choice-free proof of ftc2 26099. (Contributed by Brendan Leahy, 19-Jun-2018.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≤ 𝐵)    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))    &   ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿1)    &   ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))    ⇒   ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
Theorem	asindmre 37689	Real part of domain of differentiability of arcsine. (Contributed by Brendan Leahy, 19-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (𝐷 ∩ ℝ) = (-1(,)1)
Theorem	dvasin 37690*	Derivative of arcsine. (Contributed by Brendan Leahy, 18-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (ℂ D (arcsin ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvacos 37691*	Derivative of arccosine. (Contributed by Brendan Leahy, 18-Dec-2018.)
⊢ 𝐷 = (ℂ ∖ ((-∞(,]-1) ∪ (1[,)+∞)))    ⇒   ⊢ (ℂ D (arccos ↾ 𝐷)) = (𝑥 ∈ 𝐷 ↦ (-1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvreasin 37692	Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.)
⊢ (ℝ D (arcsin ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (1 / (√‘(1 − (𝑥↑2)))))
Theorem	dvreacos 37693	Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.) (Proof shortened by Brendan Leahy, 18-Dec-2018.)
⊢ (ℝ D (arccos ↾ (-1(,)1))) = (𝑥 ∈ (-1(,)1) ↦ (-1 / (√‘(1 − (𝑥↑2)))))
Theorem	areacirclem1 37694*	Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ (𝑅 ∈ ℝ+ → (ℝ D (𝑡 ∈ (-𝑅(,)𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2)))))))) = (𝑡 ∈ (-𝑅(,)𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2))))))
Theorem	areacirclem2 37695*	Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
Theorem	areacirclem3 37696*	Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (2 · (√‘((𝑅↑2) − (𝑡↑2))))) ∈ 𝐿1)
Theorem	areacirclem4 37697*	Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ (𝑅 ∈ ℝ+ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) · ((arcsin‘(𝑡 / 𝑅)) + ((𝑡 / 𝑅) · (√‘(1 − ((𝑡 / 𝑅)↑2))))))) ∈ ((-𝑅[,]𝑅)–cn→ℂ))
Theorem	areacirclem5 37698*	Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}    ⇒   ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅 ∧ 𝑡 ∈ ℝ) → (𝑆 “ {𝑡}) = if((abs‘𝑡) ≤ 𝑅, (-(√‘((𝑅↑2) − (𝑡↑2)))[,](√‘((𝑅↑2) − (𝑡↑2)))), ∅))
Theorem	areacirc 37699*	The area of a circle of radius 𝑅 is π · 𝑅↑2. This is Metamath 100 proof #9. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.) (Revised by Brendan Leahy, 11-Jul-2018.)
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ((𝑥↑2) + (𝑦↑2)) ≤ (𝑅↑2))}    ⇒   ⊢ ((𝑅 ∈ ℝ ∧ 0 ≤ 𝑅) → (area‘𝑆) = (π · (𝑅↑2)))
21.24  Mathbox for Jeff Madsen
21.24.1  Logic and set theory
Theorem	unirep 37700*	Define a quantity whose definition involves a choice of representative, but which is uniquely determined regardless of the choice. (Contributed by Jeff Madsen, 1-Jun-2011.)
⊢ (𝑦 = 𝐷 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑦 = 𝑧 → 𝐵 = 𝐹)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝜑 ∧ 𝜒) → 𝐵 = 𝐹) ∧ (𝐷 ∈ 𝐴 ∧ 𝜓)) → (℩𝑥∃𝑦 ∈ 𝐴 (𝜑 ∧ 𝑥 = 𝐵)) = 𝐶)