P. 478 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 47701-47800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sharhght 47701*	Let 𝐴𝐵𝐶 be a triangle, and let 𝐷 lie on the line 𝐴𝐵. Then (doubled) areas of triangles 𝐴𝐷𝐶 and 𝐶𝐷𝐵 relate as lengths of corresponding bases 𝐴𝐷 and 𝐷𝐵. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0))    ⇒   ⊢ (𝜑 → (((𝐶 − 𝐴)𝐺(𝐷 − 𝐴)) · (𝐵 − 𝐷)) = (((𝐶 − 𝐵)𝐺(𝐷 − 𝐵)) · (𝐴 − 𝐷)))
Theorem	sigaradd 47702*	Subtracting (double) area of 𝐴𝐷𝐶 from 𝐴𝐵𝐶 yields the (double) area of 𝐷𝐵𝐶. (Contributed by Saveliy Skresanov, 23-Sep-2017.)
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ ((𝐴 − 𝐷)𝐺(𝐵 − 𝐷)) = 0))    ⇒   ⊢ (𝜑 → (((𝐵 − 𝐶)𝐺(𝐴 − 𝐶)) − ((𝐷 − 𝐶)𝐺(𝐴 − 𝐶))) = ((𝐵 − 𝐶)𝐺(𝐷 − 𝐶)))
Theorem	cevathlem1 47703	Ceva's theorem first lemma. Multiplies three identities and divides by the common factors. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   ⊢ (𝜑 → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ ∧ 𝐹 ∈ ℂ))    &   ⊢ (𝜑 → (𝐺 ∈ ℂ ∧ 𝐻 ∈ ℂ ∧ 𝐾 ∈ ℂ))    &   ⊢ (𝜑 → (𝐴 ≠ 0 ∧ 𝐸 ≠ 0 ∧ 𝐶 ≠ 0))    &   ⊢ (𝜑 → ((𝐴 · 𝐵) = (𝐶 · 𝐷) ∧ (𝐸 · 𝐹) = (𝐴 · 𝐺) ∧ (𝐶 · 𝐻) = (𝐸 · 𝐾)))    ⇒   ⊢ (𝜑 → ((𝐵 · 𝐹) · 𝐻) = ((𝐷 · 𝐺) · 𝐾))
Theorem	cevathlem2 47704*	Ceva's theorem second lemma. Relate (doubled) areas of triangles 𝐶𝐴𝑂 and 𝐴𝐵𝑂 with of segments 𝐵𝐷 and 𝐷𝐶. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   ⊢ (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   ⊢ (𝜑 → 𝑂 ∈ ℂ)    &   ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐷 − 𝑂)) = 0 ∧ ((𝐵 − 𝑂)𝐺(𝐸 − 𝑂)) = 0 ∧ ((𝐶 − 𝑂)𝐺(𝐹 − 𝑂)) = 0))    &   ⊢ (𝜑 → (((𝐴 − 𝐹)𝐺(𝐵 − 𝐹)) = 0 ∧ ((𝐵 − 𝐷)𝐺(𝐶 − 𝐷)) = 0 ∧ ((𝐶 − 𝐸)𝐺(𝐴 − 𝐸)) = 0))    &   ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) ≠ 0 ∧ ((𝐵 − 𝑂)𝐺(𝐶 − 𝑂)) ≠ 0 ∧ ((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) ≠ 0))    ⇒   ⊢ (𝜑 → (((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) · (𝐵 − 𝐷)) = (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) · (𝐷 − 𝐶)))
Theorem	cevath 47705*	Ceva's theorem. Let 𝐴𝐵𝐶 be a triangle and let points 𝐹, 𝐷 and 𝐸 lie on sides 𝐴𝐵, 𝐵𝐶, 𝐶𝐴 correspondingly. Suppose that cevians 𝐴𝐷, 𝐵𝐸 and 𝐶𝐹 intersect at one point 𝑂. Then triangle's sides are partitioned into segments and their lengths satisfy a certain identity. Here we obtain a bit stronger version by using complex numbers themselves instead of their absolute values. The proof goes by applying cevathlem2 47704 three times and then using cevathlem1 47703 to multiply obtained identities and prove the theorem. In the theorem statement we are using function 𝐺 as a collinearity indicator. For justification of that use, see sigarcol 47700. This is Metamath 100 proof #61. (Contributed by Saveliy Skresanov, 24-Sep-2017.)
⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦)))    &   ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ))    &   ⊢ (𝜑 → (𝐹 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ))    &   ⊢ (𝜑 → 𝑂 ∈ ℂ)    &   ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐷 − 𝑂)) = 0 ∧ ((𝐵 − 𝑂)𝐺(𝐸 − 𝑂)) = 0 ∧ ((𝐶 − 𝑂)𝐺(𝐹 − 𝑂)) = 0))    &   ⊢ (𝜑 → (((𝐴 − 𝐹)𝐺(𝐵 − 𝐹)) = 0 ∧ ((𝐵 − 𝐷)𝐺(𝐶 − 𝐷)) = 0 ∧ ((𝐶 − 𝐸)𝐺(𝐴 − 𝐸)) = 0))    &   ⊢ (𝜑 → (((𝐴 − 𝑂)𝐺(𝐵 − 𝑂)) ≠ 0 ∧ ((𝐵 − 𝑂)𝐺(𝐶 − 𝑂)) ≠ 0 ∧ ((𝐶 − 𝑂)𝐺(𝐴 − 𝑂)) ≠ 0))    ⇒   ⊢ (𝜑 → (((𝐴 − 𝐹) · (𝐶 − 𝐸)) · (𝐵 − 𝐷)) = (((𝐹 − 𝐵) · (𝐸 − 𝐴)) · (𝐷 − 𝐶)))
21.46.2  Simple groups
Theorem	simpcntrab 47706	The center of a simple group is trivial or the group is abelian. (Contributed by SS, 3-Jan-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝑍 = (Cntr‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ SimpGrp)    ⇒   ⊢ (𝜑 → (𝑍 = { 0 } ∨ 𝐺 ∈ Abel))
21.47  Mathbox for Ender Ting
21.47.1  Interesting facts
Theorem	et-ltneverrefl 47707	Less-than class is never reflexive. (Contributed by Ender Ting, 22-Nov-2024.) Prefer to specify theorem domain and then apply ltnri 11347. (New usage is discouraged.)
⊢ ¬ 𝐴 < 𝐴
Theorem	et-equeucl 47708	Alternative proof that equality is left-Euclidean, using ax7 2049 directly instead of utility theorems; done for practice. (Contributed by Ender Ting, 21-Dec-2024.)
⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑥 = 𝑦))
Theorem	et-sqrtnegnre 47709	The square root of a negative number is not a real number. (Contributed by Ender Ting, 5-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → ¬ (√‘𝐴) ∈ ℝ)
Theorem	quantgodel 47710	There can be no formula asserting its own non-universality, in parallel to bj-babygodel 37312; proof path is shorter but relying on a property of specialization which provability predicates do not have. For a matching proof, see quantgodelALT 47711. (Contributed by Ender Ting, 9-May-2026.)
⊢ (𝜑 ↔ ¬ ∀𝑥𝜑)    ⇒   ⊢ ⊥
Theorem	quantgodelALT 47711	There can be no formula asserting its own non-universality; follows the steps of bj-babygodel 37312. (Contributed by Ender Ting, 7-May-2026.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 ↔ ¬ ∀𝑥𝜑)    ⇒   ⊢ ⊥
21.47.2  Increasing sequences and subsequences
Theorem	ormklocald 47712*	If elements of a certain sequence are ordered with respect to a certain relation, then its consecutive elements satisfy that relation (so-called "local monotonicity"). (Contributed by Ender Ting, 30-Apr-2025.)
⊢ (𝜑 → 𝑅 Or 𝑆)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡)))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))
Theorem	ormkglobd 47713*	If all adjacent elements of a certain sequence are ordered according to a relation which is a total order on S, then any element is so related to anything to right of it (so-called "global monotonicity"). Deduction form. (Contributed by Ender Ting, 30-Apr-2025.)
⊢ (𝜑 → 𝑅 Or 𝑆)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆)    &   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡)))
Theorem	chnsubseqword 47714	A subsequence of a chain is a word. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    ⇒   ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ Word 𝐴)
Theorem	chnsubseqwl 47715	A subsequence of a chain has the same length as its indexing sequence. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    ⇒   ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) = (♯‘𝐼))
Theorem	chnsubseq 47716	An order-preserving subsequence of an ordered chain is itself a chain. (Contributed by Ender Ting, 22-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    &   ⊢ (𝜑 → < Po 𝐴)    ⇒   ⊢ (𝜑 → (𝑊 ∘ 𝐼) ∈ ( < Chain 𝐴))
Theorem	chnsuslle 47717	Length of a subsequence is bounded by the length of original chain. (Contributed by Ender Ting, 30-Jan-2026.)
⊢ (𝜑 → 𝑊 ∈ ( < Chain 𝐴))    &   ⊢ (𝜑 → 𝐼 ∈ ( < Chain (0..^(♯‘𝑊))))    &   ⊢ (𝜑 → < Po 𝐴)    ⇒   ⊢ (𝜑 → (♯‘(𝑊 ∘ 𝐼)) ≤ (♯‘𝑊))
Theorem	chnerlem1 47718	In a chain constructed on an equivalence relation, the last element is equivalent to any. This theorem is a translation of chnub 18716 to equivalence relations. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    ⇒   ⊢ (𝜑 → (𝐶‘𝐽) ∼ (lastS‘𝐶))
Theorem	chnerlem2 47719	Lemma for chner 47721 where the I-th element comes before the J-th. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    ⇒   ⊢ ((𝜑 ∧ 𝐼 ∈ (0..^𝐽)) → (𝐶‘𝐼) ∼ (𝐶‘𝐽))
Theorem	chnerlem3 47720	Lemma for chner 47721- trichotomy of integers within the word's domain. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    &   ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶)))    ⇒   ⊢ (𝜑 → (𝐼 ∈ (0..^𝐽) ∨ 𝐽 ∈ (0..^𝐼) ∨ 𝐼 = 𝐽))
Theorem	chner 47721	Any two elements are equivalent in a chain constructed on an equivalence relation. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ ( ∼ Chain 𝐴))    &   ⊢ (𝜑 → 𝐽 ∈ (0..^(♯‘𝐶)))    &   ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝐶)))    ⇒   ⊢ (𝜑 → (𝐶‘𝐼) ∼ (𝐶‘𝐽))
Theorem	wrddin 47722	A word in two alphabets is also a word under their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶) → 𝐴 ∈ Word (𝐵 ∩ 𝐶))
Theorem	wrddrin 47723	A word whose alphabet is intersection of two classes is also a word in each of those alphabets. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ (𝐴 ∈ Word (𝐵 ∩ 𝐶) → (𝐴 ∈ Word 𝐵 ∧ 𝐴 ∈ Word 𝐶))
Theorem	wrddin2 47724	Distribution of word class constructor over class intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ Word (𝐵 ∩ 𝐶) = (Word 𝐵 ∩ Word 𝐶)
Theorem	wrddun 47725	Words in either of two alphabets are words in their union. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ Word 𝐵 ∨ 𝐴 ∈ Word 𝐶) → 𝐴 ∈ Word (𝐵 ∪ 𝐶))
Theorem	wrddun2 47726	Superadditivity of word constructor. Class of words over union alphabet includes all words over either alphabet in the union; if 𝐵 and 𝐶 are non-empty and different, then this subclass relation is strict because of the words which have symbols both from 𝐵 and from 𝐶. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ (Word 𝐵 ∪ Word 𝐶) ⊆ Word (𝐵 ∪ 𝐶)
Theorem	chndin 47727	A chain in two alphabets at once is also a chain in their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ (𝑅 Chain 𝐶)) → 𝐴 ∈ (𝑅 Chain (𝐵 ∩ 𝐶)))
Theorem	chndrin 47728	A chain whose alphabet is intersection of two classes is also a chain in each of those alphabets. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ (𝐴 ∈ (𝑅 Chain (𝐵 ∩ 𝐶)) → (𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ (𝑅 Chain 𝐶)))
Theorem	chndin2 47729	Distribution of chain class constructor over alphabet intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ (𝑅 Chain (𝐵 ∩ 𝐶)) = ((𝑅 Chain 𝐵) ∩ (𝑅 Chain 𝐶))
Theorem	chndun 47730	Chains in either of two alphabets are chains in their union. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ (𝑅 Chain 𝐶)) → 𝐴 ∈ (𝑅 Chain (𝐵 ∪ 𝐶)))
Theorem	chndun2 47731	Superaddivity of chain constructor over alphabet parameter. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝑅 Chain 𝐵) ∪ (𝑅 Chain 𝐶)) ⊆ (𝑅 Chain (𝐵 ∪ 𝐶))
Theorem	chnrin 47732	Satisfying two chain relations makes a chain under their intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵))
Theorem	chnrrin 47733	A chain of elements satisfying two relations at once is a chain under either of them. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ (𝐴 ∈ ((𝑅 ∩ < ) Chain 𝐵) → (𝐴 ∈ (𝑅 Chain 𝐵) ∧ 𝐴 ∈ ( < Chain 𝐵)))
Theorem	chnrin2 47734	Distribution of chain class constructor over relation intersection. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝑅 ∩ < ) Chain 𝐵) = ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵))
Theorem	chnrun 47735	Satisfying either of two chain relations is sufficient to make a chain under their union. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
Theorem	chnrun2 47736	Superadditivity of chain constructor over relation parameter. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ ((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
21.47.3  Scratchpad for number theory
Theorem	evenwodadd 47737	If an integer is multiplied by its sum with an odd number (thus changing its parity), the result is even. (Contributed by Ender Ting, 30-Apr-2025.)
⊢ (𝜑 → 𝐼 ∈ ℤ)    &   ⊢ (𝜑 → 𝐽 ∈ ℤ)    &   ⊢ (𝜑 → ¬ 2 ∥ 𝐽)    ⇒   ⊢ (𝜑 → 2 ∥ (𝐼 · (𝐼 + 𝐽)))
21.47.4  Scratchpad for math on real numbers
Theorem	squeezedltsq 47738	If a real value is squeezed between two others, its square is less than square of at least one of them. Deduction form. (Contributed by Ender Ting, 31-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    &   ⊢ (𝜑 → 𝐵 < 𝐶)    ⇒   ⊢ (𝜑 → ((𝐵 · 𝐵) < (𝐴 · 𝐴) ∨ (𝐵 · 𝐵) < (𝐶 · 𝐶)))
Theorem	sqrtnnaa 47739	Square root of a natural number is algebraic. (Contributed by Ender Ting, 21-Jul-2026.)
⊢ (𝐴 ∈ ℕ → (√‘𝐴) ∈ 𝔸)
Theorem	sqrtnzqaa 47740	Square root of a nonzero rational is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) → (√‘𝐴) ∈ 𝔸)
Theorem	sqrtqaa 47741	Square root of a rational number is algebraic. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ (𝐴 ∈ ℚ → (√‘𝐴) ∈ 𝔸)
Theorem	numtowerdt 47742	Certain number sets and fields form a tower. In particular, singleton 1, natural numbers, natural numbers with zero, integers, rationals, algebraic reals (notice that current definition allows algebraic numbers to be complex thus the restriction), reals and complex number sets are a tower of proper subsets. (Contributed by Ender Ting, 31-Jul-2026.)
⊢ ⟨“{1}ℕℕ0ℤℚ(𝔸 ∩ ℝ)ℝℂ”⟩ ∈ ( [⊊] Chain V)
Theorem	sin3t 47743	Triple-angle formula for sine, in pure sine form. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝐴 ∈ ℂ → (sin‘(3 · 𝐴)) = ((3 · (sin‘𝐴)) − (4 · ((sin‘𝐴)↑3))))
Theorem	cos3t 47744	Triple-angle formula for cosine, in pure cosine form. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝐴 ∈ ℂ → (cos‘(3 · 𝐴)) = ((4 · ((cos‘𝐴)↑3)) − (3 · (cos‘𝐴))))
Theorem	sin5tlem1 47745	Lemma 1 for quintupled angle sine calculation, expanding triple-angle sine times double-angle cosine. (Contributed by Ender Ting, 16-Mar-2026.)
⊢ (𝑁 ∈ ℂ → (((3 · 𝑁) − (4 · (𝑁↑3))) · (1 − (2 · (𝑁↑2)))) = (((8 · (𝑁↑5)) − (;10 · (𝑁↑3))) + (3 · 𝑁)))
Theorem	sin5tlem2 47746	Lemma 2 for quintupled angle sine calculation, multiplicating triple angle cosine by cosine straight and converting into sine. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · 𝑁) = ((4 · ((1 − (2 · (𝑀↑2))) + (𝑀↑4))) − (3 · (1 − (𝑀↑2)))))
Theorem	sin5tlem3 47747	Lemma 3 for quintupled angle sine calculation, multiplicating triple angle cosine by double angle sine. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁))) = (((4 · ((1 − (2 · (𝑀↑2))) + (𝑀↑4))) − (3 · (1 − (𝑀↑2)))) · (2 · 𝑀)))
Theorem	sin5tlem4 47748	Lemma 4 for quintupled angle sine calculation: expanding lemma 3 result to difference of polynomials. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁))) = ((((8 · (𝑀↑5)) − (;16 · (𝑀↑3))) + (8 · 𝑀)) − ((6 · 𝑀) − (6 · (𝑀↑3)))))
Theorem	sin5tlem5 47749	Lemma 5 for quintupled angle sine calculation: sine of triple-angle and double-angle sum, as a polynomial in sine straight. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ (𝑁↑2) = (1 − (𝑀↑2))) → ((((3 · 𝑀) − (4 · (𝑀↑3))) · (1 − (2 · (𝑀↑2)))) + (((4 · (𝑁↑3)) − (3 · 𝑁)) · (2 · (𝑀 · 𝑁)))) = (((;16 · (𝑀↑5)) − (;20 · (𝑀↑3))) + (5 · 𝑀)))
Theorem	sin5t 47750	Five-times-angle formula for sine, in pure sine form. (Contributed by Ender Ting, 17-Apr-2026.)
⊢ (𝐴 ∈ ℂ → (sin‘(5 · 𝐴)) = (((;16 · ((sin‘𝐴)↑5)) − (;20 · ((sin‘𝐴)↑3))) + (5 · (sin‘𝐴))))
Theorem	cos5t 47751	Five-times-angle formula for cosine, in pure cosine form. (Contributed by Ender Ting, 20-Apr-2026.)
⊢ (𝐴 ∈ ℂ → (cos‘(5 · 𝐴)) = (((;16 · ((cos‘𝐴)↑5)) − (;20 · ((cos‘𝐴)↑3))) + (5 · (cos‘𝐴))))
Theorem	cos5teq 47752	Five-times-angle formula for cosine, substitution helper. (Contributed by Ender Ting, 9-May-2026.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 = (5 · 𝐴) ∧ 𝐶 = (cos‘𝐴)) → (cos‘𝐵) = (((;16 · (𝐶↑5)) − (;20 · (𝐶↑3))) + (5 · 𝐶)))
Theorem	goldpolyfactor 47753	Factorization of a polynomial which has golden ratio among its roots, done by term-by-term-by-term multiplying and summing a few shorter polynomials. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ 𝐹 ∈ ℂ    ⇒   ⊢ (((((𝐹↑2) − 𝐹) − 1) · (((𝐹↑2) − 𝐹) − 1)) · (𝐹 + 2)) = ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2)
Theorem	goldrarr 47754	The golden ratio is a real value. (Contributed by Ender Ting, 15-Mar-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ 𝐹 ∈ ℝ
Theorem	goldrasin 47755	Alternative trigonometric formula for the golden ratio. (Contributed by Ender Ting, 15-Mar-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ 𝐹 = (2 · (sin‘(π · (3 / ;10))))
Theorem	goldrapos 47756	Golden ratio is positive. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ 0 < 𝐹
Theorem	goldrarp 47757	The golden ratio is a positive real. (Contributed by Ender Ting, 16-Apr-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ 𝐹 ∈ ℝ+
Theorem	goldracos5teq 47758	Lemma 1 for determining the value of golden ratio. (Contributed by Ender Ting, 9-May-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ (cos‘π) = (((;16 · ((𝐹 / 2)↑5)) − (;20 · ((𝐹 / 2)↑3))) + (5 · (𝐹 / 2)))
Theorem	goldratmolem2 47759	Lemma 2 for determining the value of golden ratio. (Contributed by Ender Ting, 9-May-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ -1 = ((((𝐹↑5) / 2) − (5 · ((𝐹↑3) / 2))) + (5 · (𝐹 / 2)))
Theorem	goldratmolem3 47760	Lemma 3 for determining the value of golden ratio. (Contributed by Ender Ting, 23-Jul-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ ((((𝐹↑5) − (5 · (𝐹↑3))) + (5 · 𝐹)) + 2) = 0
Theorem	goldratmolem4 47761	Lemma 4 for determining the value of golden ratio. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ (((𝐹↑2) − 𝐹) − 1) = 0
Theorem	goldratval 47762	Value of the golden ratio. (Contributed by Ender Ting, 24-Jul-2026.)
⊢ 𝐹 = (2 · (cos‘(π / 5)))    ⇒   ⊢ 𝐹 = ((1 + (√‘5)) / 2)
Theorem	lambert0 47763	A value of Lambert W (product logarithm) function at zero. (Contributed by Ender Ting, 13-Nov-2025.)
⊢ 𝑅 = ◡(𝑥 ∈ ℂ ↦ (𝑥 · (exp‘𝑥)))    ⇒   ⊢ 0𝑅0
Theorem	lamberte 47764	A value of Lambert W (product logarithm) function at e. (Contributed by Ender Ting, 13-Nov-2025.)
⊢ 𝑅 = ◡(𝑥 ∈ ℂ ↦ (𝑥 · (exp‘𝑥)))    ⇒   ⊢ e𝑅1
Theorem	cjnpoly 47765	Complex conjugation operator is not a polynomial with complex coefficients. Indeed; if it was, then multiplying 𝑥 conjugate by 𝑥 itself and adding 1 would yield a nowhere-zero non-constant polynomial, contrary to the fta 27324. (Contributed by Ender Ting, 8-Dec-2025.)
⊢ ¬ ∗ ∈ (Poly‘ℂ)
Theorem	tannpoly 47766	The tangent function is not a polynomial with complex coefficients, as it is not defined on the whole complex plane. (Contributed by Ender Ting, 10-Dec-2025.)
⊢ ¬ tan ∈ (Poly‘ℂ)
Theorem	sinnpoly 47767	Sine function is not a polynomial with complex coefficients. Indeed, it has infinitely many zeros but is not constant zero, contrary to fta1 26545. (Contributed by Ender Ting, 10-Dec-2025.)
⊢ ¬ sin ∈ (Poly‘ℂ)
Theorem	sqrtrrnpoly 47768	Real square root is not a polynomial with real coefficients, because its value is imaginary for negative arguments. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ ¬ √ ∈ (Poly‘ℝ)
Theorem	sqrtnpoly 47769	Square root function is not polynomial with complex coefficients either. Otherwise, its composition with a square monomial - the identity - would have to be of even degree. (Contributed by Ender Ting, 22-Jul-2026.)
⊢ ¬ √ ∈ (Poly‘ℂ)
21.47.5  Turing Machine Finite Reach theorem
Theorem	tmachlem-extapes 47770*	The class of all tapes is a set. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → 𝑇 ∈ V)
Theorem	tmachlem-finscan 47771*	Execution on any tape only scanned a finite number of cells. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆‘𝑎) ∈ Fin)
Theorem	tmachlem-agreeself 47772*	Any tape belongs to its own agreement set. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → 𝑎 ∈ (𝐴‘𝑎))
Theorem	tmachlem-agreeprod 47773*	Agreement set can be written as infinite product of acceptable values for tape's cells. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) = X𝑖 ∈ 𝐼 if(𝑖 ∈ (𝑆‘𝑎), {(𝑎‘𝑖)}, 𝑈))
Theorem	tmachlem-tpcomp 47774*	Product (discrete) topology of tapes is compact by Tychonoff's theorem. To work for infinite index sets (such as ℤ for 𝐼 which is the main interpretation), it requires Choice. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) ∈ Comp)
Theorem	tmachlem-tpbase 47775*	The base set of product topology of tapes is the set of tapes. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = 𝑇)
Theorem	tmachlem-tpitem 47776*	Topology lemma. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐼) → ((𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)‘𝑎) = 𝒫 𝑈)
Theorem	tmachlem-tpopen 47777*	Agreement sets are open in the product topology of tapes. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → X𝑏 ∈ 𝐼 if(𝑏 ∈ (𝑆‘𝑎), {(𝑎‘𝑏)}, 𝑈) ∈ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)))
Theorem	tmachlem-tpopen2 47778*	Variable-renaming lemma connecting tmachlem-agreeprod 47773 and tmachlem-tpopen 47777. (Contributed by Ender Ting, 27-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝐴‘𝑎) ∈ (∏t‘(𝑏 ∈ 𝐼 ↦ 𝒫 𝑈)))
Theorem	tmachlem-uassst 47779*	Union of all agreement sets only includes tapes. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑇)
Theorem	tmachlem-exlargecover 47780*	Product topology of tapes has an open cover. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∪ ran 𝐴 = 𝑇)
Theorem	tmachlem-extpcover 47781*	Product topology of tapes admits finite cover. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∃𝑎 ∈ (𝒫 ran 𝐴 ∩ Fin)∪ (∏t‘(𝑖 ∈ 𝐼 ↦ 𝒫 𝑈)) = ∪ 𝑎)
Theorem	tmachlem-exagreecover 47782*	Particular properties of the finite cover of agreesets. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∃𝑎(𝑎 ⊆ ran 𝐴 ∧ 𝑎 ∈ Fin ∧ 𝑇 = ∪ 𝑎))
Theorem	tmachlem-agreesn 47783*	Scans for all tapes of a single agreement set are identical. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑇) → (𝑆 “ (𝐴‘𝑎)) = {(𝑆‘𝑎)})
Theorem	tmachlem-agreefin 47784*	Any agreement set has a finite (singleton) list of possible scans. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ ((𝜑 ∧ 𝑏 ∈ ran 𝐴) → (𝑆 “ 𝑏) ∈ Fin)
Theorem	tmachlem-franscan 47785*	There is a finite number of different scan sets. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ran 𝑆 ∈ Fin)
Theorem	tmachlem-fssscan 47786*	Any scan set is finite. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ran 𝑆 ⊆ Fin)
Theorem	tmachfullfin 47787*	Folk theorem. For any algorithm deterministically processing a stream of data (essentially, an infinite tape with cell indices 𝐼 and finite alphabet 𝑈), if it terminates on every possible input, then it never looks beyond a finite portion ∪ ran 𝑆 of the input. Termination is expressed here with a weaker condition: that an execution may only look at a finite number of cells. Obviously, a program which finishes in a finite number of steps can only scan finite set of cells. This theorem has many corollaries, such as: any encoding scheme able to represent all integers has at least one non-decodable tape (in other terms, encoding of the infinity). I no longer have the source for this theorem but I believe I first read about it on LessWrong. My gratitude to Grok for suggesting that this theorem will require Axiom of Choice, and to DeepSeek for suggesting the topology-based proof route. (Contributed by Ender Ting, 28-Jul-2026.)
⊢ (𝜑 → 𝑈 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ V)    &   ⊢ (𝜑 → 𝑇 = (𝑈 ↑m 𝐼))    &   ⊢ (𝜑 → 𝑆:𝑇⟶(𝒫 𝐼 ∩ Fin))    &   ⊢ (𝜑 → 𝐴 = (𝑧 ∈ 𝑇 ↦ {𝑦 ∈ 𝑇 ∣ (𝑦 ↾ (𝑆‘𝑧)) = (𝑧 ↾ (𝑆‘𝑧))}))    &   ⊢ (𝜑 → ∀𝑧 ∈ 𝑇 ∀𝑦 ∈ (𝐴‘𝑧)(𝑆‘𝑦) = (𝑆‘𝑧))    ⇒   ⊢ (𝜑 → ∪ ran 𝑆 ∈ Fin)
21.48  Mathbox for Jarvin Udandy
Theorem	hirstL-ax3 47788	The third axiom of a system called "L" but proven to be a theorem since set.mm uses a different third axiom. This is named hirst after Holly P. Hirst and Jeffry L. Hirst. Axiom A3 of [Mendelson] p. 35. (Contributed by Jarvin Udandy, 7-Feb-2015.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → ¬ 𝜓) → ((¬ 𝜑 → 𝜓) → 𝜑))
Theorem	ax3h 47789	Recover ax-3 8 from hirstL-ax3 47788. (Contributed by Jarvin Udandy, 3-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((¬ 𝜑 → ¬ 𝜓) → (𝜓 → 𝜑))
Theorem	aibandbiaiffaiffb 47790	A closed form showing (a implies b and b implies a) same-as (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ (𝜑 ↔ 𝜓))
Theorem	aibandbiaiaiffb 47791	A closed form showing (a implies b and b implies a) implies (a same-as b). (Contributed by Jarvin Udandy, 3-Sep-2016.)
⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
Theorem	notatnand 47792	Do not use. Use intnanr instead. Given not a, there exists a proof for not (a and b). (Contributed by Jarvin Udandy, 31-Aug-2016.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ (𝜑 ∧ 𝜓)
Theorem	aistia 47793	Given a is equivalent to ⊤, there exists a proof for a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
⊢ (𝜑 ↔ ⊤)    ⇒   ⊢ 𝜑
Theorem	aisfina 47794	Given a is equivalent to ⊥, there exists a proof for not a. (Contributed by Jarvin Udandy, 30-Aug-2016.)
⊢ (𝜑 ↔ ⊥)    ⇒   ⊢ ¬ 𝜑
Theorem	bothtbothsame 47795	Given both a, b are equivalent to ⊤, there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
⊢ (𝜑 ↔ ⊤)    &   ⊢ (𝜓 ↔ ⊤)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	bothfbothsame 47796	Given both a, b are equivalent to ⊥, there exists a proof for a is the same as b. (Contributed by Jarvin Udandy, 31-Aug-2016.)
⊢ (𝜑 ↔ ⊥)    &   ⊢ (𝜓 ↔ ⊥)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	aiffbbtat 47797	Given a is equivalent to b, b is equivalent to ⊤ there exists a proof for a is equivalent to T. (Contributed by Jarvin Udandy, 29-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ ⊤)    ⇒   ⊢ (𝜑 ↔ ⊤)
Theorem	aisbbisfaisf 47798	Given a is equivalent to b, b is equivalent to ⊥ there exists a proof for a is equivalent to F. (Contributed by Jarvin Udandy, 30-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ ⊥)    ⇒   ⊢ (𝜑 ↔ ⊥)
Theorem	axorbtnotaiffb 47799	Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)); df-xor 1542 is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)
⊢ (𝜑 ⊻ 𝜓)    ⇒   ⊢ ¬ (𝜑 ↔ 𝜓)
Theorem	aiffnbandciffatnotciffb 47800	Given a is equivalent to (not b), c is equivalent to a, there exists a proof for ( not ( c iff b ) ). (Contributed by Jarvin Udandy, 7-Sep-2016.)
⊢ (𝜑 ↔ ¬ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    ⇒   ⊢ ¬ (𝜒 ↔ 𝜓)