P. 428 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42701-42800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ccatcan2d 42701	Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ Word 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ Word 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ Word 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 ++ 𝐶) = (𝐵 ++ 𝐶) ↔ 𝐴 = 𝐵))
21.30.2  Arithmetic theorems Towards the start of this section are several proofs regarding the different complex number axioms that could be used to prove some results. For example, ax-1rid 11097 is used in mulrid 11131 related theorems, so one could trade off the extra axioms in mulrid 11131 for the axioms needed to prove that something is a real number. Another example is avoiding complex number closure laws by using real number closure laws and then using ax-resscn 11084; in the other direction, real number closure laws can be avoided by using ax-resscn 11084 and then the complex number closure laws. (This only works if the result of (𝐴 + 𝐵) only needs to be a complex number). The natural numbers are especially amenable to axiom reductions, as the set ℕ is the recursive set {1, (1 + 1), ((1 + 1) + 1)}, etc., i.e. the set of numbers formed by only additions of 1. The digits 2 through 9 are defined so that they expand into additions of 1. This conveniently allows for adding natural numbers by rearranging parentheses, as shown below: (4 + 3) = 7 ((3 + 1) + (2 + 1)) = (6 + 1) ((((1 + 1) + 1) + 1) + ((1 + 1) + 1)) = ((((((1 + 1) + 1) + 1) + 1) + 1) + 1) This only requires ax-addass 11092, ax-1cn 11085, and ax-addcl 11087. (And in practice, the expression isn't fully expanded into ones.) Multiplication by 1 requires either mullidi 11139 or (ax-1rid 11097 and 1re 11133) as seen in 1t1e1 12327 and 1t1e1ALT 12221. Multiplying with greater natural numbers uses ax-distr 11094. Still, this takes fewer axioms than adding zero, which is often implicit in theorems such as (9 + 1) = ;10. Adding zero uses almost every complex number axiom, though notably not ax-mulcom 11091 (see readdrid 42853 and readdlid 42846).
Theorem	c0exALT 42702	Alternate proof of c0ex 11127 using more set theory axioms but fewer complex number axioms (add ax-10 2147, ax-11 2163, ax-13 2377, ax-nul 5241, and remove ax-1cn 11085, ax-icn 11086, ax-addcl 11087, and ax-mulcl 11089). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ V
Theorem	0cnALT3 42703	Alternate proof of 0cn 11125 using ax-resscn 11084, ax-addrcl 11088, ax-rnegex 11098, ax-cnre 11100 instead of ax-icn 11086, ax-addcl 11087, ax-mulcl 11089, ax-i2m1 11095. Version of 0cnALT 11370 using ax-1cn 11085 instead of ax-icn 11086. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	elre0re 42704	Specialized version of 0red 11136 without using ax-1cn 11085 and ax-cnre 11100. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)
Theorem	lttrii 42705	'Less than' is transitive. (Contributed by SN, 26-Aug-2025.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ 𝐴 < 𝐶
Theorem	remulcan2d 42706	mulcan2d 11773 for real numbers using fewer axioms. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐶) = (𝐵 · 𝐶) ↔ 𝐴 = 𝐵))
Theorem	readdridaddlidd 42707	Given some real number 𝐵 where 𝐴 acts like a right additive identity, derive that 𝐴 is a left additive identity. Note that the hypothesis is weaker than proving that 𝐴 is a right additive identity (for all numbers). Although, if there is a right additive identity, then by readdcan 11309, 𝐴 is the right additive identity. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐵 + 𝐴) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝐶 ∈ ℝ) → (𝐴 + 𝐶) = 𝐶)
Theorem	1p3e4 42708	1 + 3 = 4. (Contributed by SN, 19-Nov-2025.)
⊢ (1 + 3) = 4
Theorem	5ne0 42709	The number 5 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 5 ≠ 0
Theorem	6ne0 42710	The number 6 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 6 ≠ 0
Theorem	7ne0 42711	The number 7 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 7 ≠ 0
Theorem	8ne0 42712	The number 8 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 8 ≠ 0
Theorem	9ne0 42713	The number 9 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 9 ≠ 0
Theorem	sn-1ne2 42714	A proof of 1ne2 12373 without using ax-mulcom 11091, ax-mulass 11093, ax-pre-mulgt0 11104. Based on mul02lem2 11312. (Contributed by SN, 13-Dec-2023.)
⊢ 1 ≠ 2
Theorem	nnn1suc 42715*	A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)
Theorem	readdrcl2d 42716	Reverse closure for addition: the second addend is real if the first addend is real and the sum is real. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ)    ⇒   ⊢ (𝜑 → 𝐵 ∈ ℝ)
Theorem	mvrrsubd 42717	Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11549. EDITORIAL: Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → (𝐴 + 𝐶) = 𝐵)
Theorem	laddrotrd 42718	Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11550 with a commuted consequent, and of mvrladdd 11552 with a commuted hypothesis. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: ply1dg3rt0irred 33664. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐴) = 𝐵)
Theorem	raddswap12d 42719	Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11551 with a commuted consequent, and of mvlraddd 11549 with a commuted hypothesis. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 + 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐴 − 𝐶))
Theorem	lsubrotld 42720	Rotate the variables left in an equation with subtraction on the left, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐶) = 𝐴)
Theorem	rsubrotld 42721	Rotate the variables left in an equation with subtraction on the right, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 4-Jul-2025.)
⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 = (𝐵 − 𝐶))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐶 + 𝐴))
Theorem	lsubswap23d 42722	Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 23-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 − 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 − 𝐶) = 𝐵)
Theorem	addsubeq4com 42723	Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.)
⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴 − 𝐶) = (𝐷 − 𝐵)))
Theorem	sqsumi 42724	A sum squared. (Contributed by Steven Nguyen, 16-Sep-2022.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + (𝐵 · 𝐵)) + (2 · (𝐴 · 𝐵)))
Theorem	negn0nposznnd 42725	Lemma for dffltz 43078. (Contributed by Steven Nguyen, 27-Feb-2023.)
⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ¬ 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℕ)
Theorem	sqmid3api 42726	Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝑁 ∈ ℂ    &   ⊢ (𝐴 + 𝑁) = 𝐵    &   ⊢ (𝐵 + 𝑁) = 𝐶    ⇒   ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁))
Theorem	decaddcom 42727	Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    ⇒   ⊢ (;𝐴𝐵 + 𝐶) = (;𝐴𝐶 + 𝐵)
Theorem	sqn5i 42728	The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
⊢ 𝐴 ∈ ℕ0    ⇒   ⊢ (;𝐴5 · ;𝐴5) = ;;(𝐴 · (𝐴 + 1))25
Theorem	sqn5ii 42729	The square of a number ending in 5. This shortcut only works because 5 is half of 10. (Contributed by Steven Nguyen, 16-Sep-2022.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ (𝐴 + 1) = 𝐵    &   ⊢ (𝐴 · 𝐵) = 𝐶    ⇒   ⊢ (;𝐴5 · ;𝐴5) = ;;𝐶25
Theorem	decpmulnc 42730	Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11590. (Contributed by Steven Nguyen, 9-Dec-2022.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ (𝐴 · 𝐶) = 𝐸    &   ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   ⊢ (𝐵 · 𝐷) = 𝐺    ⇒   ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;;𝐸𝐹𝐺
Theorem	decpmul 42731	Partial products algorithm for two digit multiplication. (Contributed by Steven Nguyen, 10-Dec-2022.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ 𝐷 ∈ ℕ0    &   ⊢ (𝐴 · 𝐶) = 𝐸    &   ⊢ ((𝐴 · 𝐷) + (𝐵 · 𝐶)) = 𝐹    &   ⊢ (𝐵 · 𝐷) = ;𝐺𝐻    &   ⊢ (;𝐸𝐺 + 𝐹) = 𝐼    &   ⊢ 𝐺 ∈ ℕ0    &   ⊢ 𝐻 ∈ ℕ0    ⇒   ⊢ (;𝐴𝐵 · ;𝐶𝐷) = ;𝐼𝐻
Theorem	sqdeccom12 42732	The square of a number in terms of its digits switched. (Contributed by Steven Nguyen, 3-Jan-2023.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    ⇒   ⊢ ((;𝐴𝐵 · ;𝐴𝐵) − (;𝐵𝐴 · ;𝐵𝐴)) = (;99 · ((𝐴 · 𝐴) − (𝐵 · 𝐵)))
Theorem	sq3deccom12 42733	Variant of sqdeccom12 42732 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    &   ⊢ (𝐴 + 𝐶) = 𝐷    ⇒   ⊢ ((;;𝐴𝐵𝐶 · ;;𝐴𝐵𝐶) − (;𝐷𝐵 · ;𝐷𝐵)) = (;99 · ((;𝐴𝐵 · ;𝐴𝐵) − (𝐶 · 𝐶)))
Theorem	4t5e20 42734	4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
⊢ (4 · 5) = ;20
Theorem	3rdpwhole 42735	A third of a number plus the number is four thirds of the number. (Contributed by SN, 19-Nov-2025.)
⊢ (𝐴 ∈ ℂ → ((𝐴 / 3) + 𝐴) = (4 · (𝐴 / 3)))
Theorem	sq4 42736	The square of 4 is 16. (Contributed by SN, 26-Aug-2025.)
⊢ (4↑2) = ;16
Theorem	sq5 42737	The square of 5 is 25. (Contributed by SN, 26-Aug-2025.)
⊢ (5↑2) = ;25
Theorem	sq6 42738	The square of 6 is 36. (Contributed by SN, 26-Aug-2025.)
⊢ (6↑2) = ;36
Theorem	sq7 42739	The square of 7 is 49. (Contributed by SN, 26-Aug-2025.)
⊢ (7↑2) = ;49
Theorem	sq8 42740	The square of 8 is 64. (Contributed by SN, 26-Aug-2025.)
⊢ (8↑2) = ;64
Theorem	sq9 42741	The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
⊢ (9↑2) = ;81
Theorem	rpsscn 42742	The positive reals are a subset of the complex numbers. (Contributed by SN, 1-Oct-2025.)
⊢ ℝ+ ⊆ ℂ
Theorem	4rp 42743	4 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 4 ∈ ℝ+
Theorem	6rp 42744	6 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 6 ∈ ℝ+
Theorem	7rp 42745	7 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 7 ∈ ℝ+
Theorem	8rp 42746	8 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 8 ∈ ℝ+
Theorem	9rp 42747	9 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 9 ∈ ℝ+
Theorem	235t711 42748	Calculate a product by long multiplication as a base comparison with other multiplication algorithms. Conveniently, 711 has two ones which greatly simplifies calculations like 235 · 1. There isn't a higher level mulcomli 11143 saving the lower level uses of mulcomli 11143 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12742 are added then this proof would benefit more than ex-decpmul 42749. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12300 or 8t7e56 12753. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)
⊢ (;;235 · ;;711) = ;;;;;167085
Theorem	ex-decpmul 42749	Example usage of decpmul 42731. This proof is significantly longer than 235t711 42748. There is more unnecessary carrying compared to 235t711 42748. Although saving 5 visual steps, using mulcomli 11143 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (;;235 · ;;711) = ;;;;;167085
Theorem	eluzp1 42750	Membership in a successor upper set of integers. (Contributed by SN, 5-Jul-2025.)
⊢ (𝑀 ∈ ℤ → (𝑁 ∈ (ℤ≥‘(𝑀 + 1)) ↔ (𝑁 ∈ ℤ ∧ 𝑀 < 𝑁)))
Theorem	sn-eluzp1l 42751	Shorter proof of eluzp1l 12804. (Contributed by NM, 12-Sep-2005.) (Revised by SN, 5-Jul-2025.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁)
Theorem	fz1sumconst 42752*	The sum of 𝑁 constant terms (𝑘 is not free in 𝐶). (Contributed by SN, 21-Mar-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)𝐶 = (𝑁 · 𝐶))
Theorem	fz1sump1 42753*	Add one more term to a sum. Special case of fsump1 15707 generalized to 𝑁 ∈ ℕ0. (Contributed by SN, 22-Mar-2025.)
⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵))
Theorem	oddnumth 42754*	The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15814. (Contributed by SN, 21-Mar-2025.)
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2))
Theorem	nicomachus 42755*	Nicomachus's Theorem. The sum of the odd numbers from 𝑁↑2 − 𝑁 + 1 to 𝑁↑2 + 𝑁 − 1 is 𝑁↑3. Proof 2 from https://proofwiki.org/wiki/Nicomachus%27s_Theorem. (Contributed by SN, 21-Mar-2025.)
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(((𝑁↑2) − 𝑁) + ((2 · 𝑘) − 1)) = (𝑁↑3))
Theorem	sumcubes 42756*	The sum of the first 𝑁 perfect cubes is the sum of the first 𝑁 nonnegative integers, squared. This is the Proof by Nicomachus from https://proofwiki.org/wiki/Sum_of_Sequence_of_Cubes using induction and index shifting to collect all the odd numbers. (Contributed by SN, 22-Mar-2025.)
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)(𝑘↑3) = (Σ𝑘 ∈ (1...𝑁)𝑘↑2))
Theorem	ine1 42757	i is not 1. (Contributed by SN, 25-Apr-2025.)
⊢ i ≠ 1
Theorem	0tie0 42758	0 times i equals 0. (Contributed by SN, 25-Apr-2025.)
⊢ (0 · i) = 0
Theorem	it1ei 42759	i times 1 equals i. (Contributed by SN, 25-Apr-2025.)
⊢ (i · 1) = i
Theorem	1tiei 42760	1 times i equals i. (Contributed by SN, 25-Apr-2025.)
⊢ (1 · i) = i
Theorem	itrere 42761	i times a real is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
⊢ (𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
Theorem	retire 42762	A real times i is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
Theorem	iocioodisjd 42763	Adjacent intervals where the lower interval is right-closed and the upper interval is open are disjoint. (Contributed by SN, 1-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)
Theorem	rpabsid 42764	A positive real is its own absolute value. (Contributed by SN, 1-Oct-2025.)
⊢ (𝑅 ∈ ℝ+ → (abs‘𝑅) = 𝑅)
21.30.3  Exponents and divisibility
Theorem	oexpreposd 42765	Lemma for dffltz 43078. For a more standard version, see expgt0b 32910. TODO-SN?: This can be used to show exp11d 42769 holds for all integers when the exponent is odd. (Contributed by SN, 4-Mar-2023.)
⊢ (𝜑 → 𝑁 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)    ⇒   ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))
Theorem	explt1d 42766	A nonnegative real number less than one raised to a positive integer is less than one. (Contributed by SN, 3-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐴 < 1)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) < 1)
Theorem	expeq1d 42767	A nonnegative real number is one if and only if it is one when raised to a positive integer. (Contributed by SN, 3-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑁) = 1 ↔ 𝐴 = 1))
Theorem	expeqidd 42768	A nonnegative real number is zero or one if and only if it is itself when raised to an integer greater than one. (Contributed by SN, 3-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2))    &   ⊢ (𝜑 → 0 ≤ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑁) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))
Theorem	exp11d 42769	exp11nnd 14212 for nonzero integer exponents. (Contributed by SN, 14-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ≠ 0)    &   ⊢ (𝜑 → (𝐴↑𝑁) = (𝐵↑𝑁))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	0dvds0 42770	0 divides 0. (Contributed by SN, 15-Sep-2024.)
⊢ 0 ∥ 0
Theorem	absdvdsabsb 42771	Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ 𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
Theorem	gcdnn0id 42772	The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
⊢ (𝑁 ∈ ℕ0 → (𝑁 gcd 𝑁) = 𝑁)
Theorem	gcdle1d 42773	The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑀)
Theorem	gcdle2d 42774	The greatest common divisor of a positive integer and another integer is less than or equal to the positive integer. (Contributed by SN, 25-Aug-2024.)
⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑁)
Theorem	dvdsexpad 42775	Deduction associated with dvdsexpim 16513. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))
Theorem	dvdsexpnn 42776	dvdssqlem 16524 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
Theorem	dvdsexpnn0 42777	dvdsexpnn 42776 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
Theorem	dvdsexpb 42778	dvdssq 16525 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∥ 𝐵 ↔ (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
Theorem	posqsqznn 42779	When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16717 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
⊢ (𝜑 → (𝐴↑2) ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℚ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 ∈ ℕ)
Theorem	zdivgd 42780*	Two ways to express "𝑁 is an integer multiple of 𝑀". Originally a subproof of zdiv 12588. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝑀 ∈ ℂ)    &   ⊢ (𝜑 → 𝑁 ∈ ℂ)    &   ⊢ (𝜑 → 𝑀 ≠ 0)    ⇒   ⊢ (𝜑 → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
Theorem	efsubd 42781	Difference of exponents law for exponential function, deduction form. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (exp‘(𝐴 − 𝐵)) = ((exp‘𝐴) / (exp‘𝐵)))
Theorem	ef11d 42782*	General condition for the exponential function to be one-to-one. efper 26459 shows that exponentiation is periodic. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((exp‘𝐴) = (exp‘𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + ((i · (2 · π)) · 𝑛))))
Theorem	logccne0d 42783	The logarithm isn't 0 if its argument isn't 0 or 1, deduction form. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐴 ≠ 1)    ⇒   ⊢ (𝜑 → (log‘𝐴) ≠ 0)
Theorem	cxp112d 42784*	General condition for complex exponentiation to be one-to-one with respect to the second argument. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    &   ⊢ (𝜑 → 𝐶 ≠ 1)    ⇒   ⊢ (𝜑 → ((𝐶↑𝑐𝐴) = (𝐶↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘𝐶)))))
Theorem	cxp111d 42785*	General condition for complex exponentiation to be one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶) ↔ ∃𝑛 ∈ ℤ (log‘𝐴) = ((log‘𝐵) + (((i · (2 · π)) · 𝑛) / 𝐶))))
Theorem	cxpi11d 42786*	i to the powers of 𝐴 and 𝐵 are equal iff 𝐴 and 𝐵 are a multiple of 4 apart. EDITORIAL: This theorem may be revised to a more convenient form. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛))))
Theorem	logne0d 42787	Deduction form of logne0 26559. See logccne0d 42783 for a more general version. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ≠ 1)    ⇒   ⊢ (𝜑 → (log‘𝐴) ≠ 0)
Theorem	rxp112d 42788	Real exponentiation is one-to-one with respect to the second argument. (TODO: Note that the base 𝐶 must be positive since -𝐶↑𝐴 is 𝐶↑𝐴 · e↑iπ𝐴, so in the negative case 𝐴 = 𝐵 + 2𝑘). (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐶 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 1)    &   ⊢ (𝜑 → (𝐶↑𝑐𝐴) = (𝐶↑𝑐𝐵))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	log11d 42789	The natural logarithm is one-to-one. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	rplog11d 42790	The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    ⇒   ⊢ (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
Theorem	rxp11d 42791	Real exponentiation is one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    &   ⊢ (𝜑 → (𝐴↑𝑐𝐶) = (𝐵↑𝑐𝐶))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
21.30.4  Trigonometry and Calculus
Theorem	tanhalfpim 42792	The tangent of π / 2 minus a number is the cotangent, here represented by cos𝐴 / sin𝐴. (Contributed by SN, 2-Sep-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (sin‘𝐴) ≠ 0)    ⇒   ⊢ (𝜑 → (tan‘((π / 2) − 𝐴)) = ((cos‘𝐴) / (sin‘𝐴)))
Theorem	sinpim 42793	Sine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
⊢ (𝐴 ∈ ℂ → (sin‘(π − 𝐴)) = (sin‘𝐴))
Theorem	cospim 42794	Cosine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
⊢ (𝐴 ∈ ℂ → (cos‘(π − 𝐴)) = -(cos‘𝐴))
Theorem	tan3rdpi 42795	The tangent of π / 3 is √3. (Contributed by SN, 2-Sep-2025.)
⊢ (tan‘(π / 3)) = (√‘3)
Theorem	sin2t3rdpi 42796	The sine of 2 · (π / 3) is (√3) / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (sin‘(2 · (π / 3))) = ((√‘3) / 2)
Theorem	cos2t3rdpi 42797	The cosine of 2 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (cos‘(2 · (π / 3))) = -(1 / 2)
Theorem	sin4t3rdpi 42798	The sine of 4 · (π / 3) is -(√3) / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (sin‘(4 · (π / 3))) = -((√‘3) / 2)
Theorem	cos4t3rdpi 42799	The cosine of 4 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (cos‘(4 · (π / 3))) = -(1 / 2)
Theorem	asin1half 42800	The arcsine of 1 / 2 is π / 6. (Contributed by SN, 31-Aug-2025.)
⊢ (arcsin‘(1 / 2)) = (π / 6)