P. 429 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ovmpogad 42801*	Value of an operation given by a maps-to rule. Deduction form of ovmpoga 7539. (Contributed by SN, 14-Mar-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)
Theorem	ofun 42802	A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)
⊢ (𝜑 → 𝐴 Fn 𝑀)    &   ⊢ (𝜑 → 𝐵 Fn 𝑀)    &   ⊢ (𝜑 → 𝐶 Fn 𝑁)    &   ⊢ (𝜑 → 𝐷 Fn 𝑁)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑁 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑀 ∩ 𝑁) = ∅)    ⇒   ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)))
Theorem	dfqs3 42803*	Alternate definition of quotient set. (Contributed by Steven Nguyen, 7-Jun-2023.)
⊢ (𝐴 / 𝑅) = ∪ 𝑥 ∈ 𝐴 {[𝑥]𝑅}
Theorem	qseq12d 42804	Equality theorem for quotient set, deduction form. (Contributed by Steven Nguyen, 30-Apr-2023.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 / 𝐶) = (𝐵 / 𝐷))
Theorem	qsalrel 42805*	The quotient set is equal to the singleton of 𝐴 when all elements are related and 𝐴 is nonempty. (Contributed by SN, 8-Jun-2023.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∼ 𝑦)    &   ⊢ (𝜑 → ∼ Er 𝐴)    &   ⊢ (𝜑 → 𝑁 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐴 / ∼ ) = {𝐴})
Theorem	supinf 42806*	The supremum is the infimum of the upper bounds. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → < Or 𝐴)    &   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐵 𝑦 < 𝑧)))    ⇒   ⊢ (𝜑 → sup(𝐵, 𝐴, < ) = inf({𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐵 ¬ 𝑥 < 𝑤}, 𝐴, < ))
Theorem	mapcod 42807	Compose two mappings. (Contributed by SN, 11-Mar-2025.)
⊢ (𝜑 → 𝐹 ∈ (𝐴 ↑m 𝐵))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐵 ↑m 𝐶))    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (𝐴 ↑m 𝐶))
Theorem	fisdomnn 42808	A finite set is dominated by the set of natural numbers. (Contributed by SN, 6-Jul-2025.)
⊢ (𝐴 ∈ Fin → 𝐴 ≺ ℕ)
Theorem	ltex 42809	The less-than relation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ < ∈ V
Theorem	leex 42810	The less-than-or-equal-to relation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ ≤ ∈ V
Theorem	subex 42811	The subtraction operation is a set. (Contributed by SN, 5-Jun-2025.)
⊢ − ∈ V
Theorem	absex 42812	The absolute value function is a set. (Contributed by SN, 5-Jun-2025.)
⊢ abs ∈ V
Theorem	cjex 42813	The conjugate function is a set. (Contributed by SN, 5-Jun-2025.)
⊢ ∗ ∈ V
Theorem	fzosumm1 42814*	Separate out the last term in a finite sum. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘𝑀))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝐴 ∈ ℂ)    &   ⊢ (𝑘 = (𝑁 − 1) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ℤ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝐴 = (Σ𝑘 ∈ (𝑀..^(𝑁 − 1))𝐴 + 𝐵))
Theorem	ccatcan2d 42815	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 11133 is used in mulrid 11169 related theorems, so one could trade off the extra axioms in mulrid 11169 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 11120; in the other direction, real number closure laws can be avoided by using ax-resscn 11120 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 11128, ax-1cn 11121, and ax-addcl 11123. (And in practice, the expression isn't fully expanded into ones.) Multiplication by 1 requires either mullidi 11177 or (ax-1rid 11133 and 1re 11171) as seen in 1t1e1 12369 and 1t1e1ALT 12258. Multiplying with greater natural numbers uses ax-distr 11130. 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 11127 (see readdrid 42967 and readdlid 42960).
Theorem	c0exALT 42816	Alternate proof of c0ex 11163 using more set theory axioms but fewer complex number axioms (add ax-10 2169, ax-11 2185, ax-13 2397, ax-nul 5250, and remove ax-1cn 11121, ax-icn 11122, ax-addcl 11123, and ax-mulcl 11125). (Contributed by Steven Nguyen, 4-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ V
Theorem	0cnALT3 42817	Alternate proof of 0cn 11161 using ax-resscn 11120, ax-addrcl 11124, ax-rnegex 11134, ax-cnre 11136 instead of ax-icn 11122, ax-addcl 11123, ax-mulcl 11125, ax-i2m1 11131. Version of 0cnALT 11408 using ax-1cn 11121 instead of ax-icn 11122. (Contributed by Steven Nguyen, 7-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 0 ∈ ℂ
Theorem	elre0re 42818	Specialized version of 0red 11174 without using ax-1cn 11121 and ax-cnre 11136. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝐴 ∈ ℝ → 0 ∈ ℝ)
Theorem	lttrii 42819	'Less than' is transitive. (Contributed by SN, 26-Aug-2025.)
⊢ 𝐴 ∈ ℝ    &   ⊢ 𝐵 ∈ ℝ    &   ⊢ 𝐶 ∈ ℝ    &   ⊢ 𝐴 < 𝐵    &   ⊢ 𝐵 < 𝐶    ⇒   ⊢ 𝐴 < 𝐶
Theorem	remulcan2d 42820	mulcan2d 11811 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 42821	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 11347, 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 42822	1 + 3 = 4. (Contributed by SN, 19-Nov-2025.)
⊢ (1 + 3) = 4
Theorem	5ne0 42823	The number 5 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 5 ≠ 0
Theorem	6ne0 42824	The number 6 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 6 ≠ 0
Theorem	7ne0 42825	The number 7 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 7 ≠ 0
Theorem	8ne0 42826	The number 8 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 8 ≠ 0
Theorem	9ne0 42827	The number 9 is nonzero. (Contributed by SN, 22-Oct-2025.)
⊢ 9 ≠ 0
Theorem	sn-1ne2 42828	A proof of 1ne2 12418 without using ax-mulcom 11127, ax-mulass 11129, ax-pre-mulgt0 11140. Based on mul02lem2 11350. (Contributed by SN, 13-Dec-2023.)
⊢ 1 ≠ 2
Theorem	nnn1suc 42829*	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 42830	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 42831	Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11587. 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 42832	Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11588 with a commuted consequent, and of mvrladdd 11590 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 33734. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐶 − 𝐴) = 𝐵)
Theorem	raddswap12d 42833	Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11589 with a commuted consequent, and of mvlraddd 11587 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 42834	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 42835	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 42836	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 42837	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 42838	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 42839	Lemma for dffltz 43164. (Contributed by Steven Nguyen, 27-Feb-2023.)
⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → ¬ 0 < 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ ℤ)    ⇒   ⊢ (𝜑 → -𝐴 ∈ ℕ)
Theorem	sqmid3api 42840	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 42841	Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
⊢ 𝐴 ∈ ℕ0    &   ⊢ 𝐵 ∈ ℕ0    &   ⊢ 𝐶 ∈ ℕ0    ⇒   ⊢ (;𝐴𝐵 + 𝐶) = (;𝐴𝐶 + 𝐵)
Theorem	sqn5i 42842	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 42843	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 42844	Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11628. (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 42845	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 42846	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 42847	Variant of sqdeccom12 42846 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 42848	4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
⊢ (4 · 5) = ;20
Theorem	3rdpwhole 42849	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 42850	The square of 4 is 16. (Contributed by SN, 26-Aug-2025.)
⊢ (4↑2) = ;16
Theorem	sq5 42851	The square of 5 is 25. (Contributed by SN, 26-Aug-2025.)
⊢ (5↑2) = ;25
Theorem	sq6 42852	The square of 6 is 36. (Contributed by SN, 26-Aug-2025.)
⊢ (6↑2) = ;36
Theorem	sq7 42853	The square of 7 is 49. (Contributed by SN, 26-Aug-2025.)
⊢ (7↑2) = ;49
Theorem	sq8 42854	The square of 8 is 64. (Contributed by SN, 26-Aug-2025.)
⊢ (8↑2) = ;64
Theorem	sq9 42855	The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
⊢ (9↑2) = ;81
Theorem	rpsscn 42856	The positive reals are a subset of the complex numbers. (Contributed by SN, 1-Oct-2025.)
⊢ ℝ+ ⊆ ℂ
Theorem	4rp 42857	4 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 4 ∈ ℝ+
Theorem	6rp 42858	6 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 6 ∈ ℝ+
Theorem	7rp 42859	7 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 7 ∈ ℝ+
Theorem	8rp 42860	8 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 8 ∈ ℝ+
Theorem	9rp 42861	9 is a positive real. (Contributed by SN, 26-Aug-2025.)
⊢ 9 ∈ ℝ+
Theorem	235t711 42862	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 11181 saving the lower level uses of mulcomli 11181 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12792 are added then this proof would benefit more than ex-decpmul 42863. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12342 or 8t7e56 12803. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)
⊢ (;;235 · ;;711) = ;;;;;167085
Theorem	ex-decpmul 42863	Example usage of decpmul 42845. This proof is significantly longer than 235t711 42862. There is more unnecessary carrying compared to 235t711 42862. Although saving 5 visual steps, using mulcomli 11181 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 42864	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 42865	Shorter proof of eluzp1l 12856. (Contributed by NM, 12-Sep-2005.) (Revised by SN, 5-Jul-2025.)
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑁)
Theorem	fz1sumconst 42866*	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 42867*	Add one more term to a sum. Special case of fsump1 15759 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 42868*	The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15866. (Contributed by SN, 21-Mar-2025.)
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2))
Theorem	nicomachus 42869*	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 42870*	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 42871	i is not 1. (Contributed by SN, 25-Apr-2025.)
⊢ i ≠ 1
Theorem	0tie0 42872	0 times i equals 0. (Contributed by SN, 25-Apr-2025.)
⊢ (0 · i) = 0
Theorem	it1ei 42873	i times 1 equals i. (Contributed by SN, 25-Apr-2025.)
⊢ (i · 1) = i
Theorem	1tiei 42874	1 times i equals i. (Contributed by SN, 25-Apr-2025.)
⊢ (1 · i) = i
Theorem	itrere 42875	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 42876	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 42877	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 42878	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 42879	Lemma for dffltz 43164. For a more standard version, see expgt0b 32962. TODO-SN?: This can be used to show exp11d 42883 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 42880	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 42881	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 42882	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 42883	exp11nnd 14264 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 42884	0 divides 0. (Contributed by SN, 15-Sep-2024.)
⊢ 0 ∥ 0
Theorem	absdvdsabsb 42885	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 42886	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 42887	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 42888	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 42889	Deduction associated with dvdsexpim 16565. (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 42890	dvdssqlem 16576 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 42891	dvdsexpnn 42890 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 42892	dvdssq 16577 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 42893	When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16769 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 42894*	Two ways to express "𝑁 is an integer multiple of 𝑀". Originally a subproof of zdiv 12633. (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 42895	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 42896*	General condition for the exponential function to be one-to-one. efper 26514 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 42897	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 42898*	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 42899*	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 42900*	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))))