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Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnn1suc 42301* 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)
 
Theoremnnadd1com 42302 Addition with 1 is commutative for natural numbers. (Contributed by Steven Nguyen, 9-Dec-2022.)
(𝐴 ∈ ℕ → (𝐴 + 1) = (1 + 𝐴))
 
Theoremnnaddcom 42303 Addition is commutative for natural numbers. Uses fewer axioms than addcom 11447. (Contributed by Steven Nguyen, 9-Dec-2022.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
 
Theoremnnaddcomli 42304 Version of addcomli 11453 for natural numbers. (Contributed by Steven Nguyen, 1-Aug-2023.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   (𝐴 + 𝐵) = 𝐶       (𝐵 + 𝐴) = 𝐶
 
Theoremnnadddir 42305 Right-distributivity for natural numbers without ax-mulcom 11219. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
 
Theoremnnmul1com 42306 Multiplication with 1 is commutative for natural numbers, without ax-mulcom 11219. Since (𝐴 · 1) is 𝐴 by ax-1rid 11225, this is equivalent to remullid 42463 for natural numbers, but using fewer axioms (avoiding ax-resscn 11212, ax-addass 11220, ax-mulass 11221, ax-rnegex 11226, ax-pre-lttri 11229, ax-pre-lttrn 11230, ax-pre-ltadd 11231). (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℕ → (1 · 𝐴) = (𝐴 · 1))
 
Theoremnnmulcom 42307 Multiplication is commutative for natural numbers. (Contributed by SN, 5-Feb-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 
Theoremreaddrcl2d 42308 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) ∈ ℝ)       (𝜑𝐵 ∈ ℝ)
 
Theoremmvrrsubd 42309 Move a subtraction in the RHS to a right-addition in the LHS. Converse of mvlraddd 11673.

EDITORIAL: Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 21-Aug-2024.)

(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵𝐶))       (𝜑 → (𝐴 + 𝐶) = 𝐵)
 
Theoremladdrotrd 42310 Rotate the variables right in an equation with addition on the left, converting it into a subtraction. Version of mvlladdd 11674 with a commuted consequent, and of mvrladdd 11676 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 33607. (Contributed by SN, 21-Aug-2024.)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑 → (𝐶𝐴) = 𝐵)
 
Theoremraddswap12d 42311 Swap the first two variables in an equation with addition on the right, converting it into a subtraction. Version of mvrraddd 11675 with a commuted consequent, and of mvlraddd 11673 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.)

(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵 + 𝐶))       (𝜑𝐵 = (𝐴𝐶))
 
Theoremlsubrotld 42312 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.)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐵 + 𝐶) = 𝐴)
 
Theoremrsubrotld 42313 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.)

(𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 = (𝐵𝐶))       (𝜑𝐵 = (𝐶 + 𝐴))
 
Theoremlsubswap23d 42314 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.)

(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐴𝐶) = 𝐵)
 
Theoremaddsubeq4com 42315 Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023.)
(((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) = (𝐶 + 𝐷) ↔ (𝐴𝐶) = (𝐷𝐵)))
 
Theoremsqsumi 42316 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)))
 
Theoremnegn0nposznnd 42317 Lemma for dffltz 42644. (Contributed by Steven Nguyen, 27-Feb-2023.)
(𝜑𝐴 ≠ 0)    &   (𝜑 → ¬ 0 < 𝐴)    &   (𝜑𝐴 ∈ ℤ)       (𝜑 → -𝐴 ∈ ℕ)
 
Theoremsqmid3api 42318 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))
 
Theoremdecaddcom 42319 Commute ones place in addition. (Contributed by Steven Nguyen, 29-Jan-2023.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0       (𝐴𝐵 + 𝐶) = (𝐴𝐶 + 𝐵)
 
Theoremsqn5i 42320 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
 
Theoremsqn5ii 42321 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
 
Theoremdecpmulnc 42322 Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11714. (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
 
Theoremdecpmul 42323 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
 
Theoremsqdeccom12 42324 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)))
 
Theoremsq3deccom12 42325 Variant of sqdeccom12 42324 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)))
 
Theorem4t5e20 42326 4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
(4 · 5) = 20
 
Theoremsq4 42327 The square of 4 is 16. (Contributed by SN, 26-Aug-2025.)
(4↑2) = 16
 
Theoremsq5 42328 The square of 5 is 25. (Contributed by SN, 26-Aug-2025.)
(5↑2) = 25
 
Theoremsq6 42329 The square of 6 is 36. (Contributed by SN, 26-Aug-2025.)
(6↑2) = 36
 
Theoremsq7 42330 The square of 7 is 49. (Contributed by SN, 26-Aug-2025.)
(7↑2) = 49
 
Theoremsq8 42331 The square of 8 is 64. (Contributed by SN, 26-Aug-2025.)
(8↑2) = 64
 
Theoremsq9 42332 The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
(9↑2) = 81
 
Theoremrpsscn 42333 The positive reals are a subset of the complex numbers. (Contributed by SN, 1-Oct-2025.)
+ ⊆ ℂ
 
Theorem4rp 42334 4 is a positive real. (Contributed by SN, 26-Aug-2025.)
4 ∈ ℝ+
 
Theorem6rp 42335 6 is a positive real. (Contributed by SN, 26-Aug-2025.)
6 ∈ ℝ+
 
Theorem7rp 42336 7 is a positive real. (Contributed by SN, 26-Aug-2025.)
7 ∈ ℝ+
 
Theorem8rp 42337 8 is a positive real. (Contributed by SN, 26-Aug-2025.)
8 ∈ ℝ+
 
Theorem9rp 42338 9 is a positive real. (Contributed by SN, 26-Aug-2025.)
9 ∈ ℝ+
 
Theorem235t711 42339 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 11270 saving the lower level uses of mulcomli 11270 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12842 are added then this proof would benefit more than ex-decpmul 42340.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12401 or 8t7e56 12853. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 · 711) = 167085
 
Theoremex-decpmul 42340 Example usage of decpmul 42323. This proof is significantly longer than 235t711 42339. There is more unnecessary carrying compared to 235t711 42339. Although saving 5 visual steps, using mulcomli 11270 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
 
Theoremeluzp1 42341 Membership in a successor upper set of integers. (Contributed by SN, 5-Jul-2025.)
(𝑀 ∈ ℤ → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑁 ∈ ℤ ∧ 𝑀 < 𝑁)))
 
Theoremsn-eluzp1l 42342 Shorter proof of eluzp1l 12905. (Contributed by NM, 12-Sep-2005.) (Revised by SN, 5-Jul-2025.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → 𝑀 < 𝑁)
 
Theoremfz1sumconst 42343* The sum of 𝑁 constant terms (𝑘 is not free in 𝐶). (Contributed by SN, 21-Mar-2025.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → Σ𝑘 ∈ (1...𝑁)𝐶 = (𝑁 · 𝐶))
 
Theoremfz1sump1 42344* Add one more term to a sum. Special case of fsump1 15792 generalized to 𝑁 ∈ ℕ0. (Contributed by SN, 22-Mar-2025.)
(𝜑𝑁 ∈ ℕ0)    &   ((𝜑𝑘 ∈ (1...(𝑁 + 1))) → 𝐴 ∈ ℂ)    &   (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵)       (𝜑 → Σ𝑘 ∈ (1...(𝑁 + 1))𝐴 = (Σ𝑘 ∈ (1...𝑁)𝐴 + 𝐵))
 
Theoremoddnumth 42345* The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15896. (Contributed by SN, 21-Mar-2025.)
(𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2))
 
Theoremnicomachus 42346* 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))
 
Theoremsumcubes 42347* 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))
 
Theorempine0 42348 π is nonzero. (Contributed by SN, 25-Apr-2025.)
π ≠ 0
 
Theoremine1 42349 i is not 1. (Contributed by SN, 25-Apr-2025.)
i ≠ 1
 
Theorem0tie0 42350 0 times i equals 0. (Contributed by SN, 25-Apr-2025.)
(0 · i) = 0
 
Theoremit1ei 42351 i times 1 equals i. (Contributed by SN, 25-Apr-2025.)
(i · 1) = i
 
Theorem1tiei 42352 1 times i equals i. (Contributed by SN, 25-Apr-2025.)
(1 · i) = i
 
Theoremitrere 42353 i times a real is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
(𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
 
Theoremretire 42354 A real times i is real iff the real is zero. (Contributed by SN, 25-Apr-2025.)
(𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
 
Theoremiocioodisjd 42355 Adjacent intervals where the lower interval is right-closed and the upper interval is open are disjoint. (Contributed by SN, 1-Oct-2025.)
(𝜑𝐴 ∈ ℝ*)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)       (𝜑 → ((𝐴(,]𝐵) ∩ (𝐵(,)𝐶)) = ∅)
 
Theoremrpabsid 42356 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
 
Theoremoexpreposd 42357 Lemma for dffltz 42644. For a more standard version, see expgt0b 32818. TODO-SN?: This can be used to show exp11d 42361 holds for all integers when the exponent is odd. (Contributed by SN, 4-Mar-2023.)
(𝜑𝑁 ∈ ℝ)    &   (𝜑𝑀 ∈ ℕ)    &   (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)       (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁𝑀)))
 
Theoremexplt1d 42358 A nonnegative real number less than one raised to a positive integer is less than one. (Contributed by SN, 3-Jul-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐴 < 1)       (𝜑 → (𝐴𝑁) < 1)
 
Theoremexpeq1d 42359 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ ℕ)    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ((𝐴𝑁) = 1 ↔ 𝐴 = 1))
 
Theoremexpeqidd 42360 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.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝑁 ∈ (ℤ‘2))    &   (𝜑 → 0 ≤ 𝐴)       (𝜑 → ((𝐴𝑁) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1)))
 
Theoremexp11d 42361 exp11nnd 14300 for nonzero integer exponents. (Contributed by SN, 14-Sep-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)    &   (𝜑 → (𝐴𝑁) = (𝐵𝑁))       (𝜑𝐴 = 𝐵)
 
Theorem0dvds0 42362 0 divides 0. (Contributed by SN, 15-Sep-2024.)
0 ∥ 0
 
Theoremabsdvdsabsb 42363 Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀𝑁 ↔ (abs‘𝑀) ∥ (abs‘𝑁)))
 
Theoremgcdnn0id 42364 The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
(𝑁 ∈ ℕ0 → (𝑁 gcd 𝑁) = 𝑁)
 
Theoremgcdle1d 42365 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.)
(𝜑𝑀 ∈ ℕ)    &   (𝜑𝑁 ∈ ℤ)       (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑀)
 
Theoremgcdle2d 42366 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.)
(𝜑𝑀 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ)       (𝜑 → (𝑀 gcd 𝑁) ≤ 𝑁)
 
Theoremdvdsexpad 42367 Deduction associated with dvdsexpim 16592. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝑁) ∥ (𝐵𝑁))
 
Theoremdvdsexpnn 42368 dvdssqlem 16603 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremdvdsexpnn0 42369 dvdsexpnn 42368 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremdvdsexpb 42370 dvdssq 16604 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremposqsqznn 42371 When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16795 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
(𝜑 → (𝐴↑2) ∈ ℤ)    &   (𝜑𝐴 ∈ ℚ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 ∈ ℕ)
 
Theoremzdivgd 42372* Two ways to express "𝑁 is an integer multiple of 𝑀". Originally a subproof of zdiv 12688. (Contributed by SN, 25-Apr-2025.)
(𝜑𝑀 ∈ ℂ)    &   (𝜑𝑁 ∈ ℂ)    &   (𝜑𝑀 ≠ 0)       (𝜑 → (∃𝑘 ∈ ℤ (𝑀 · 𝑘) = 𝑁 ↔ (𝑁 / 𝑀) ∈ ℤ))
 
Theoremefne0d 42373 The exponential of a complex number is nonzero, deduction form. EDITORIAL: Using efne0d 42373 in efne0 16133 is shorter than vice versa. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)       (𝜑 → (exp‘𝐴) ≠ 0)
 
Theoremefsubd 42374 Difference of exponents law for exponential function, deduction form. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → (exp‘(𝐴𝐵)) = ((exp‘𝐴) / (exp‘𝐵)))
 
Theoremef11d 42375* General condition for the exponential function to be one-to-one. efper 26521 shows that exponentiation is periodic. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((exp‘𝐴) = (exp‘𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + ((i · (2 · π)) · 𝑛))))
 
Theoremlogccne0d 42376 The logarithm isn't 0 if its argument isn't 0 or 1, deduction form. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐴 ≠ 1)       (𝜑 → (log‘𝐴) ≠ 0)
 
Theoremcxp112d 42377* General condition for complex exponentiation to be one-to-one with respect to the second argument. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑𝐶 ≠ 1)       (𝜑 → ((𝐶𝑐𝐴) = (𝐶𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (((i · (2 · π)) · 𝑛) / (log‘𝐶)))))
 
Theoremcxp111d 42378* General condition for complex exponentiation to be one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐴𝑐𝐶) = (𝐵𝑐𝐶) ↔ ∃𝑛 ∈ ℤ (log‘𝐴) = ((log‘𝐵) + (((i · (2 · π)) · 𝑛) / 𝐶))))
 
Theoremcxpi11d 42379* 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.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)       (𝜑 → ((i↑𝑐𝐴) = (i↑𝑐𝐵) ↔ ∃𝑛 ∈ ℤ 𝐴 = (𝐵 + (4 · 𝑛))))
 
Theoremlogne0d 42380 Deduction form of logne0 26621. See logccne0d 42376 for a more general version. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐴 ≠ 1)       (𝜑 → (log‘𝐴) ≠ 0)
 
Theoremrxp112d 42381 Real exponentiation is one-to-one with respect to the second argument. (TODO: Note that the base 𝐶 must be positive since -𝐶𝐴 is 𝐶𝐴 · e↑iπ𝐴, so in the negative case 𝐴 = 𝐵 + 2𝑘). (Contributed by SN, 25-Apr-2025.)
(𝜑𝐶 ∈ ℝ+)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ≠ 1)    &   (𝜑 → (𝐶𝑐𝐴) = (𝐶𝑐𝐵))       (𝜑𝐴 = 𝐵)
 
Theoremlog11d 42382 The natural logarithm is one-to-one. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)       (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
 
Theoremrplog11d 42383 The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → ((log‘𝐴) = (log‘𝐵) ↔ 𝐴 = 𝐵))
 
Theoremrxp11d 42384 Real exponentiation is one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → (𝐴𝑐𝐶) = (𝐵𝑐𝐶))       (𝜑𝐴 = 𝐵)
 
21.30.4  Trigonometry and Calculus
 
Theoremtanhalfpim 42385 The tangent of π / 2 minus a number is the cotangent, here represented by cos𝐴 / sin𝐴. (Contributed by SN, 2-Sep-2025.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (sin‘𝐴) ≠ 0)       (𝜑 → (tan‘((π / 2) − 𝐴)) = ((cos‘𝐴) / (sin‘𝐴)))
 
Theoremtan3rdpi 42386 The tangent of π / 3 is √3. (Contributed by SN, 2-Sep-2025.)
(tan‘(π / 3)) = (√‘3)
 
Theoremasin1half 42387 The arcsine of 1 / 2 is π / 6. (Contributed by SN, 31-Aug-2025.)
(arcsin‘(1 / 2)) = (π / 6)
 
Theoremacos1half 42388 The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
(arccos‘(1 / 2)) = (π / 3)
 
Theoremdvun 42389 Condition for the union of the derivatives of two disjoint functions to be equal to the derivative of the union of the two functions. If 𝐴 and 𝐵 are open sets, this condition (dvun.n) is satisfied by isopn3i 23090. (Contributed by SN, 30-Sep-2025.)
𝐽 = (𝐾t 𝑆)    &   𝐾 = (TopOpen‘ℂfld)    &   (𝜑𝑆 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝐺:𝐵⟶ℂ)    &   (𝜑𝐴𝑆)    &   (𝜑𝐵𝑆)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑 → (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵)) = ((int‘𝐽)‘(𝐴𝐵)))       (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹𝐺)))
 
Theoremredvmptabs 42390* The derivative of the absolute value, for real numbers. (Contributed by SN, 30-Sep-2025.)
𝐷 = (ℝ ∖ {0})       (ℝ D (𝑥𝐷 ↦ (abs‘𝑥))) = (𝑥𝐷 ↦ if(𝑥 < 0, -1, 1))
 
Theoremreadvrec2 42391* The antiderivative of 1/x in real numbers, without using the absolute value function. (Contributed by SN, 1-Oct-2025.)
𝐷 = (ℝ ∖ {0})       (ℝ D (𝑥𝐷 ↦ ((log‘(𝑥↑2)) / 2))) = (𝑥𝐷 ↦ (1 / 𝑥))
 
Theoremreadvrec 42392* For real numbers, the antiderivative of 1/x is ln|x|. (Contributed by SN, 30-Sep-2025.)
𝐷 = (ℝ ∖ {0})       (ℝ D (𝑥𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥𝐷 ↦ (1 / 𝑥))
 
Theoremresuppsinopn 42393 The support of sin (df-supp 8186) restricted to the reals is an open set. (Contributed by SN, 7-Oct-2025.)
𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}       𝐷 ∈ (topGen‘ran (,))
 
Theoremreadvcot 42394* Real antiderivative of cotangent. (Contributed by SN, 7-Oct-2025.)
𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}       (ℝ D (𝑥𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))
 
21.30.5  Real subtraction
 
Syntaxcresub 42395 Real number subtraction.
class
 
Definitiondf-resub 42396* Define subtraction between real numbers. This operator saves a few axioms over df-sub 11494 in certain situations. Theorem resubval 42397 shows its value, resubadd 42409 relates it to addition, and rersubcl 42408 proves its closure. It is the restriction of df-sub 11494 to the reals: subresre 42460. (Contributed by Steven Nguyen, 7-Jan-2023.)
= (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
 
Theoremresubval 42397* Value of real subtraction, which is the (unique) real 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
 
Theoremrenegeulemv 42398* Lemma for renegeu 42400 and similar. Derive existential uniqueness from existence. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴)
 
Theoremrenegeulem 42399* Lemma for renegeu 42400 and similar. Remove a change in bound variables from renegeulemv 42398. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
 
Theoremrenegeu 42400* Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
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