P. 416 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41501-41600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	decpmulnc 41501	Partial products algorithm for two digit multiplication, no carry. Compare muladdi 11669. (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 41502	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 41503	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 41504	Variant of sqdeccom12 41503 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 41505	4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
⊢ (4 · 5) = ;20
Theorem	sq9 41506	The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
⊢ (9↑2) = ;81
Theorem	235t711 41507	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 11227 saving the lower level uses of mulcomli 11227 within 235 · 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12790 are added then this proof would benefit more than ex-decpmul 41508. For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12351 or 8t7e56 12801. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)
⊢ (;;235 · ;;711) = ;;;;;167085
Theorem	ex-decpmul 41508	Example usage of decpmul 41502. This proof is significantly longer than 235t711 41507. There is more unnecessary carrying compared to 235t711 41507. Although saving 5 visual steps, using mulcomli 11227 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	fz1sumconst 41509*	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 41510*	Add one more term to a sum. Special case of fsump1 15706 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 41511*	The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15810. (Contributed by SN, 21-Mar-2025.)
⊢ (𝑁 ∈ ℕ0 → Σ𝑘 ∈ (1...𝑁)((2 · 𝑘) − 1) = (𝑁↑2))
Theorem	nicomachus 41512*	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 41513*	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))
21.28.4  Exponents and divisibility
Theorem	oexpreposd 41514	Lemma for dffltz 41678. TODO-SN?: This can be used to show exp11d 41518 holds for all integers when the exponent is odd. The more standard ¬ 2 ∥ 𝑀 should be used. (Contributed by SN, 4-Mar-2023.)
⊢ (𝜑 → 𝑁 ∈ ℝ)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ (𝜑 → ¬ (𝑀 / 2) ∈ ℕ)    ⇒   ⊢ (𝜑 → (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))
Theorem	ltexp1d 41515	ltmul1d 13061 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → (𝐴 < 𝐵 ↔ (𝐴↑𝑁) < (𝐵↑𝑁)))
Theorem	ltexp1dd 41516	Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐴 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) < (𝐵↑𝑁))
Theorem	exp11nnd 41517	sq11d 14225 for positive real bases and positive integer exponents. The base cannot be generalized much further, since if 𝑁 is even then we have 𝐴↑𝑁 = -𝐴↑𝑁. (Contributed by SN, 14-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → (𝐴↑𝑁) = (𝐵↑𝑁))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	exp11d 41518	exp11nnd 41517 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 41519	0 divides 0. (Contributed by SN, 15-Sep-2024.)
⊢ 0 ∥ 0
Theorem	absdvdsabsb 41520	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	dvdsexpim 41521	dvdssqim 16500 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝐴 ∥ 𝐵 → (𝐴↑𝑁) ∥ (𝐵↑𝑁)))
Theorem	gcdnn0id 41522	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 41523	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 41524	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 41525	Deduction associated with dvdsexpim 41521. (Contributed by SN, 21-Aug-2024.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝐴 ∥ 𝐵)    ⇒   ⊢ (𝜑 → (𝐴↑𝑁) ∥ (𝐵↑𝑁))
Theorem	nn0rppwr 41526	If 𝐴 and 𝐵 are relatively prime, then so are 𝐴↑𝑁 and 𝐵↑𝑁. rppwr 16505 extended to nonnegative integers. Less general than rpexp12i 16665. (Contributed by Steven Nguyen, 4-Apr-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴↑𝑁) gcd (𝐵↑𝑁)) = 1))
Theorem	expgcd 41527	Exponentiation distributes over GCD. sqgcd 16506 extended to nonnegative exponents. (Contributed by Steven Nguyen, 4-Apr-2023.)
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
Theorem	nn0expgcd 41528	Exponentiation distributes over GCD. nn0gcdsq 16692 extended to nonnegative exponents. expgcd 41527 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
Theorem	zexpgcd 41529	Exponentiation distributes over GCD. zgcdsq 16693 extended to nonnegative exponents. nn0expgcd 41528 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴↑𝑁) gcd (𝐵↑𝑁)))
Theorem	numdenexp 41530	numdensq 16694 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → ((numer‘(𝐴↑𝑁)) = ((numer‘𝐴)↑𝑁) ∧ (denom‘(𝐴↑𝑁)) = ((denom‘𝐴)↑𝑁)))
Theorem	numexp 41531	numsq 16695 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴↑𝑁)) = ((numer‘𝐴)↑𝑁))
Theorem	denexp 41532	densq 16696 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
⊢ ((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴↑𝑁)) = ((denom‘𝐴)↑𝑁))
Theorem	dvdsexpnn 41533	dvdssqlem 16507 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 41534	dvdsexpnn 41533 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 41535	dvdssq 16508 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 41536	When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16698 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	zrtelqelz 41537	zsqrtelqelz 16698 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴↑𝑐(1 / 𝑁)) ∈ ℤ)
Theorem	zrtdvds 41538	A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023.)
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ ℕ) → (𝐴↑𝑐(1 / 𝑁)) ∥ 𝐴)
Theorem	rtprmirr 41539	The root of a prime number is irrational. (Contributed by Steven Nguyen, 6-Apr-2023.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘2)) → (𝑃↑𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ))
21.28.5  Real subtraction
Syntax	cresub 41540	Real number subtraction.
class −ℝ
Definition	df-resub 41541*	Define subtraction between real numbers. This operator saves a few axioms over df-sub 11450 in certain situations. Theorem resubval 41542 shows its value, resubadd 41554 relates it to addition, and rersubcl 41553 proves its closure. It is the restriction of df-sub 11450 to the reals: subresre 41605. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
Theorem	resubval 41542*	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 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
Theorem	renegeulemv 41543*	Lemma for renegeu 41545 and similar. Derive existential uniqueness from existence. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴)
Theorem	renegeulem 41544*	Lemma for renegeu 41545 and similar. Remove a change in bound variables from renegeulemv 41543. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
Theorem	renegeu 41545*	Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	rernegcl 41546	Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ)
Theorem	renegadd 41547	Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 −ℝ 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0))
Theorem	renegid 41548	Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0)
Theorem	reneg0addlid 41549	Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) + 𝐴) = 𝐴)
Theorem	resubeulem1 41550	Lemma for resubeu 41552. A value which when added to zero, results in negative zero. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (0 + (0 −ℝ (0 + 0))) = (0 −ℝ 0))
Theorem	resubeulem2 41551	Lemma for resubeu 41552. A value which when added to 𝐴, results in 𝐵. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 −ℝ 𝐴) + ((0 −ℝ (0 + 0)) + 𝐵))) = 𝐵)
Theorem	resubeu 41552*	Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
Theorem	rersubcl 41553	Closure for real subtraction. Based on subcl 11463. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ)
Theorem	resubadd 41554	Relation between real subtraction and addition. Based on subadd 11467. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	resubaddd 41555	Relationship between subtraction and addition. Based on subaddd 11593. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	resubf 41556	Real subtraction is an operation on the real numbers. Based on subf 11466. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ :(ℝ × ℝ)⟶ℝ
Theorem	repncan2 41557	Addition and subtraction of equals. Compare pncan2 11471. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ 𝐴) = 𝐵)
Theorem	repncan3 41558	Addition and subtraction of equals. Based on pncan3 11472. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵)
Theorem	readdsub 41559	Law for addition and subtraction. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ 𝐶) = ((𝐴 −ℝ 𝐶) + 𝐵))
Theorem	reladdrsub 41560	Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11629 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 = (𝐶 −ℝ 𝐴))
Theorem	reltsub1 41561	Subtraction from both sides of 'less than'. Compare ltsub1 11714. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 −ℝ 𝐶) < (𝐵 −ℝ 𝐶)))
Theorem	reltsubadd2 41562	'Less than' relationship between addition and subtraction. Compare ltsubadd2 11689. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	resubcan2 41563	Cancellation law for real subtraction. Compare subcan2 11489. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	resubsub4 41564	Law for double subtraction. Compare subsub4 11497. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶)))
Theorem	rennncan2 41565	Cancellation law for real subtraction. Compare nnncan2 11501. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) −ℝ (𝐵 −ℝ 𝐶)) = (𝐴 −ℝ 𝐵))
Theorem	renpncan3 41566	Cancellation law for real subtraction. Compare npncan3 11502. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵))
Theorem	repnpcan 41567	Cancellation law for addition and real subtraction. Compare pnpcan 11503. (Contributed by Steven Nguyen, 19-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ (𝐴 + 𝐶)) = (𝐵 −ℝ 𝐶))
Theorem	reppncan 41568	Cancellation law for mixed addition and real subtraction. Compare ppncan 11506. (Contributed by SN, 3-Sep-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 −ℝ 𝐶)) = (𝐴 + 𝐵))
Theorem	resubidaddlidlem 41569	Lemma for resubidaddlid 41570. A special case of npncan 11485. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 −ℝ 𝐵) = (𝐵 −ℝ 𝐶))    ⇒   ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶)) = (𝐴 −ℝ 𝐶))
Theorem	resubidaddlid 41570	Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 −ℝ 𝐴) + 𝐵) = 𝐵)
Theorem	resubdi 41571	Distribution of multiplication over real subtraction. (Contributed by Steven Nguyen, 3-Jun-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · (𝐵 −ℝ 𝐶)) = ((𝐴 · 𝐵) −ℝ (𝐴 · 𝐶)))
Theorem	re1m1e0m0 41572	Equality of two left-additive identities. See resubidaddlid 41570. Uses ax-i2m1 11180. (Contributed by SN, 25-Dec-2023.)
⊢ (1 −ℝ 1) = (0 −ℝ 0)
Theorem	sn-00idlem1 41573	Lemma for sn-00id 41576. (Contributed by SN, 25-Dec-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴))
Theorem	sn-00idlem2 41574	Lemma for sn-00id 41576. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1)
Theorem	sn-00idlem3 41575	Lemma for sn-00id 41576. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0)
Theorem	sn-00id 41576	00id 11393 proven without ax-mulcom 11176 but using ax-1ne0 11181. (Though note that the current version of 00id 11393 can be changed to avoid ax-icn 11171, ax-addcl 11172, ax-mulcl 11174, ax-i2m1 11180, ax-cnre 11185. Most of this is by using 0cnALT3 41476 instead of 0cn 11210). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
⊢ (0 + 0) = 0
Theorem	re0m0e0 41577	Real number version of 0m0e0 12336 proven without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (0 −ℝ 0) = 0
Theorem	readdlid 41578	Real number version of addlid 11401. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
Theorem	sn-addlid 41579	addlid 11401 without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	remul02 41580	Real number version of mul02 11396 proven without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
Theorem	sn-0ne2 41581	0ne2 12423 without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ 0 ≠ 2
Theorem	remul01 41582	Real number version of mul01 11397 proven without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
Theorem	resubid 41583	Subtraction of a real number from itself (compare subid 11483). (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
Theorem	readdrid 41584	Real number version of addrid 11398 without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
Theorem	resubid1 41585	Real number version of subid1 11484 without ax-mulcom 11176. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
Theorem	renegneg 41586	A real number is equal to the negative of its negative. Compare negneg 11514. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 41587	Commuted version of readdcan 11392 without ax-mulcom 11176. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 41588	Commuted version of renegid 41548. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 41589	Product with negative is negative of product. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 · (0 −ℝ 𝐵)) = (0 −ℝ (𝐴 · 𝐵)))
Theorem	sn-it0e0 41590	Proof of it0e0 12438 without ax-mulcom 11176. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11185 and renegid2 41588. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 41591*	A combination of cnegex 11399 and cnegex2 11400, this proof takes cnre 11215 𝐴 = 𝑟 + i · 𝑠 and shows that i · -𝑠 + -𝑟 is both a left and right inverse. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ ((𝐴 + 𝑏) = 0 ∧ (𝑏 + 𝐴) = 0))
Theorem	sn-negex 41592*	Proof of cnegex 11399 without ax-mulcom 11176. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 41593*	Proof of cnegex2 11400 without ax-mulcom 11176. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 41594	addcand 11421 without ax-mulcom 11176. Note how the proof is almost identical to addcan 11402. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
Theorem	sn-addrid 41595	addrid 11398 without ax-mulcom 11176. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 41596	addcan2d 11422 without ax-mulcom 11176. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 41597	ixi 11847 without ax-mulcom 11176. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 41598	i4 14172 without ax-mulcom 11176. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 41599	A number that sums to itself is zero. Compare addid0 11637, readdridaddlidd 41480. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 41600	mul01 11397 without ax-mulcom 11176. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)