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Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsqdeccom12 41201 The square of a number in terms of its digits switched. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ β„•0    &   π΅ ∈ β„•0    β‡’   ((𝐴𝐡 Β· 𝐴𝐡) βˆ’ (𝐡𝐴 Β· 𝐡𝐴)) = (99 Β· ((𝐴 Β· 𝐴) βˆ’ (𝐡 Β· 𝐡)))
 
Theoremsq3deccom12 41202 Variant of sqdeccom12 41201 with a three digit square. (Contributed by Steven Nguyen, 3-Jan-2023.)
𝐴 ∈ β„•0    &   π΅ ∈ β„•0    &   πΆ ∈ β„•0    &   (𝐴 + 𝐢) = 𝐷    β‡’   ((𝐴𝐡𝐢 Β· 𝐴𝐡𝐢) βˆ’ (𝐷𝐡 Β· 𝐷𝐡)) = (99 Β· ((𝐴𝐡 Β· 𝐴𝐡) βˆ’ (𝐢 Β· 𝐢)))
 
Theorem4t5e20 41203 4 times 5 equals 20. (Contributed by SN, 30-Mar-2025.)
(4 Β· 5) = 20
 
Theoremsq9 41204 The square of 9 is 81. (Contributed by SN, 30-Mar-2025.)
(9↑2) = 81
 
Theorem235t711 41205 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 Β· 1. There isn't a higher level mulcomli 11223 saving the lower level uses of mulcomli 11223 within 235 Β· 7 since mulcom2 doesn't exist, but if commuted versions of theorems like 7t2e14 12786 are added then this proof would benefit more than ex-decpmul 41206.

For practicality, this proof doesn't have "e167085" at the end of its name like 2p2e4 12347 or 8t7e56 12797. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.)

(235 Β· 711) = 167085
 
Theoremex-decpmul 41206 Example usage of decpmul 41200. This proof is significantly longer than 235t711 41205. There is more unnecessary carrying compared to 235t711 41205. Although saving 5 visual steps, using mulcomli 11223 early on increases the compressed proof length. (Contributed by Steven Nguyen, 10-Dec-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(235 Β· 711) = 167085
 
Theoremfz1sumconst 41207* The sum of 𝑁 constant terms (π‘˜ is not free in 𝐢). (Contributed by SN, 21-Mar-2025.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ (1...𝑁)𝐢 = (𝑁 Β· 𝐢))
 
Theoremfz1sump1 41208* Add one more term to a sum. Special case of fsump1 15702 generalized to 𝑁 ∈ β„•0. (Contributed by SN, 22-Mar-2025.)
(πœ‘ β†’ 𝑁 ∈ β„•0)    &   ((πœ‘ ∧ π‘˜ ∈ (1...(𝑁 + 1))) β†’ 𝐴 ∈ β„‚)    &   (π‘˜ = (𝑁 + 1) β†’ 𝐴 = 𝐡)    β‡’   (πœ‘ β†’ Ξ£π‘˜ ∈ (1...(𝑁 + 1))𝐴 = (Ξ£π‘˜ ∈ (1...𝑁)𝐴 + 𝐡))
 
Theoremoddnumth 41209* The Odd Number Theorem. The sum of the first 𝑁 odd numbers is 𝑁↑2. A corollary of arisum 15806. (Contributed by SN, 21-Mar-2025.)
(𝑁 ∈ β„•0 β†’ Ξ£π‘˜ ∈ (1...𝑁)((2 Β· π‘˜) βˆ’ 1) = (𝑁↑2))
 
Theoremnicomachus 41210* Nicomachus's Theorem. The sum of the odd numbers from 𝑁↑2 βˆ’ 𝑁 + 1 to 𝑁↑2 + 𝑁 βˆ’ 1 is 𝑁↑3. Proof 2 from https://proofwiki.org/wiki/Nicomachus%27s_Theorem. (Contributed by SN, 21-Mar-2025.)
(𝑁 ∈ β„•0 β†’ Ξ£π‘˜ ∈ (1...𝑁)(((𝑁↑2) βˆ’ 𝑁) + ((2 Β· π‘˜) βˆ’ 1)) = (𝑁↑3))
 
Theoremsumcubes 41211* 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 ∈ β„•0 β†’ Ξ£π‘˜ ∈ (1...𝑁)(π‘˜β†‘3) = (Ξ£π‘˜ ∈ (1...𝑁)π‘˜β†‘2))
 
21.28.4  Exponents and divisibility
 
Theoremoexpreposd 41212 Lemma for dffltz 41376. TODO-SN?: This can be used to show exp11d 41216 holds for all integers when the exponent is odd. The more standard Β¬ 2 βˆ₯ 𝑀 should be used. (Contributed by SN, 4-Mar-2023.)
(πœ‘ β†’ 𝑁 ∈ ℝ)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ Β¬ (𝑀 / 2) ∈ β„•)    β‡’   (πœ‘ β†’ (0 < 𝑁 ↔ 0 < (𝑁↑𝑀)))
 
Theoremltexp1d 41213 ltmul1d 13057 for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝐴 < 𝐡 ↔ (𝐴↑𝑁) < (𝐡↑𝑁)))
 
Theoremltexp1dd 41214 Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ (𝐴↑𝑁) < (𝐡↑𝑁))
 
Theoremexp11nnd 41215 sq11d 14221 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.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ (𝐴↑𝑁) = (𝐡↑𝑁))    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theoremexp11d 41216 exp11nnd 41215 for nonzero integer exponents. (Contributed by SN, 14-Sep-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ+)    &   (πœ‘ β†’ 𝐡 ∈ ℝ+)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑁 β‰  0)    &   (πœ‘ β†’ (𝐴↑𝑁) = (𝐡↑𝑁))    β‡’   (πœ‘ β†’ 𝐴 = 𝐡)
 
Theorem0dvds0 41217 0 divides 0. (Contributed by SN, 15-Sep-2024.)
0 βˆ₯ 0
 
Theoremabsdvdsabsb 41218 Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024.)
((𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝑀 βˆ₯ 𝑁 ↔ (absβ€˜π‘€) βˆ₯ (absβ€˜π‘)))
 
Theoremdvdsexpim 41219 dvdssqim 16496 generalized to nonnegative exponents. (Contributed by Steven Nguyen, 2-Apr-2023.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ (𝐴 βˆ₯ 𝐡 β†’ (𝐴↑𝑁) βˆ₯ (𝐡↑𝑁)))
 
Theoremgcdnn0id 41220 The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
(𝑁 ∈ β„•0 β†’ (𝑁 gcd 𝑁) = 𝑁)
 
Theoremgcdle1d 41221 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.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) ≀ 𝑀)
 
Theoremgcdle2d 41222 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.)
(πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    β‡’   (πœ‘ β†’ (𝑀 gcd 𝑁) ≀ 𝑁)
 
Theoremdvdsexpad 41223 Deduction associated with dvdsexpim 41219. (Contributed by SN, 21-Aug-2024.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„•0)    &   (πœ‘ β†’ 𝐴 βˆ₯ 𝐡)    β‡’   (πœ‘ β†’ (𝐴↑𝑁) βˆ₯ (𝐡↑𝑁))
 
Theoremnn0rppwr 41224 If 𝐴 and 𝐡 are relatively prime, then so are 𝐴↑𝑁 and 𝐡↑𝑁. rppwr 16501 extended to nonnegative integers. Less general than rpexp12i 16661. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 gcd 𝐡) = 1 β†’ ((𝐴↑𝑁) gcd (𝐡↑𝑁)) = 1))
 
Theoremexpgcd 41225 Exponentiation distributes over GCD. sqgcd 16502 extended to nonnegative exponents. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 gcd 𝐡)↑𝑁) = ((𝐴↑𝑁) gcd (𝐡↑𝑁)))
 
Theoremnn0expgcd 41226 Exponentiation distributes over GCD. nn0gcdsq 16688 extended to nonnegative exponents. expgcd 41225 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ β„•0 ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 gcd 𝐡)↑𝑁) = ((𝐴↑𝑁) gcd (𝐡↑𝑁)))
 
Theoremzexpgcd 41227 Exponentiation distributes over GCD. zgcdsq 16689 extended to nonnegative exponents. nn0expgcd 41226 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 gcd 𝐡)↑𝑁) = ((𝐴↑𝑁) gcd (𝐡↑𝑁)))
 
Theoremnumdenexp 41228 numdensq 16690 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ β„š ∧ 𝑁 ∈ β„•0) β†’ ((numerβ€˜(𝐴↑𝑁)) = ((numerβ€˜π΄)↑𝑁) ∧ (denomβ€˜(𝐴↑𝑁)) = ((denomβ€˜π΄)↑𝑁)))
 
Theoremnumexp 41229 numsq 16691 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ β„š ∧ 𝑁 ∈ β„•0) β†’ (numerβ€˜(𝐴↑𝑁)) = ((numerβ€˜π΄)↑𝑁))
 
Theoremdenexp 41230 densq 16692 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ β„š ∧ 𝑁 ∈ β„•0) β†’ (denomβ€˜(𝐴↑𝑁)) = ((denomβ€˜π΄)↑𝑁))
 
Theoremdvdsexpnn 41231 dvdssqlem 16503 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
((𝐴 ∈ β„• ∧ 𝐡 ∈ β„• ∧ 𝑁 ∈ β„•) β†’ (𝐴 βˆ₯ 𝐡 ↔ (𝐴↑𝑁) βˆ₯ (𝐡↑𝑁)))
 
Theoremdvdsexpnn0 41232 dvdsexpnn 41231 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ β„•0 ∧ 𝑁 ∈ β„•) β†’ (𝐴 βˆ₯ 𝐡 ↔ (𝐴↑𝑁) βˆ₯ (𝐡↑𝑁)))
 
Theoremdvdsexpb 41233 dvdssq 16504 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 ∈ β„•) β†’ (𝐴 βˆ₯ 𝐡 ↔ (𝐴↑𝑁) βˆ₯ (𝐡↑𝑁)))
 
Theoremposqsqznn 41234 When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16694 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
(πœ‘ β†’ (𝐴↑2) ∈ β„€)    &   (πœ‘ β†’ 𝐴 ∈ β„š)    &   (πœ‘ β†’ 0 < 𝐴)    β‡’   (πœ‘ β†’ 𝐴 ∈ β„•)
 
Theoremzrtelqelz 41235 zsqrtelqelz 16694 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„• ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ β„š) β†’ (𝐴↑𝑐(1 / 𝑁)) ∈ β„€)
 
Theoremzrtdvds 41236 A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„• ∧ (𝐴↑𝑐(1 / 𝑁)) ∈ β„•) β†’ (𝐴↑𝑐(1 / 𝑁)) βˆ₯ 𝐴)
 
Theoremrtprmirr 41237 The root of a prime number is irrational. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝑃 ∈ β„™ ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑃↑𝑐(1 / 𝑁)) ∈ (ℝ βˆ– β„š))
 
21.28.5  Real subtraction
 
Syntaxcresub 41238 Real number subtraction.
class βˆ’β„
 
Definitiondf-resub 41239* Define subtraction between real numbers. This operator saves a few axioms over df-sub 11446 in certain situations. Theorem resubval 41240 shows its value, resubadd 41252 relates it to addition, and rersubcl 41251 proves its closure. It is the restriction of df-sub 11446 to the reals: subresre 41303. (Contributed by Steven Nguyen, 7-Jan-2023.)
βˆ’β„ = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = π‘₯))
 
Theoremresubval 41240* Value of real subtraction, which is the (unique) real π‘₯ such that 𝐡 + π‘₯ = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 βˆ’β„ 𝐡) = (β„©π‘₯ ∈ ℝ (𝐡 + π‘₯) = 𝐴))
 
Theoremrenegeulemv 41241* Lemma for renegeu 41243 and similar. Derive existential uniqueness from existence. (Contributed by Steven Nguyen, 28-Jan-2023.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ (𝐡 + 𝑦) = 𝐴)    β‡’   (πœ‘ β†’ βˆƒ!π‘₯ ∈ ℝ (𝐡 + π‘₯) = 𝐴)
 
Theoremrenegeulem 41242* Lemma for renegeu 41243 and similar. Remove a change in bound variables from renegeulemv 41241. (Contributed by Steven Nguyen, 28-Jan-2023.)
(πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ βˆƒπ‘¦ ∈ ℝ (𝐡 + 𝑦) = 𝐴)    β‡’   (πœ‘ β†’ βˆƒ!𝑦 ∈ ℝ (𝐡 + 𝑦) = 𝐴)
 
Theoremrenegeu 41243* Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ β†’ βˆƒ!π‘₯ ∈ ℝ (𝐴 + π‘₯) = 0)
 
Theoremrernegcl 41244 Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ β†’ (0 βˆ’β„ 𝐴) ∈ ℝ)
 
Theoremrenegadd 41245 Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((0 βˆ’β„ 𝐴) = 𝐡 ↔ (𝐴 + 𝐡) = 0))
 
Theoremrenegid 41246 Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ β†’ (𝐴 + (0 βˆ’β„ 𝐴)) = 0)
 
Theoremreneg0addlid 41247 Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ β†’ ((0 βˆ’β„ 0) + 𝐴) = 𝐴)
 
Theoremresubeulem1 41248 Lemma for resubeu 41250. A value which when added to zero, results in negative zero. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ β†’ (0 + (0 βˆ’β„ (0 + 0))) = (0 βˆ’β„ 0))
 
Theoremresubeulem2 41249 Lemma for resubeu 41250. A value which when added to 𝐴, results in 𝐡. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + ((0 βˆ’β„ 𝐴) + ((0 βˆ’β„ (0 + 0)) + 𝐡))) = 𝐡)
 
Theoremresubeu 41250* Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ βˆƒ!π‘₯ ∈ ℝ (𝐴 + π‘₯) = 𝐡)
 
Theoremrersubcl 41251 Closure for real subtraction. Based on subcl 11459. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 βˆ’β„ 𝐡) ∈ ℝ)
 
Theoremresubadd 41252 Relation between real subtraction and addition. Based on subadd 11463. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐡) = 𝐢 ↔ (𝐡 + 𝐢) = 𝐴))
 
Theoremresubaddd 41253 Relationship between subtraction and addition. Based on subaddd 11589. (Contributed by Steven Nguyen, 8-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    β‡’   (πœ‘ β†’ ((𝐴 βˆ’β„ 𝐡) = 𝐢 ↔ (𝐡 + 𝐢) = 𝐴))
 
Theoremresubf 41254 Real subtraction is an operation on the real numbers. Based on subf 11462. (Contributed by Steven Nguyen, 7-Jan-2023.)
βˆ’β„ :(ℝ Γ— ℝ)βŸΆβ„
 
Theoremrepncan2 41255 Addition and subtraction of equals. Compare pncan2 11467. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((𝐴 + 𝐡) βˆ’β„ 𝐴) = 𝐡)
 
Theoremrepncan3 41256 Addition and subtraction of equals. Based on pncan3 11468. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 + (𝐡 βˆ’β„ 𝐴)) = 𝐡)
 
Theoremreaddsub 41257 Law for addition and subtraction. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 + 𝐡) βˆ’β„ 𝐢) = ((𝐴 βˆ’β„ 𝐢) + 𝐡))
 
Theoremreladdrsub 41258 Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11625 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ (𝐴 + 𝐡) = 𝐢)    β‡’   (πœ‘ β†’ 𝐡 = (𝐢 βˆ’β„ 𝐴))
 
Theoremreltsub1 41259 Subtraction from both sides of 'less than'. Compare ltsub1 11710. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (𝐴 βˆ’β„ 𝐢) < (𝐡 βˆ’β„ 𝐢)))
 
Theoremreltsubadd2 41260 'Less than' relationship between addition and subtraction. Compare ltsubadd2 11685. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐡) < 𝐢 ↔ 𝐴 < (𝐡 + 𝐢)))
 
Theoremresubcan2 41261 Cancellation law for real subtraction. Compare subcan2 11485. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐢) = (𝐡 βˆ’β„ 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremresubsub4 41262 Law for double subtraction. Compare subsub4 11493. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐡) βˆ’β„ 𝐢) = (𝐴 βˆ’β„ (𝐡 + 𝐢)))
 
Theoremrennncan2 41263 Cancellation law for real subtraction. Compare nnncan2 11497. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐢) βˆ’β„ (𝐡 βˆ’β„ 𝐢)) = (𝐴 βˆ’β„ 𝐡))
 
Theoremrenpncan3 41264 Cancellation law for real subtraction. Compare npncan3 11498. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐡) + (𝐢 βˆ’β„ 𝐴)) = (𝐢 βˆ’β„ 𝐡))
 
Theoremrepnpcan 41265 Cancellation law for addition and real subtraction. Compare pnpcan 11499. (Contributed by Steven Nguyen, 19-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 + 𝐡) βˆ’β„ (𝐴 + 𝐢)) = (𝐡 βˆ’β„ 𝐢))
 
Theoremreppncan 41266 Cancellation law for mixed addition and real subtraction. Compare ppncan 11502. (Contributed by SN, 3-Sep-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 + 𝐢) + (𝐡 βˆ’β„ 𝐢)) = (𝐴 + 𝐡))
 
Theoremresubidaddlidlem 41267 Lemma for resubidaddlid 41268. A special case of npncan 11481. (Contributed by Steven Nguyen, 8-Jan-2023.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    &   (πœ‘ β†’ 𝐢 ∈ ℝ)    &   (πœ‘ β†’ (𝐴 βˆ’β„ 𝐡) = (𝐡 βˆ’β„ 𝐢))    β‡’   (πœ‘ β†’ ((𝐴 βˆ’β„ 𝐡) + (𝐡 βˆ’β„ 𝐢)) = (𝐴 βˆ’β„ 𝐢))
 
Theoremresubidaddlid 41268 Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ ((𝐴 βˆ’β„ 𝐴) + 𝐡) = 𝐡)
 
Theoremresubdi 41269 Distribution of multiplication over real subtraction. (Contributed by Steven Nguyen, 3-Jun-2023.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ (𝐴 Β· (𝐡 βˆ’β„ 𝐢)) = ((𝐴 Β· 𝐡) βˆ’β„ (𝐴 Β· 𝐢)))
 
Theoremre1m1e0m0 41270 Equality of two left-additive identities. See resubidaddlid 41268. Uses ax-i2m1 11178. (Contributed by SN, 25-Dec-2023.)
(1 βˆ’β„ 1) = (0 βˆ’β„ 0)
 
Theoremsn-00idlem1 41271 Lemma for sn-00id 41274. (Contributed by SN, 25-Dec-2023.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· (0 βˆ’β„ 0)) = (𝐴 βˆ’β„ 𝐴))
 
Theoremsn-00idlem2 41272 Lemma for sn-00id 41274. (Contributed by SN, 25-Dec-2023.)
((0 βˆ’β„ 0) β‰  0 β†’ (0 βˆ’β„ 0) = 1)
 
Theoremsn-00idlem3 41273 Lemma for sn-00id 41274. (Contributed by SN, 25-Dec-2023.)
((0 βˆ’β„ 0) = 1 β†’ (0 + 0) = 0)
 
Theoremsn-00id 41274 00id 11389 proven without ax-mulcom 11174 but using ax-1ne0 11179. (Though note that the current version of 00id 11389 can be changed to avoid ax-icn 11169, ax-addcl 11170, ax-mulcl 11172, ax-i2m1 11178, ax-cnre 11183. Most of this is by using 0cnALT3 41174 instead of 0cn 11206). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
(0 + 0) = 0
 
Theoremre0m0e0 41275 Real number version of 0m0e0 12332 proven without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(0 βˆ’β„ 0) = 0
 
Theoremreaddlid 41276 Real number version of addlid 11397. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (0 + 𝐴) = 𝐴)
 
Theoremsn-addlid 41277 addlid 11397 without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ β„‚ β†’ (0 + 𝐴) = 𝐴)
 
Theoremremul02 41278 Real number version of mul02 11392 proven without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (0 Β· 𝐴) = 0)
 
Theoremsn-0ne2 41279 0ne2 12419 without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
0 β‰  2
 
Theoremremul01 41280 Real number version of mul01 11393 proven without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (𝐴 Β· 0) = 0)
 
Theoremresubid 41281 Subtraction of a real number from itself (compare subid 11479). (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (𝐴 βˆ’β„ 𝐴) = 0)
 
Theoremreaddrid 41282 Real number version of addrid 11394 without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (𝐴 + 0) = 𝐴)
 
Theoremresubid1 41283 Real number version of subid1 11480 without ax-mulcom 11174. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ β†’ (𝐴 βˆ’β„ 0) = 𝐴)
 
Theoremrenegneg 41284 A real number is equal to the negative of its negative. Compare negneg 11510. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ β†’ (0 βˆ’β„ (0 βˆ’β„ 𝐴)) = 𝐴)
 
Theoremreaddcan2 41285 Commuted version of readdcan 11388 without ax-mulcom 11174. (Contributed by SN, 21-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 + 𝐢) = (𝐡 + 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremrenegid2 41286 Commuted version of renegid 41246. (Contributed by SN, 4-May-2024.)
(𝐴 ∈ ℝ β†’ ((0 βˆ’β„ 𝐴) + 𝐴) = 0)
 
Theoremremulneg2d 41287 Product with negative is negative of product. (Contributed by SN, 25-Jan-2025.)
(πœ‘ β†’ 𝐴 ∈ ℝ)    &   (πœ‘ β†’ 𝐡 ∈ ℝ)    β‡’   (πœ‘ β†’ (𝐴 Β· (0 βˆ’β„ 𝐡)) = (0 βˆ’β„ (𝐴 Β· 𝐡)))
 
Theoremsn-it0e0 41288 Proof of it0e0 12434 without ax-mulcom 11174. Informally, a real number times 0 is 0, and βˆƒπ‘Ÿ ∈ β„π‘Ÿ = i Β· 𝑠 by ax-cnre 11183 and renegid2 41286. (Contributed by SN, 30-Apr-2024.)
(i Β· 0) = 0
 
Theoremsn-negex12 41289* A combination of cnegex 11395 and cnegex2 11396, this proof takes cnre 11211 𝐴 = π‘Ÿ + i Β· 𝑠 and shows that i Β· -𝑠 + -π‘Ÿ is both a left and right inverse. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘ ∈ β„‚ ((𝐴 + 𝑏) = 0 ∧ (𝑏 + 𝐴) = 0))
 
Theoremsn-negex 41290* Proof of cnegex 11395 without ax-mulcom 11174. (Contributed by SN, 30-Apr-2024.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘ ∈ β„‚ (𝐴 + 𝑏) = 0)
 
Theoremsn-negex2 41291* Proof of cnegex2 11396 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ β„‚ β†’ βˆƒπ‘ ∈ β„‚ (𝑏 + 𝐴) = 0)
 
Theoremsn-addcand 41292 addcand 11417 without ax-mulcom 11174. Note how the proof is almost identical to addcan 11398. (Contributed by SN, 5-May-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐡) = (𝐴 + 𝐢) ↔ 𝐡 = 𝐢))
 
Theoremsn-addrid 41293 addrid 11394 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ β„‚ β†’ (𝐴 + 0) = 𝐴)
 
Theoremsn-addcan2d 41294 addcan2d 11418 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    β‡’   (πœ‘ β†’ ((𝐴 + 𝐢) = (𝐡 + 𝐢) ↔ 𝐴 = 𝐡))
 
Theoremreixi 41295 ixi 11843 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
(i Β· i) = (0 βˆ’β„ 1)
 
Theoremrei4 41296 i4 14168 without ax-mulcom 11174. (Contributed by SN, 27-May-2024.)
((i Β· i) Β· (i Β· i)) = 1
 
Theoremsn-addid0 41297 A number that sums to itself is zero. Compare addid0 11633, readdridaddlidd 41178. (Contributed by SN, 5-May-2024.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ (𝐴 + 𝐴) = 𝐴)    β‡’   (πœ‘ β†’ 𝐴 = 0)
 
Theoremsn-mul01 41298 mul01 11393 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ β„‚ β†’ (𝐴 Β· 0) = 0)
 
Theoremsn-subeu 41299* negeu 11450 without ax-mulcom 11174 and complex number version of resubeu 41250. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ βˆƒ!π‘₯ ∈ β„‚ (𝐴 + π‘₯) = 𝐡)
 
Theoremsn-subcl 41300 subcl 11459 without ax-mulcom 11174. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 βˆ’ 𝐡) ∈ β„‚)
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