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Theorem List for Metamath Proof Explorer - 40801-40900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgcdnn0id 40801 The gcd of a nonnegative integer and itself is the integer. (Contributed by SN, 25-Aug-2024.)
(𝑁 ∈ ℕ0 → (𝑁 gcd 𝑁) = 𝑁)
 
Theoremgcdle1d 40802 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 40803 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 40804 Deduction associated with dvdsexpim 40800. (Contributed by SN, 21-Aug-2024.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴𝐵)       (𝜑 → (𝐴𝑁) ∥ (𝐵𝑁))
 
Theoremnn0rppwr 40805 If 𝐴 and 𝐵 are relatively prime, then so are 𝐴𝑁 and 𝐵𝑁. rppwr 16440 extended to nonnegative integers. Less general than rpexp12i 16600. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵) = 1 → ((𝐴𝑁) gcd (𝐵𝑁)) = 1))
 
Theoremexpgcd 40806 Exponentiation distributes over GCD. sqgcd 16441 extended to nonnegative exponents. (Contributed by Steven Nguyen, 4-Apr-2023.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremnn0expgcd 40807 Exponentiation distributes over GCD. nn0gcdsq 16627 extended to nonnegative exponents. expgcd 40806 extended to nonnegative bases. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremzexpgcd 40808 Exponentiation distributes over GCD. zgcdsq 16628 extended to nonnegative exponents. nn0expgcd 40807 extended to integer bases by symmetry. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 gcd 𝐵)↑𝑁) = ((𝐴𝑁) gcd (𝐵𝑁)))
 
Theoremnumdenexp 40809 numdensq 16629 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → ((numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁) ∧ (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁)))
 
Theoremnumexp 40810 numsq 16630 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (numer‘(𝐴𝑁)) = ((numer‘𝐴)↑𝑁))
 
Theoremdenexp 40811 densq 16631 extended to nonnegative exponents. (Contributed by Steven Nguyen, 5-Apr-2023.)
((𝐴 ∈ ℚ ∧ 𝑁 ∈ ℕ0) → (denom‘(𝐴𝑁)) = ((denom‘𝐴)↑𝑁))
 
Theoremdvdsexpnn 40812 dvdssqlem 16442 generalized to positive integer exponents. (Contributed by SN, 20-Aug-2024.)
((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremdvdsexpnn0 40813 dvdsexpnn 40812 generalized to include zero bases. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremdvdsexpb 40814 dvdssq 16443 generalized to positive integer exponents. (Contributed by SN, 15-Sep-2024.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴𝐵 ↔ (𝐴𝑁) ∥ (𝐵𝑁)))
 
Theoremposqsqznn 40815 When a positive rational squared is an integer, the rational is a positive integer. zsqrtelqelz 16633 with all terms squared and positive. (Contributed by SN, 23-Aug-2024.)
(𝜑 → (𝐴↑2) ∈ ℤ)    &   (𝜑𝐴 ∈ ℚ)    &   (𝜑 → 0 < 𝐴)       (𝜑𝐴 ∈ ℕ)
 
Theoremcxpgt0d 40816 A positive real raised to a real power is positive. (Contributed by SN, 6-Apr-2023.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝑁 ∈ ℝ)       (𝜑 → 0 < (𝐴𝑐𝑁))
 
Theoremzrtelqelz 40817 zsqrtelqelz 16633 generalized to positive integer roots. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℚ) → (𝐴𝑐(1 / 𝑁)) ∈ ℤ)
 
Theoremzrtdvds 40818 A positive integer root divides its integer. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴𝑐(1 / 𝑁)) ∈ ℕ) → (𝐴𝑐(1 / 𝑁)) ∥ 𝐴)
 
Theoremrtprmirr 40819 The root of a prime number is irrational. (Contributed by Steven Nguyen, 6-Apr-2023.)
((𝑃 ∈ ℙ ∧ 𝑁 ∈ (ℤ‘2)) → (𝑃𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ))
 
21.26.6  Real subtraction
 
Syntaxcresub 40820 Real number subtraction.
class
 
Definitiondf-resub 40821* Define subtraction between real numbers. This operator saves a few axioms over df-sub 11387 in certain situations. Theorem resubval 40822 shows its value, resubadd 40834 relates it to addition, and rersubcl 40833 proves its closure. It is the restriction of df-sub 11387 to the reals: subresre 40885. (Contributed by Steven Nguyen, 7-Jan-2023.)
= (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
 
Theoremresubval 40822* 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 40823* Lemma for renegeu 40825 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 40824* Lemma for renegeu 40825 and similar. Remove a change in bound variables from renegeulemv 40823. (Contributed by Steven Nguyen, 28-Jan-2023.)
(𝜑𝐵 ∈ ℝ)    &   (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)       (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
 
Theoremrenegeu 40825* Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
 
Theoremrernegcl 40826 Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (0 − 𝐴) ∈ ℝ)
 
Theoremrenegadd 40827 Relationship between real negation and addition. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((0 − 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 0))
 
Theoremrenegid 40828 Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (𝐴 + (0 − 𝐴)) = 0)
 
Theoremreneg0addid2 40829 Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → ((0 − 0) + 𝐴) = 𝐴)
 
Theoremresubeulem1 40830 Lemma for resubeu 40832. A value which when added to zero, results in negative zero. (Contributed by Steven Nguyen, 7-Jan-2023.)
(𝐴 ∈ ℝ → (0 + (0 − (0 + 0))) = (0 − 0))
 
Theoremresubeulem2 40831 Lemma for resubeu 40832. A value which when added to 𝐴, results in 𝐵. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + ((0 − 𝐴) + ((0 − (0 + 0)) + 𝐵))) = 𝐵)
 
Theoremresubeu 40832* Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
 
Theoremrersubcl 40833 Closure for real subtraction. Based on subcl 11400. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) ∈ ℝ)
 
Theoremresubadd 40834 Relation between real subtraction and addition. Based on subadd 11404. (Contributed by Steven Nguyen, 7-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubaddd 40835 Relationship between subtraction and addition. Based on subaddd 11530. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)       (𝜑 → ((𝐴 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
 
Theoremresubf 40836 Real subtraction is an operation on the real numbers. Based on subf 11403. (Contributed by Steven Nguyen, 7-Jan-2023.)
:(ℝ × ℝ)⟶ℝ
 
Theoremrepncan2 40837 Addition and subtraction of equals. Compare pncan2 11408. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
 
Theoremrepncan3 40838 Addition and subtraction of equals. Based on pncan3 11409. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 𝐴)) = 𝐵)
 
Theoremreaddsub 40839 Law for addition and subtraction. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 𝐶) + 𝐵))
 
Theoremreladdrsub 40840 Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11566 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 + 𝐵) = 𝐶)       (𝜑𝐵 = (𝐶 𝐴))
 
Theoremreltsub1 40841 Subtraction from both sides of 'less than'. Compare ltsub1 11651. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 𝐶) < (𝐵 𝐶)))
 
Theoremreltsubadd2 40842 'Less than' relationship between addition and subtraction. Compare ltsubadd2 11626. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) < 𝐶𝐴 < (𝐵 + 𝐶)))
 
Theoremresubcan2 40843 Cancellation law for real subtraction. Compare subcan2 11426. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) = (𝐵 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremresubsub4 40844 Law for double subtraction. Compare subsub4 11434. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) − 𝐶) = (𝐴 (𝐵 + 𝐶)))
 
Theoremrennncan2 40845 Cancellation law for real subtraction. Compare nnncan2 11438. (Contributed by Steven Nguyen, 14-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐶) − (𝐵 𝐶)) = (𝐴 𝐵))
 
Theoremrenpncan3 40846 Cancellation law for real subtraction. Compare npncan3 11439. (Contributed by Steven Nguyen, 28-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 𝐵) + (𝐶 𝐴)) = (𝐶 𝐵))
 
Theoremrepnpcan 40847 Cancellation law for addition and real subtraction. Compare pnpcan 11440. (Contributed by Steven Nguyen, 19-May-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) − (𝐴 + 𝐶)) = (𝐵 𝐶))
 
Theoremreppncan 40848 Cancellation law for mixed addition and real subtraction. Compare ppncan 11443. (Contributed by SN, 3-Sep-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 𝐶)) = (𝐴 + 𝐵))
 
Theoremresubidaddid1lem 40849 Lemma for resubidaddid1 40850. A special case of npncan 11422. (Contributed by Steven Nguyen, 8-Jan-2023.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑 → (𝐴 𝐵) = (𝐵 𝐶))       (𝜑 → ((𝐴 𝐵) + (𝐵 𝐶)) = (𝐴 𝐶))
 
Theoremresubidaddid1 40850 Any real number subtracted from itself forms a left additive identity. (Contributed by Steven Nguyen, 8-Jan-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 𝐴) + 𝐵) = 𝐵)
 
Theoremresubdi 40851 Distribution of multiplication over real subtraction. (Contributed by Steven Nguyen, 3-Jun-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 · (𝐵 𝐶)) = ((𝐴 · 𝐵) − (𝐴 · 𝐶)))
 
Theoremre1m1e0m0 40852 Equality of two left-additive identities. See resubidaddid1 40850. Uses ax-i2m1 11119. (Contributed by SN, 25-Dec-2023.)
(1 − 1) = (0 − 0)
 
Theoremsn-00idlem1 40853 Lemma for sn-00id 40856. (Contributed by SN, 25-Dec-2023.)
(𝐴 ∈ ℝ → (𝐴 · (0 − 0)) = (𝐴 𝐴))
 
Theoremsn-00idlem2 40854 Lemma for sn-00id 40856. (Contributed by SN, 25-Dec-2023.)
((0 − 0) ≠ 0 → (0 − 0) = 1)
 
Theoremsn-00idlem3 40855 Lemma for sn-00id 40856. (Contributed by SN, 25-Dec-2023.)
((0 − 0) = 1 → (0 + 0) = 0)
 
Theoremsn-00id 40856 00id 11330 proven without ax-mulcom 11115 but using ax-1ne0 11120. (Though note that the current version of 00id 11330 can be changed to avoid ax-icn 11110, ax-addcl 11111, ax-mulcl 11113, ax-i2m1 11119, ax-cnre 11124. Most of this is by using 0cnALT3 40762 instead of 0cn 11147). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
(0 + 0) = 0
 
Theoremre0m0e0 40857 Real number version of 0m0e0 12273 proven without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(0 − 0) = 0
 
Theoremreaddid2 40858 Real number version of addid2 11338. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
 
Theoremsn-addid2 40859 addid2 11338 without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
 
Theoremremul02 40860 Real number version of mul02 11333 proven without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (0 · 𝐴) = 0)
 
Theoremsn-0ne2 40861 0ne2 12360 without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
0 ≠ 2
 
Theoremremul01 40862 Real number version of mul01 11334 proven without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 · 0) = 0)
 
Theoremresubid 40863 Subtraction of a real number from itself (compare subid 11420). (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 𝐴) = 0)
 
Theoremreaddid1 40864 Real number version of addid1 11335 without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
 
Theoremresubid1 40865 Real number version of subid1 11421 without ax-mulcom 11115. (Contributed by SN, 23-Jan-2024.)
(𝐴 ∈ ℝ → (𝐴 0) = 𝐴)
 
Theoremrenegneg 40866 A real number is equal to the negative of its negative. Compare negneg 11451. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 − (0 − 𝐴)) = 𝐴)
 
Theoremreaddcan2 40867 Commuted version of readdcan 11329 without ax-mulcom 11115. (Contributed by SN, 21-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremrenegid2 40868 Commuted version of renegid 40828. (Contributed by SN, 4-May-2024.)
(𝐴 ∈ ℝ → ((0 − 𝐴) + 𝐴) = 0)
 
Theoremremulneg2d 40869 Product with negative is negative of product. (Contributed by SN, 25-Jan-2025.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)       (𝜑 → (𝐴 · (0 − 𝐵)) = (0 − (𝐴 · 𝐵)))
 
Theoremsn-it0e0 40870 Proof of it0e0 12375 without ax-mulcom 11115. Informally, a real number times 0 is 0, and 𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11124 and renegid2 40868. (Contributed by SN, 30-Apr-2024.)
(i · 0) = 0
 
Theoremsn-negex12 40871* A combination of cnegex 11336 and cnegex2 11337, this proof takes cnre 11152 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))
 
Theoremsn-negex 40872* Proof of cnegex 11336 without ax-mulcom 11115. (Contributed by SN, 30-Apr-2024.)
(𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
 
Theoremsn-negex2 40873* Proof of cnegex2 11337 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
 
Theoremsn-addcand 40874 addcand 11358 without ax-mulcom 11115. Note how the proof is almost identical to addcan 11339. (Contributed by SN, 5-May-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐵) = (𝐴 + 𝐶) ↔ 𝐵 = 𝐶))
 
Theoremsn-addid1 40875 addid1 11335 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
 
Theoremsn-addcan2d 40876 addcan2d 11359 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)       (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
 
Theoremreixi 40877 ixi 11784 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
(i · i) = (0 − 1)
 
Theoremrei4 40878 i4 14108 without ax-mulcom 11115. (Contributed by SN, 27-May-2024.)
((i · i) · (i · i)) = 1
 
Theoremsn-addid0 40879 A number that sums to itself is zero. Compare addid0 11574, readdid1addid2d 40766. (Contributed by SN, 5-May-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐴) = 𝐴)       (𝜑𝐴 = 0)
 
Theoremsn-mul01 40880 mul01 11334 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
(𝐴 ∈ ℂ → (𝐴 · 0) = 0)
 
Theoremsn-subeu 40881* negeu 11391 without ax-mulcom 11115 and complex number version of resubeu 40832. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
 
Theoremsn-subcl 40882 subcl 11400 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) ∈ ℂ)
 
Theoremsn-subf 40883 subf 11403 without ax-mulcom 11115. (Contributed by SN, 5-May-2024.)
− :(ℂ × ℂ)⟶ℂ
 
Theoremresubeqsub 40884 Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 𝐵) = (𝐴𝐵))
 
Theoremsubresre 40885 Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
= ( − ↾ (ℝ × ℝ))
 
Theoremaddinvcom 40886 A number commutes with its additive inverse. Compare remulinvcom 40887. (Contributed by SN, 5-May-2024.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑 → (𝐴 + 𝐵) = 0)       (𝜑 → (𝐵 + 𝐴) = 0)
 
Theoremremulinvcom 40887 A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11115. (Contributed by SN, 5-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑 → (𝐴 · 𝐵) = 1)       (𝜑 → (𝐵 · 𝐴) = 1)
 
Theoremremulid2 40888 Commuted version of ax-1rid 11121 without ax-mulcom 11115. (Contributed by SN, 5-Feb-2024.)
(𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
 
Theoremsn-1ticom 40889 Lemma for sn-mulid2 40890 and it1ei 40891. (Contributed by SN, 27-May-2024.)
(1 · i) = (i · 1)
 
Theoremsn-mulid2 40890 mulid2 11154 without ax-mulcom 11115. (Contributed by SN, 27-May-2024.)
(𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
 
Theoremit1ei 40891 1 is a multiplicative identity for i (see sn-mulid2 40890 for commuted version). (Contributed by SN, 1-Jun-2024.)
(i · 1) = i
 
Theoremipiiie0 40892 The multiplicative inverse of i (per i4 14108) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
(i + (i · (i · i))) = 0
 
Theoremremulcand 40893 Commuted version of remulcan2d 40765 without ax-mulcom 11115. (Contributed by SN, 21-Feb-2024.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐶 ≠ 0)       (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
 
Theoremsn-0tie0 40894 Lemma for sn-mul02 40895. Commuted version of sn-it0e0 40870. (Contributed by SN, 30-Jun-2024.)
(0 · i) = 0
 
Theoremsn-mul02 40895 mul02 11333 without ax-mulcom 11115. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11115 for an outline. (Contributed by SN, 30-Jun-2024.)
(𝐴 ∈ ℂ → (0 · 𝐴) = 0)
 
Theoremsn-ltaddpos 40896 ltaddpos 11645 without ax-mulcom 11115. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴𝐵 < (𝐵 + 𝐴)))
 
Theoremsn-ltaddneg 40897 ltaddneg 11370 without ax-mulcom 11115. (Contributed by SN, 25-Jan-2025.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
 
Theoremreposdif 40898 Comparison of two numbers whose difference is positive. Compare posdif 11648. (Contributed by SN, 13-Feb-2024.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 𝐴)))
 
Theoremrelt0neg1 40899 Comparison of a real and its negative to zero. Compare lt0neg1 11661. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 − 𝐴)))
 
Theoremrelt0neg2 40900 Comparison of a real and its negative to zero. Compare lt0neg2 11662. (Contributed by SN, 13-Feb-2024.)
(𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 − 𝐴) < 0))
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