P. 427 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42601-42700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sinpim 42601	Sine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
⊢ (𝐴 ∈ ℂ → (sin‘(π − 𝐴)) = (sin‘𝐴))
Theorem	cospim 42602	Cosine of a number subtracted from π. (Contributed by SN, 19-Nov-2025.)
⊢ (𝐴 ∈ ℂ → (cos‘(π − 𝐴)) = -(cos‘𝐴))
Theorem	tan3rdpi 42603	The tangent of π / 3 is √3. (Contributed by SN, 2-Sep-2025.)
⊢ (tan‘(π / 3)) = (√‘3)
Theorem	sin2t3rdpi 42604	The sine of 2 · (π / 3) is (√3) / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (sin‘(2 · (π / 3))) = ((√‘3) / 2)
Theorem	cos2t3rdpi 42605	The cosine of 2 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (cos‘(2 · (π / 3))) = -(1 / 2)
Theorem	sin4t3rdpi 42606	The sine of 4 · (π / 3) is -(√3) / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (sin‘(4 · (π / 3))) = -((√‘3) / 2)
Theorem	cos4t3rdpi 42607	The cosine of 4 · (π / 3) is -1 / 2. (Contributed by SN, 19-Nov-2025.)
⊢ (cos‘(4 · (π / 3))) = -(1 / 2)
Theorem	asin1half 42608	The arcsine of 1 / 2 is π / 6. (Contributed by SN, 31-Aug-2025.)
⊢ (arcsin‘(1 / 2)) = (π / 6)
Theorem	acos1half 42609	The arccosine of 1 / 2 is π / 3. (Contributed by SN, 31-Aug-2024.)
⊢ (arccos‘(1 / 2)) = (π / 3)
Theorem	dvun 42610	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 23026. (Contributed by SN, 30-Sep-2025.)
⊢ 𝐽 = (𝐾 ↾t 𝑆)    &   ⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝑆 ⊆ ℂ)    &   ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)    &   ⊢ (𝜑 → 𝐺:𝐵⟶ℂ)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑆)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵)) = ((int‘𝐽)‘(𝐴 ∪ 𝐵)))    ⇒   ⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺)))
Theorem	redvmptabs 42611*	The derivative of the absolute value, for real numbers. (Contributed by SN, 30-Sep-2025.)
⊢ 𝐷 = (ℝ ∖ {0})    ⇒   ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (abs‘𝑥))) = (𝑥 ∈ 𝐷 ↦ if(𝑥 < 0, -1, 1))
Theorem	readvrec2 42612*	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 (𝑥 ∈ 𝐷 ↦ ((log‘(𝑥↑2)) / 2))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
Theorem	readvrec 42613*	For real numbers, the antiderivative of 1/x is ln|x|. (Contributed by SN, 30-Sep-2025.)
⊢ 𝐷 = (ℝ ∖ {0})    ⇒   ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (1 / 𝑥))
Theorem	resuppsinopn 42614	The support of sin (df-supp 8103) restricted to the reals is an open set. (Contributed by SN, 7-Oct-2025.)
⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}    ⇒   ⊢ 𝐷 ∈ (topGen‘ran (,))
Theorem	readvcot 42615*	Real antiderivative of cotangent. (Contributed by SN, 7-Oct-2025.)
⊢ 𝐷 = {𝑦 ∈ ℝ ∣ (sin‘𝑦) ≠ 0}    ⇒   ⊢ (ℝ D (𝑥 ∈ 𝐷 ↦ (log‘(abs‘(sin‘𝑥))))) = (𝑥 ∈ 𝐷 ↦ ((cos‘𝑥) / (sin‘𝑥)))
21.30.5  Independence of ax-mulcom This section mainly concerns the independence of ax-mulcom 11090, which is the only real and complex number axiom whose independence is open ( https://us.metamath.org/mpeuni/mmcomplex.html 11090). In particular, this is a combination of attempts to prove more and more properties of real and complex numbers without ax-mulcom 11090. Completing this direction would show that ax-mulcom 11090 is not independent. Alternatively, one could search for a model satisfying all axioms except ax-mulcom 11090, thus showing it is independent. A few models satisfying non-commutativity which only violate one other axiom are provided at https://gist.github.com/icecream17/933f95d820e0b8f1cab0d4293b68eaf9 11090. I conjecture that if it is possible to prove ax-mulcom 11090 from the other axioms, then all the other axioms are needed. In abstract terms, the symbol ℝ would have to correspond to an infinite non-commutative left-near-field with a Dedekind-complete order compatible with its ring operations. (Note: https://en.wikipedia.org/wiki/Near-field_(mathematics) 11090 does not require commutativity despite having "field" in the name.) Needless to say, this is a very undeveloped area of math. In addition, such a structure for ℝ would have to, together with the structure for the symbol ℂ, satisfy ax-resscn 11083, ax-icn 11085, ax-i2m1 11094, and most crucially ax-cnre 11099. None of the theorems in this section should be moved to main. If there is a naming conflict, feel free to add the prefix "sn-".
Syntax	cresub 42616	Real number subtraction.
class −ℝ
Definition	df-resub 42617*	Define subtraction between real numbers. This operator saves a few axioms over df-sub 11366 in certain situations. Theorem resubval 42618 shows its value, resubadd 42630 relates it to addition, and rersubcl 42629 proves its closure. It is the restriction of df-sub 11366 to the reals: subresre 42682. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
Theorem	resubval 42618*	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 42619*	Lemma for renegeu 42621 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 42620*	Lemma for renegeu 42621 and similar. Remove a change in bound variables from renegeulemv 42619. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
Theorem	renegeu 42621*	Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	rernegcl 42622	Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ)
Theorem	renegadd 42623	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 42624	Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0)
Theorem	reneg0addlid 42625	Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) + 𝐴) = 𝐴)
Theorem	resubeulem1 42626	Lemma for resubeu 42628. 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 42627	Lemma for resubeu 42628. 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 42628*	Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
Theorem	rersubcl 42629	Closure for real subtraction. Based on subcl 11379. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ)
Theorem	resubadd 42630	Relation between real subtraction and addition. Based on subadd 11383. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	resubaddd 42631	Relationship between subtraction and addition. Based on subaddd 11510. (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 42632	Real subtraction is an operation on the real numbers. Based on subf 11382. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ :(ℝ × ℝ)⟶ℝ
Theorem	repncan2 42633	Addition and subtraction of equals. Compare pncan2 11387. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ 𝐴) = 𝐵)
Theorem	repncan3 42634	Addition and subtraction of equals. Based on pncan3 11388. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵)
Theorem	readdsub 42635	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 42636	Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11548 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 42637	Subtraction from both sides of 'less than'. Compare ltsub1 11633. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 −ℝ 𝐶) < (𝐵 −ℝ 𝐶)))
Theorem	reltsubadd2 42638	'Less than' relationship between addition and subtraction. Compare ltsubadd2 11608. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	resubcan2 42639	Cancellation law for real subtraction. Compare subcan2 11406. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	resubsub4 42640	Law for double subtraction. Compare subsub4 11414. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶)))
Theorem	rennncan2 42641	Cancellation law for real subtraction. Compare nnncan2 11418. (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 42642	Cancellation law for real subtraction. Compare npncan3 11419. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵))
Theorem	repnpcan 42643	Cancellation law for addition and real subtraction. Compare pnpcan 11420. (Contributed by Steven Nguyen, 19-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ (𝐴 + 𝐶)) = (𝐵 −ℝ 𝐶))
Theorem	reppncan 42644	Cancellation law for mixed addition and real subtraction. Compare ppncan 11423. (Contributed by SN, 3-Sep-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 −ℝ 𝐶)) = (𝐴 + 𝐵))
Theorem	resubidaddlidlem 42645	Lemma for resubidaddlid 42646. A special case of npncan 11402. (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 42646	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 42647	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 42648	Equality of two left-additive identities. See resubidaddlid 42646. Uses ax-i2m1 11094. (Contributed by SN, 25-Dec-2023.)
⊢ (1 −ℝ 1) = (0 −ℝ 0)
Theorem	sn-00idlem1 42649	Lemma for sn-00id 42652. (Contributed by SN, 25-Dec-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴))
Theorem	sn-00idlem2 42650	Lemma for sn-00id 42652. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1)
Theorem	sn-00idlem3 42651	Lemma for sn-00id 42652. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0)
Theorem	sn-00id 42652	00id 11308 proven without ax-mulcom 11090 but using ax-1ne0 11095. (Though note that the current version of 00id 11308 can be changed to avoid ax-icn 11085, ax-addcl 11086, ax-mulcl 11088, ax-i2m1 11094, ax-cnre 11099. Most of this is by using 0cnALT3 42504 instead of 0cn 11124). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
⊢ (0 + 0) = 0
Theorem	re0m0e0 42653	Real number version of 0m0e0 12260 proven without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (0 −ℝ 0) = 0
Theorem	readdlid 42654	Real number version of addlid 11316. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
Theorem	sn-addlid 42655	addlid 11316 without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	remul02 42656	Real number version of mul02 11311 proven without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
Theorem	sn-0ne2 42657	0ne2 12347 without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ 0 ≠ 2
Theorem	remul01 42658	Real number version of mul01 11312 proven without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
Theorem	sn-remul0ord 42659	A product is zero iff one of its factors are zero. (Contributed by SN, 24-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 0 ∨ 𝐵 = 0)))
Theorem	resubid 42660	Subtraction of a real number from itself (compare subid 11400). (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
Theorem	readdrid 42661	Real number version of addrid 11313 without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
Theorem	resubid1 42662	Real number version of subid1 11401 without ax-mulcom 11090. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
Theorem	renegneg 42663	A real number is equal to the negative of its negative. Compare negneg 11431. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 42664	Commuted version of readdcan 11307 without ax-mulcom 11090. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 42665	Commuted version of renegid 42624. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 42666	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 42667	Proof of it0e0 12364 without ax-mulcom 11090. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11099 and renegid2 42665. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 42668*	A combination of cnegex 11314 and cnegex2 11315, this proof takes cnre 11129 𝐴 = 𝑟 + i · 𝑠 and shows that i · -𝑠 + -𝑟 is both a left and right inverse. (Contributed by SN, 5-May-2024.) (Proof shortened by SN, 4-Jul-2025.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ ((𝐴 + 𝑏) = 0 ∧ (𝑏 + 𝐴) = 0))
Theorem	sn-negex 42669*	Proof of cnegex 11314 without ax-mulcom 11090. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 42670*	Proof of cnegex2 11315 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 42671	addcand 11336 without ax-mulcom 11090. Note how the proof is almost identical to addcan 11317. (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 42672	addrid 11313 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 42673	addcan2d 11337 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 42674	ixi 11766 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 42675	i4 14127 without ax-mulcom 11090. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 42676	A number that sums to itself is zero. Compare addid0 11556, readdridaddlidd 42509. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 42677	mul01 11312 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	sn-subeu 42678*	negeu 11370 without ax-mulcom 11090 and complex number version of resubeu 42628. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	sn-subcl 42679	subcl 11379 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 42680	subf 11382 without ax-mulcom 11090. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 42681	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 42682	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 42683	A number commutes with its additive inverse. Compare remulinvcom 42684. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 42684	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11090. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 42685	Commuted version of ax-1rid 11096 without ax-mulcom 11090. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 42686	Lemma for sn-mullid 42687 and sn-it1ei 42688. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 42687	mullid 11131 without ax-mulcom 11090. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 42688	it1ei 42567 without ax-mulcom 11090. (See sn-mullid 42687 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 42689	The multiplicative inverse of i (per i4 14127) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 42690	Commuted version of remulcan2d 42508 without ax-mulcom 11090. (Contributed by SN, 21-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐶 · 𝐴) = (𝐶 · 𝐵) ↔ 𝐴 = 𝐵))
Syntax	crediv 42691	Real number division.
class /ℝ
Definition	df-rediv 42692*	Define division between real numbers. This operator saves ax-mulcom 11090 over df-div 11795 in certain situations. (Contributed by SN, 25-Nov-2025.)
⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
Theorem	redivvald 42693*	Value of real division, which is the (unique) real 𝑥 such that (𝐵 · 𝑥) = 𝐴. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
Theorem	rediveud 42694*	Existential uniqueness of real quotients. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ∃!𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴)
Theorem	sn-redivcld 42695	Closure law for real division. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 𝐵) ∈ ℝ)
Theorem	redivmuld 42696	Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 /ℝ 𝐶) = 𝐵 ↔ (𝐶 · 𝐵) = 𝐴))
Theorem	redivcan2d 42697	A cancellation law for division. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐵 · (𝐴 /ℝ 𝐵)) = 𝐴)
Theorem	redivcan3d 42698	A cancellation law for division. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐵 · 𝐴) /ℝ 𝐵) = 𝐴)
Theorem	sn-rereccld 42699	Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
Theorem	rerecid 42700	Multiplication of a number and its reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 · (1 /ℝ 𝐴)) = 1)