P. 436 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43501-43600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	reneg0addlid 43501	Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) + 𝐴) = 𝐴)
Theorem	resubeulem1 43502	Lemma for resubeu 43504. 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 43503	Lemma for resubeu 43504. 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 43504*	Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
Theorem	rersubcl 43505	Closure for real subtraction. Based on subcl 11557. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ)
Theorem	resubadd 43506	Relation between real subtraction and addition. Based on subadd 11561. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	resubaddd 43507	Relationship between subtraction and addition. Based on subaddd 11688. (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 43508	Real subtraction is an operation on the real numbers. Based on subf 11560. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ :(ℝ × ℝ)⟶ℝ
Theorem	repncan2 43509	Addition and subtraction of equals. Compare pncan2 11565. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ 𝐴) = 𝐵)
Theorem	repncan3 43510	Addition and subtraction of equals. Based on pncan3 11566. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵)
Theorem	readdsub 43511	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 43512	Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11726 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 43513	Subtraction from both sides of 'less than'. Compare ltsub1 11813. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 −ℝ 𝐶) < (𝐵 −ℝ 𝐶)))
Theorem	reltsubadd2 43514	'Less than' relationship between addition and subtraction. Compare ltsubadd2 11788. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	resubcan2 43515	Cancellation law for real subtraction. Compare subcan2 11584. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	resubsub4 43516	Law for double subtraction. Compare subsub4 11592. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶)))
Theorem	rennncan2 43517	Cancellation law for real subtraction. Compare nnncan2 11596. (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 43518	Cancellation law for real subtraction. Compare npncan3 11597. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵))
Theorem	repnpcan 43519	Cancellation law for addition and real subtraction. Compare pnpcan 11598. (Contributed by Steven Nguyen, 19-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ (𝐴 + 𝐶)) = (𝐵 −ℝ 𝐶))
Theorem	reppncan 43520	Cancellation law for mixed addition and real subtraction. Compare ppncan 11601. (Contributed by SN, 3-Sep-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 −ℝ 𝐶)) = (𝐴 + 𝐵))
Theorem	resubidaddlidlem 43521	Lemma for resubidaddlid 43522. A special case of npncan 11580. (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 43522	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 43523	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 43524	Equality of two left-additive identities. See resubidaddlid 43522. Uses ax-i2m1 11269. (Contributed by SN, 25-Dec-2023.)
⊢ (1 −ℝ 1) = (0 −ℝ 0)
Theorem	sn-00idlem1 43525	Lemma for sn-00id 43528. (Contributed by SN, 25-Dec-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴))
Theorem	sn-00idlem2 43526	Lemma for sn-00id 43528. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1)
Theorem	sn-00idlem3 43527	Lemma for sn-00id 43528. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0)
Theorem	sn-00id 43528	00id 11486 proven without ax-mulcom 11265 but using ax-1ne0 11270. (Though note that the current version of 00id 11486 can be changed to avoid ax-icn 11260, ax-addcl 11261, ax-mulcl 11263, ax-i2m1 11269, ax-cnre 11274. Most of this is by using 0cnALT3 43380 instead of 0cn 11299). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
⊢ (0 + 0) = 0
Theorem	re0m0e0 43529	Real number version of 0m0e0 12462 proven without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (0 −ℝ 0) = 0
Theorem	readdlid 43530	Real number version of addlid 11494. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
Theorem	sn-addlid 43531	addlid 11494 without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	remul02 43532	Real number version of mul02 11489 proven without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
Theorem	sn-0ne2 43533	0ne2 12553 without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ 0 ≠ 2
Theorem	remul01 43534	Real number version of mul01 11490 proven without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
Theorem	sn-remul0ord 43535	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 43536	Subtraction of a real number from itself (compare subid 11578). (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
Theorem	readdrid 43537	Real number version of addrid 11491 without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
Theorem	resubid1 43538	Real number version of subid1 11579 without ax-mulcom 11265. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
Theorem	renegneg 43539	A real number is equal to the negative of its negative. Compare negneg 11609. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 43540	Commuted version of readdcan 11485 without ax-mulcom 11265. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 43541	Commuted version of renegid 43500. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 43542	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 43543	Proof of it0e0 12570 without ax-mulcom 11265. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11274 and renegid2 43541. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 43544*	A combination of cnegex 11492 and cnegex2 11493, this proof takes cnre 11306 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 43545*	Proof of cnegex 11492 without ax-mulcom 11265. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 43546*	Proof of cnegex2 11493 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 43547	addcand 11514 without ax-mulcom 11265. Note how the proof is almost identical to addcan 11495. (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 43548	addrid 11491 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 43549	addcan2d 11515 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 43550	ixi 11946 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 43551	i4 14349 without ax-mulcom 11265. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 43552	A number that sums to itself is zero. Compare addid0 11736, readdridaddlidd 43384. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 43553	mul01 11490 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	sn-subeu 43554*	negeu 11548 without ax-mulcom 11265 and complex number version of resubeu 43504. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	sn-subcl 43555	subcl 11557 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 43556	subf 11560 without ax-mulcom 11265. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 43557	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 43558	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 43559	A number commutes with its additive inverse. Compare remulinvcom 43560. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 43560	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11265. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 43561	Commuted version of ax-1rid 11271 without ax-mulcom 11265. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 43562	Lemma for sn-mullid 43563 and sn-it1ei 43564. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 43563	mullid 11308 without ax-mulcom 11265. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 43564	it1ei 43449 without ax-mulcom 11265. (See sn-mullid 43563 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 43565	The multiplicative inverse of i (per i4 14349) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 43566	Commuted version of remulcan2d 43383 without ax-mulcom 11265. (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 43567	Real number division.
class /ℝ
Definition	df-rediv 43568*	Define division between real numbers. This operator saves ax-mulcom 11265 over df-div 11975 in certain situations. (Contributed by SN, 25-Nov-2025.)
⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
Theorem	redivvald 43569*	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 43570*	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 43571	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 43572	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	redivmul2d 43573	Relationship between division and multiplication. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 /ℝ 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵)))
Theorem	redivcan2d 43574	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 43575	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	rediveq0d 43576	A ratio is zero iff the numerator is zero. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 /ℝ 𝐵) = 0 ↔ 𝐴 = 0))
Theorem	redivne0bd 43577	The ratio of nonzero numbers is nonzero. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 ≠ 0 ↔ (𝐴 /ℝ 𝐵) ≠ 0))
Theorem	rediveq1d 43578	Equality in terms of unit ratio. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 /ℝ 𝐵) = 1 ↔ 𝐴 = 𝐵))
Theorem	sn-rediv1d 43579	A number divided by 1 is itself. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 1) = 𝐴)
Theorem	sn-rediv0d 43580	Division into zero is zero. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (0 /ℝ 𝐴) = 0)
Theorem	sn-redividd 43581	A number divided by itself is 1. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 𝐴) = 1)
Theorem	sn-rereccld 43582	Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
Theorem	rerecne0d 43583	The reciprocal of a nonzero number is nonzero. (Contributed by SN, 4-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ≠ 0)
Theorem	rerecidd 43584	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)
Theorem	rerecid2d 43585	Multiplication of a number and its reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → ((1 /ℝ 𝐴) · 𝐴) = 1)
Theorem	rerecrecd 43586	A number is equal to the reciprocal of its reciprocal. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ (1 /ℝ 𝐴)) = 𝐴)
Theorem	redivrec2d 43587	Relationship between division and reciprocal. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ≠ 0)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 𝐵) = ((1 /ℝ 𝐵) · 𝐴))
Theorem	rediv23d 43588	A "commutative"/associative law for division. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 · 𝐵) /ℝ 𝐶) = ((𝐴 /ℝ 𝐶) · 𝐵))
Theorem	redivdird 43589	Distribution of division over addition. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐵) /ℝ 𝐶) = ((𝐴 /ℝ 𝐶) + (𝐵 /ℝ 𝐶)))
Theorem	rediv11d 43590	One-to-one relationship for division. (Contributed by SN, 9-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ≠ 0)    ⇒   ⊢ (𝜑 → ((𝐴 /ℝ 𝐶) = (𝐵 /ℝ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	sn-0tie0 43591	Lemma for sn-mul02 43592. Commuted version of sn-it0e0 43543. (Contributed by SN, 30-Jun-2024.)
⊢ (0 · i) = 0
Theorem	sn-mul02 43592	mul02 11489 without ax-mulcom 11265. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11265 for an outline. (Contributed by SN, 30-Jun-2024.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	sn-ltaddpos 43593	ltaddpos 11807 without ax-mulcom 11265. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	sn-ltaddneg 43594	ltaddneg 11527 without ax-mulcom 11265. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	reposdif 43595	Comparison of two numbers whose difference is positive. Compare posdif 11810. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴)))
Theorem	relt0neg1 43596	Comparison of a real and its negative to zero. Compare lt0neg1 11823. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴)))
Theorem	relt0neg2 43597	Comparison of a real and its negative to zero. Compare lt0neg2 11824. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0))
Theorem	sn-addlt0d 43598	The sum of negative numbers is negative. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    &   ⊢ (𝜑 → 𝐵 < 0)    ⇒   ⊢ (𝜑 → (𝐴 + 𝐵) < 0)
Theorem	sn-addgt0d 43599	The sum of positive numbers is positive. Proof of addgt0d 11892 without ax-mulcom 11265. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → 0 < (𝐴 + 𝐵))
Theorem	sn-nnne0 43600	nnne0 12373 without ax-mulcom 11265. (Contributed by SN, 25-Jan-2025.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)