P. 429 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42801-42900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	renegeulem 42801*	Lemma for renegeu 42802 and similar. Remove a change in bound variables from renegeulemv 42800. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → ∃𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)    ⇒   ⊢ (𝜑 → ∃!𝑦 ∈ ℝ (𝐵 + 𝑦) = 𝐴)
Theorem	renegeu 42802*	Existential uniqueness of real negatives. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 0)
Theorem	rernegcl 42803	Closure law for negative reals. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ 𝐴) ∈ ℝ)
Theorem	renegadd 42804	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 42805	Addition of a real number and its negative. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 + (0 −ℝ 𝐴)) = 0)
Theorem	reneg0addlid 42806	Negative zero is a left additive identity. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 0) + 𝐴) = 𝐴)
Theorem	resubeulem1 42807	Lemma for resubeu 42809. 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 42808	Lemma for resubeu 42809. 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 42809*	Existential uniqueness of real differences. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐴 + 𝑥) = 𝐵)
Theorem	rersubcl 42810	Closure for real subtraction. Based on subcl 11392. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ)
Theorem	resubadd 42811	Relation between real subtraction and addition. Based on subadd 11396. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) = 𝐶 ↔ (𝐵 + 𝐶) = 𝐴))
Theorem	resubaddd 42812	Relationship between subtraction and addition. Based on subaddd 11523. (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 42813	Real subtraction is an operation on the real numbers. Based on subf 11395. (Contributed by Steven Nguyen, 7-Jan-2023.)
⊢ −ℝ :(ℝ × ℝ)⟶ℝ
Theorem	repncan2 42814	Addition and subtraction of equals. Compare pncan2 11400. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ 𝐴) = 𝐵)
Theorem	repncan3 42815	Addition and subtraction of equals. Based on pncan3 11401. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵)
Theorem	readdsub 42816	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 42817	Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11561 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 42818	Subtraction from both sides of 'less than'. Compare ltsub1 11646. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 ↔ (𝐴 −ℝ 𝐶) < (𝐵 −ℝ 𝐶)))
Theorem	reltsubadd2 42819	'Less than' relationship between addition and subtraction. Compare ltsubadd2 11621. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) < 𝐶 ↔ 𝐴 < (𝐵 + 𝐶)))
Theorem	resubcan2 42820	Cancellation law for real subtraction. Compare subcan2 11419. (Contributed by Steven Nguyen, 8-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐶) = (𝐵 −ℝ 𝐶) ↔ 𝐴 = 𝐵))
Theorem	resubsub4 42821	Law for double subtraction. Compare subsub4 11427. (Contributed by Steven Nguyen, 14-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) −ℝ 𝐶) = (𝐴 −ℝ (𝐵 + 𝐶)))
Theorem	rennncan2 42822	Cancellation law for real subtraction. Compare nnncan2 11431. (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 42823	Cancellation law for real subtraction. Compare npncan3 11432. (Contributed by Steven Nguyen, 28-Jan-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 −ℝ 𝐵) + (𝐶 −ℝ 𝐴)) = (𝐶 −ℝ 𝐵))
Theorem	repnpcan 42824	Cancellation law for addition and real subtraction. Compare pnpcan 11433. (Contributed by Steven Nguyen, 19-May-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐵) −ℝ (𝐴 + 𝐶)) = (𝐵 −ℝ 𝐶))
Theorem	reppncan 42825	Cancellation law for mixed addition and real subtraction. Compare ppncan 11436. (Contributed by SN, 3-Sep-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) + (𝐵 −ℝ 𝐶)) = (𝐴 + 𝐵))
Theorem	resubidaddlidlem 42826	Lemma for resubidaddlid 42827. A special case of npncan 11415. (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 42827	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 42828	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 42829	Equality of two left-additive identities. See resubidaddlid 42827. Uses ax-i2m1 11106. (Contributed by SN, 25-Dec-2023.)
⊢ (1 −ℝ 1) = (0 −ℝ 0)
Theorem	sn-00idlem1 42830	Lemma for sn-00id 42833. (Contributed by SN, 25-Dec-2023.)
⊢ (𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴))
Theorem	sn-00idlem2 42831	Lemma for sn-00id 42833. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1)
Theorem	sn-00idlem3 42832	Lemma for sn-00id 42833. (Contributed by SN, 25-Dec-2023.)
⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0)
Theorem	sn-00id 42833	00id 11321 proven without ax-mulcom 11102 but using ax-1ne0 11107. (Though note that the current version of 00id 11321 can be changed to avoid ax-icn 11097, ax-addcl 11098, ax-mulcl 11100, ax-i2m1 11106, ax-cnre 11111. Most of this is by using 0cnALT3 42692 instead of 0cn 11136). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.)
⊢ (0 + 0) = 0
Theorem	re0m0e0 42834	Real number version of 0m0e0 12296 proven without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (0 −ℝ 0) = 0
Theorem	readdlid 42835	Real number version of addlid 11329. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴)
Theorem	sn-addlid 42836	addlid 11329 without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴)
Theorem	remul02 42837	Real number version of mul02 11324 proven without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0)
Theorem	sn-0ne2 42838	0ne2 12383 without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ 0 ≠ 2
Theorem	remul01 42839	Real number version of mul01 11325 proven without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
Theorem	sn-remul0ord 42840	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 42841	Subtraction of a real number from itself (compare subid 11413). (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
Theorem	readdrid 42842	Real number version of addrid 11326 without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
Theorem	resubid1 42843	Real number version of subid1 11414 without ax-mulcom 11102. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
Theorem	renegneg 42844	A real number is equal to the negative of its negative. Compare negneg 11444. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 42845	Commuted version of readdcan 11320 without ax-mulcom 11102. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 42846	Commuted version of renegid 42805. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 42847	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 42848	Proof of it0e0 12400 without ax-mulcom 11102. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11111 and renegid2 42846. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 42849*	A combination of cnegex 11327 and cnegex2 11328, this proof takes cnre 11141 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 42850*	Proof of cnegex 11327 without ax-mulcom 11102. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 42851*	Proof of cnegex2 11328 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 42852	addcand 11349 without ax-mulcom 11102. Note how the proof is almost identical to addcan 11330. (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 42853	addrid 11326 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 42854	addcan2d 11350 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 42855	ixi 11779 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 42856	i4 14166 without ax-mulcom 11102. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 42857	A number that sums to itself is zero. Compare addid0 11569, readdridaddlidd 42696. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 42858	mul01 11325 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	sn-subeu 42859*	negeu 11383 without ax-mulcom 11102 and complex number version of resubeu 42809. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	sn-subcl 42860	subcl 11392 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 42861	subf 11395 without ax-mulcom 11102. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 42862	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 42863	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 42864	A number commutes with its additive inverse. Compare remulinvcom 42865. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 42865	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11102. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 42866	Commuted version of ax-1rid 11108 without ax-mulcom 11102. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 42867	Lemma for sn-mullid 42868 and sn-it1ei 42869. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 42868	mullid 11143 without ax-mulcom 11102. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 42869	it1ei 42748 without ax-mulcom 11102. (See sn-mullid 42868 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 42870	The multiplicative inverse of i (per i4 14166) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 42871	Commuted version of remulcan2d 42695 without ax-mulcom 11102. (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 42872	Real number division.
class /ℝ
Definition	df-rediv 42873*	Define division between real numbers. This operator saves ax-mulcom 11102 over df-div 11808 in certain situations. (Contributed by SN, 25-Nov-2025.)
⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
Theorem	redivvald 42874*	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 42875*	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 42876	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 42877	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 42878	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 42879	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 42880	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 42881	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 42882	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 42883	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 42884	A number divided by 1 is itself. (Contributed by SN, 2-Apr-2026.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    ⇒   ⊢ (𝜑 → (𝐴 /ℝ 1) = 𝐴)
Theorem	sn-rediv0d 42885	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 42886	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 42887	Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
Theorem	rerecne0d 42888	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 42889	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 42890	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 42891	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 42892	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 42893	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 42894	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 42895	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 42896	Lemma for sn-mul02 42897. Commuted version of sn-it0e0 42848. (Contributed by SN, 30-Jun-2024.)
⊢ (0 · i) = 0
Theorem	sn-mul02 42897	mul02 11324 without ax-mulcom 11102. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11102 for an outline. (Contributed by SN, 30-Jun-2024.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	sn-ltaddpos 42898	ltaddpos 11640 without ax-mulcom 11102. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	sn-ltaddneg 42899	ltaddneg 11362 without ax-mulcom 11102. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	reposdif 42900	Comparison of two numbers whose difference is positive. Compare posdif 11643. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴)))