P. 425 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42401-42500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	sn-0ne2 42401	0ne2 12395 without ax-mulcom 11139. (Contributed by SN, 23-Jan-2024.)
⊢ 0 ≠ 2
Theorem	remul01 42402	Real number version of mul01 11360 proven without ax-mulcom 11139. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0)
Theorem	sn-remul0ord 42403	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 42404	Subtraction of a real number from itself (compare subid 11448). (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0)
Theorem	readdrid 42405	Real number version of addrid 11361 without ax-mulcom 11139. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
Theorem	resubid1 42406	Real number version of subid1 11449 without ax-mulcom 11139. (Contributed by SN, 23-Jan-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴)
Theorem	renegneg 42407	A real number is equal to the negative of its negative. Compare negneg 11479. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 42408	Commuted version of readdcan 11355 without ax-mulcom 11139. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 42409	Commuted version of renegid 42368. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 42410	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 42411	Proof of it0e0 12412 without ax-mulcom 11139. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11148 and renegid2 42409. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 42412*	A combination of cnegex 11362 and cnegex2 11363, this proof takes cnre 11178 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 42413*	Proof of cnegex 11362 without ax-mulcom 11139. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 42414*	Proof of cnegex2 11363 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 42415	addcand 11384 without ax-mulcom 11139. Note how the proof is almost identical to addcan 11365. (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 42416	addrid 11361 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 42417	addcan2d 11385 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 42418	ixi 11814 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 42419	i4 14176 without ax-mulcom 11139. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 42420	A number that sums to itself is zero. Compare addid0 11604, readdridaddlidd 42253. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 42421	mul01 11360 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	sn-subeu 42422*	negeu 11418 without ax-mulcom 11139 and complex number version of resubeu 42372. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	sn-subcl 42423	subcl 11427 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 42424	subf 11430 without ax-mulcom 11139. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 42425	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 42426	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 42427	A number commutes with its additive inverse. Compare remulinvcom 42428. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 42428	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11139. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 42429	Commuted version of ax-1rid 11145 without ax-mulcom 11139. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 42430	Lemma for sn-mullid 42431 and sn-it1ei 42432. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 42431	mullid 11180 without ax-mulcom 11139. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 42432	it1ei 42311 without ax-mulcom 11139. (See sn-mullid 42431 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 42433	The multiplicative inverse of i (per i4 14176) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 42434	Commuted version of remulcan2d 42252 without ax-mulcom 11139. (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 42435	Real number division.
class /ℝ
Definition	df-rediv 42436*	Define division between real numbers. This operator saves ax-mulcom 11139 over df-div 11843 in certain situations. (Contributed by SN, 25-Nov-2025.)
⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
Theorem	redivvald 42437*	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 42438*	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 42439	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 42440	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 42441	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 42442	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 42443	Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
Theorem	rerecid 42444	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	rerecid2 42445	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	sn-0tie0 42446	Lemma for sn-mul02 42447. Commuted version of sn-it0e0 42411. (Contributed by SN, 30-Jun-2024.)
⊢ (0 · i) = 0
Theorem	sn-mul02 42447	mul02 11359 without ax-mulcom 11139. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11139 for an outline. (Contributed by SN, 30-Jun-2024.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	sn-ltaddpos 42448	ltaddpos 11675 without ax-mulcom 11139. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	sn-ltaddneg 42449	ltaddneg 11397 without ax-mulcom 11139. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	reposdif 42450	Comparison of two numbers whose difference is positive. Compare posdif 11678. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴)))
Theorem	relt0neg1 42451	Comparison of a real and its negative to zero. Compare lt0neg1 11691. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴)))
Theorem	relt0neg2 42452	Comparison of a real and its negative to zero. Compare lt0neg2 11692. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0))
Theorem	sn-addlt0d 42453	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 42454	The sum of positive numbers is positive. Proof of addgt0d 11760 without ax-mulcom 11139. (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 42455	nnne0 12227 without ax-mulcom 11139. (Contributed by SN, 25-Jan-2025.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	reelznn0nn 42456	elznn0nn 12550 restated using df-resub 42361. (Contributed by SN, 25-Jan-2025.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)))
Theorem	nn0addcom 42457	Addition is commutative for nonnegative integers. Proven without ax-mulcom 11139. (Contributed by SN, 1-Feb-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcomlem 42458	Lemma for zaddcom 42459. (Contributed by SN, 1-Feb-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcom 42459	Addition is commutative for integers. Proven without ax-mulcom 11139. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	renegmulnnass 42460	Move multiplication by a natural number inside and outside negation. (Contributed by SN, 25-Jan-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ)    ⇒   ⊢ (𝜑 → ((0 −ℝ 𝐴) · 𝑁) = (0 −ℝ (𝐴 · 𝑁)))
Theorem	nn0mulcom 42461	Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11139. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcomlem 42462	Lemma for zmulcom 42463. (Contributed by SN, 25-Jan-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcom 42463	Multiplication is commutative for integers. Proven without ax-mulcom 11139. From this result and grpcominv1 42503, we can show that rationals commute under multiplication without using ax-mulcom 11139. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	mulgt0con1dlem 42464	Lemma for mulgt0con1d 42465. Contraposes a positive deduction to a negative deduction. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (0 < 𝐴 → 0 < 𝐵))    &   ⊢ (𝜑 → (𝐴 = 0 → 𝐵 = 0))    ⇒   ⊢ (𝜑 → (𝐵 < 0 → 𝐴 < 0))
Theorem	mulgt0con1d 42465	Counterpart to mulgt0con2d 42466, though not a lemma. This is the first use of ax-pre-mulgt0 11152. One direction of mulgt0b2d 42473. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐴 < 0)
Theorem	mulgt0con2d 42466	Lemma for mulgt0b1d 42467 and contrapositive of mulgt0 11258. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐵 < 0)
Theorem	mulgt0b1d 42467	Biconditional, deductive form of mulgt0 11258. The second factor is positive iff the product is. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → (0 < 𝐵 ↔ 0 < (𝐴 · 𝐵)))
Theorem	sn-ltmul2d 42468	ltmul2d 13044 without ax-mulcom 11139. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐶)    ⇒   ⊢ (𝜑 → ((𝐶 · 𝐴) < (𝐶 · 𝐵) ↔ 𝐴 < 𝐵))
Theorem	sn-ltmulgt11d 42469	ltmulgt11d 13037 without ax-mulcom 11139. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴)))
Theorem	sn-0lt1 42470	0lt1 11707 without ax-mulcom 11139. (Contributed by SN, 13-Feb-2024.)
⊢ 0 < 1
Theorem	sn-ltp1 42471	ltp1 12029 without ax-mulcom 11139. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
Theorem	sn-recgt0d 42472	The reciprocal of a positive real is positive. (Contributed by SN, 26-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → 0 < (1 /ℝ 𝐴))
Theorem	mulgt0b2d 42473	Biconditional, deductive form of mulgt0 11258. The first factor is positive iff the product is. (Contributed by SN, 24-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ 0 < (𝐴 · 𝐵)))
Theorem	sn-mulgt1d 42474	mulgt1d 12126 without ax-mulcom 11139. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 1 < 𝐵)    ⇒   ⊢ (𝜑 → 1 < (𝐴 · 𝐵))
Theorem	reneg1lt0 42475	Negative one is a negative number. (Contributed by SN, 1-Jun-2024.)
⊢ (0 −ℝ 1) < 0
Theorem	sn-reclt0d 42476	The reciprocal of a negative real is negative. (Contributed by SN, 26-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) < 0)
Theorem	mulltgt0d 42477	Negative times positive is negative. (Contributed by SN, 26-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)
Theorem	mullt0b1d 42478	When the first term is negative, the second term is positive iff the product is negative. (Contributed by SN, 26-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    ⇒   ⊢ (𝜑 → (0 < 𝐵 ↔ (𝐴 · 𝐵) < 0))
Theorem	mullt0b2d 42479	When the second term is negative, the first term is positive iff the product is negative. (Contributed by SN, 26-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 < 0)    ⇒   ⊢ (𝜑 → (0 < 𝐴 ↔ (𝐴 · 𝐵) < 0))
Theorem	sn-mullt0d 42480	The product of two negative numbers is positive. (Contributed by SN, 1-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 < 0)    &   ⊢ (𝜑 → 𝐵 < 0)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐵))
Theorem	sn-msqgt0d 42481	A nonzero square is positive. (Contributed by SN, 1-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐴))
Theorem	sn-inelr 42482	inelr 12183 without ax-mulcom 11139. (Contributed by SN, 1-Jun-2024.)
⊢ ¬ i ∈ ℝ
Theorem	sn-itrere 42483	i times a real is real iff the real is zero. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((i · 𝑅) ∈ ℝ ↔ 𝑅 = 0))
Theorem	sn-retire 42484	Commuted version of sn-itrere 42483. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
Theorem	cnreeu 42485	The reals in the expression given by cnre 11178 uniquely define a complex number. (Contributed by SN, 27-Jun-2024.)
⊢ (𝜑 → 𝑟 ∈ ℝ)    &   ⊢ (𝜑 → 𝑠 ∈ ℝ)    &   ⊢ (𝜑 → 𝑡 ∈ ℝ)    &   ⊢ (𝜑 → 𝑢 ∈ ℝ)    ⇒   ⊢ (𝜑 → ((𝑟 + (i · 𝑠)) = (𝑡 + (i · 𝑢)) ↔ (𝑟 = 𝑡 ∧ 𝑠 = 𝑢)))
Theorem	sn-sup2 42486*	sup2 12146 with exactly the same proof except for using sn-ltp1 42471 instead of ltp1 12029, saving ax-mulcom 11139. (Contributed by SN, 26-Jun-2024.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	sn-sup3d 42487*	sup3 12147 without ax-mulcom 11139, proven trivially from sn-sup2 42486. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
Theorem	sn-suprcld 42488*	suprcld 12153 without ax-mulcom 11139, proven trivially from sn-sup3d 42487. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    ⇒   ⊢ (𝜑 → sup(𝐴, ℝ, < ) ∈ ℝ)
Theorem	sn-suprubd 42489*	suprubd 12152 without ax-mulcom 11139, proven trivially from sn-suprcld 42488. (Contributed by SN, 29-Jun-2025.)
⊢ (𝜑 → 𝐴 ⊆ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝐵 ≤ sup(𝐴, ℝ, < ))
21.30.6  Structures
Theorem	sn-base0 42490	Avoid axioms in base0 17191 by using the discouraged df-base 17187. This kind of axiom save is probably not worth it. (Contributed by SN, 16-Sep-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∅ = (Base‘∅)
Theorem	nelsubginvcld 42491	The inverse of a non-subgroup-member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) ∈ (𝐵 ∖ 𝑆))
Theorem	nelsubgcld 42492	A non-subgroup-member plus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝐵 ∖ 𝑆))
Theorem	nelsubgsubcld 42493	A non-subgroup-member minus a subgroup member is a non-subgroup-member. (Contributed by Steven Nguyen, 15-Apr-2023.)
⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑆))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝜑 → (𝑋 − 𝑌) ∈ (𝐵 ∖ 𝑆))
Theorem	rnasclg 42494	The set of injected scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 1 = (1r‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring) → ran 𝐴 = (𝑁‘{ 1 }))
Theorem	frlmfielbas 42495	The vectors of a finite free module are the functions from 𝐼 to 𝑁. (Contributed by SN, 31-Aug-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑋 ∈ 𝐵 ↔ 𝑋:𝐼⟶𝑁))
Theorem	frlmfzwrd 42496	A vector of a module with indices from 0 to 𝑁 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆)
Theorem	frlmfzowrd 42497	A vector of a module with indices from 0 to 𝑁 − 1 is a word over the scalars of the module. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ Word 𝑆)
Theorem	frlmfzolen 42498	The dimension of a vector of a module with indices from 0 to 𝑁 − 1. (Contributed by SN, 1-Sep-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (♯‘𝑋) = 𝑁)
Theorem	frlmfzowrdb 42499	The vectors of a module with indices 0 to 𝑁 − 1 are the length- 𝑁 words over the scalars of the module. (Contributed by SN, 1-Sep-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝑆 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ Word 𝑆 ∧ (♯‘𝑋) = 𝑁)))
Theorem	frlmfzoccat 42500	The concatenation of two vectors of dimension 𝑁 and 𝑀 forms a vector of dimension 𝑁 + 𝑀. (Contributed by SN, 31-Aug-2023.)
⊢ 𝑊 = (𝐾 freeLMod (0..^𝐿))    &   ⊢ 𝑋 = (𝐾 freeLMod (0..^𝑀))    &   ⊢ 𝑌 = (𝐾 freeLMod (0..^𝑁))    &   ⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐶 = (Base‘𝑋)    &   ⊢ 𝐷 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑍)    &   ⊢ (𝜑 → (𝑀 + 𝑁) = 𝐿)    &   ⊢ (𝜑 → 𝑀 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑉 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝑈 ++ 𝑉) ∈ 𝐵)