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-subcl 42401	subcl 11380 without ax-mulcom 11092. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 42402	subf 11383 without ax-mulcom 11092. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 42403	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 42404	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 42405	A number commutes with its additive inverse. Compare remulinvcom 42406. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 42406	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11092. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 42407	Commuted version of ax-1rid 11098 without ax-mulcom 11092. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 42408	Lemma for sn-mullid 42409 and sn-it1ei 42410. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 42409	mullid 11133 without ax-mulcom 11092. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 42410	it1ei 42289 without ax-mulcom 11092. (See sn-mullid 42409 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 42411	The multiplicative inverse of i (per i4 14129) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 42412	Commuted version of remulcan2d 42230 without ax-mulcom 11092. (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 42413	Real number division.
class /ℝ
Definition	df-rediv 42414*	Define division between real numbers. This operator saves ax-mulcom 11092 over df-div 11796 in certain situations. (Contributed by SN, 25-Nov-2025.)
⊢ /ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑧 ∈ ℝ (𝑦 · 𝑧) = 𝑥))
Theorem	redivvald 42415*	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 42416*	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 42417	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 42418	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 42419	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 42420	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 42421	Closure law for reciprocal. (Contributed by SN, 25-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → (1 /ℝ 𝐴) ∈ ℝ)
Theorem	rerecid 42422	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 42423	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 42424	Lemma for sn-mul02 42425. Commuted version of sn-it0e0 42389. (Contributed by SN, 30-Jun-2024.)
⊢ (0 · i) = 0
Theorem	sn-mul02 42425	mul02 11312 without ax-mulcom 11092. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11092 for an outline. (Contributed by SN, 30-Jun-2024.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	sn-ltaddpos 42426	ltaddpos 11628 without ax-mulcom 11092. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	sn-ltaddneg 42427	ltaddneg 11350 without ax-mulcom 11092. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	reposdif 42428	Comparison of two numbers whose difference is positive. Compare posdif 11631. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴)))
Theorem	relt0neg1 42429	Comparison of a real and its negative to zero. Compare lt0neg1 11644. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴)))
Theorem	relt0neg2 42430	Comparison of a real and its negative to zero. Compare lt0neg2 11645. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0))
Theorem	sn-addlt0d 42431	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 42432	The sum of positive numbers is positive. Proof of addgt0d 11713 without ax-mulcom 11092. (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 42433	nnne0 12180 without ax-mulcom 11092. (Contributed by SN, 25-Jan-2025.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	reelznn0nn 42434	elznn0nn 12503 restated using df-resub 42339. (Contributed by SN, 25-Jan-2025.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)))
Theorem	nn0addcom 42435	Addition is commutative for nonnegative integers. Proven without ax-mulcom 11092. (Contributed by SN, 1-Feb-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcomlem 42436	Lemma for zaddcom 42437. (Contributed by SN, 1-Feb-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcom 42437	Addition is commutative for integers. Proven without ax-mulcom 11092. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	renegmulnnass 42438	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 42439	Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11092. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcomlem 42440	Lemma for zmulcom 42441. (Contributed by SN, 25-Jan-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcom 42441	Multiplication is commutative for integers. Proven without ax-mulcom 11092. From this result and grpcominv1 42481, we can show that rationals commute under multiplication without using ax-mulcom 11092. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	mulgt0con1dlem 42442	Lemma for mulgt0con1d 42443. 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 42443	Counterpart to mulgt0con2d 42444, though not a lemma. This is the first use of ax-pre-mulgt0 11105. One direction of mulgt0b2d 42451. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐴 < 0)
Theorem	mulgt0con2d 42444	Lemma for mulgt0b1d 42445 and contrapositive of mulgt0 11211. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐵 < 0)
Theorem	mulgt0b1d 42445	Biconditional, deductive form of mulgt0 11211. 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 42446	ltmul2d 12997 without ax-mulcom 11092. (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 42447	ltmulgt11d 12990 without ax-mulcom 11092. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴)))
Theorem	sn-0lt1 42448	0lt1 11660 without ax-mulcom 11092. (Contributed by SN, 13-Feb-2024.)
⊢ 0 < 1
Theorem	sn-ltp1 42449	ltp1 11982 without ax-mulcom 11092. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
Theorem	sn-recgt0d 42450	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 42451	Biconditional, deductive form of mulgt0 11211. 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 42452	mulgt1d 12079 without ax-mulcom 11092. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 1 < 𝐵)    ⇒   ⊢ (𝜑 → 1 < (𝐴 · 𝐵))
Theorem	reneg1lt0 42453	Negative one is a negative number. (Contributed by SN, 1-Jun-2024.)
⊢ (0 −ℝ 1) < 0
Theorem	sn-reclt0d 42454	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 42455	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 42456	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 42457	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 42458	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 42459	A nonzero square is positive. (Contributed by SN, 1-Dec-2025.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    ⇒   ⊢ (𝜑 → 0 < (𝐴 · 𝐴))
Theorem	sn-inelr 42460	inelr 12136 without ax-mulcom 11092. (Contributed by SN, 1-Jun-2024.)
⊢ ¬ i ∈ ℝ
Theorem	sn-itrere 42461	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 42462	Commuted version of sn-itrere 42461. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
Theorem	cnreeu 42463	The reals in the expression given by cnre 11131 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 42464*	sup2 12099 with exactly the same proof except for using sn-ltp1 42449 instead of ltp1 11982, saving ax-mulcom 11092. (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 42465*	sup3 12100 without ax-mulcom 11092, proven trivially from sn-sup2 42464. (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 42466*	suprcld 12106 without ax-mulcom 11092, proven trivially from sn-sup3d 42465. (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 42467*	suprubd 12105 without ax-mulcom 11092, proven trivially from sn-suprcld 42466. (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 42468	Avoid axioms in base0 17143 by using the discouraged df-base 17139. 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 42469	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 42470	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 42471	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 42472	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 42473	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 42474	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 42475	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 42476	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 42477	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 42478	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)
Theorem	frlmvscadiccat 42479	Scalar multiplication distributes over concatenation. (Contributed by SN, 6-Sep-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)    &   ⊢ 𝑂 = ( ·𝑠 ‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝑋)    &   ⊢ · = ( ·𝑠 ‘𝑌)    &   ⊢ 𝑆 = (Base‘𝐾)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐴𝑂(𝑈 ++ 𝑉)) = ((𝐴 ∙ 𝑈) ++ (𝐴 · 𝑉)))
Theorem	grpasscan2d 42480	An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
Theorem	grpcominv1 42481	If two elements commute, then they commute with each other's inverses (case of the first element commuting with the inverse of the second element). (Contributed by SN, 29-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   ⊢ (𝜑 → (𝑋 + (𝑁‘𝑌)) = ((𝑁‘𝑌) + 𝑋))
Theorem	grpcominv2 42482	If two elements commute, then they commute with each other's inverses (case of the second element commuting with the inverse of the first element). (Contributed by SN, 1-Feb-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))    ⇒   ⊢ (𝜑 → (𝑌 + (𝑁‘𝑋)) = ((𝑁‘𝑋) + 𝑌))
Theorem	finsubmsubg 42483	A submonoid of a finite group is a subgroup. This does not extend to infinite groups, as the submonoid ℕ0 of the group (ℤ, + ) shows. Note also that the union of a submonoid and its inverses need not be a submonoid, as the submonoid (ℕ0 ∖ {1}) of the group (ℤ, + ) shows: 3 is in that submonoid, -2 is the inverse of 2, but 1 is not in their union. Or simply, the subgroup generated by (ℕ0 ∖ {1}) is ℤ, not (ℤ ∖ {1, -1}). (Contributed by SN, 31-Jan-2025.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ Grp)    &   ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺))    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝐺))
Theorem	opprmndb 42484	A class is a monoid if and only if its opposite (ring) is a monoid. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
Theorem	opprgrpb 42485	A class is a group if and only if its opposite (ring) is a group. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
Theorem	opprablb 42486	A class is an Abelian group if and only if its opposite (ring) is an Abelian group. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Abel ↔ 𝑂 ∈ Abel)
Theorem	imacrhmcl 42487	The image of a commutative ring homomorphism is a commutative ring. (Contributed by SN, 10-Jan-2025.)
⊢ 𝐶 = (𝑁 ↾s (𝐹 “ 𝑆))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑀 RingHom 𝑁))    &   ⊢ (𝜑 → 𝑀 ∈ CRing)    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑀))    ⇒   ⊢ (𝜑 → 𝐶 ∈ CRing)
Theorem	rimrcl1 42488	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
Theorem	rimrcl2 42489	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
Theorem	rimcnv 42490	The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → ◡𝐹 ∈ (𝑆 RingIso 𝑅))
Theorem	rimco 42491	The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝐹 ∈ (𝑆 RingIso 𝑇) ∧ 𝐺 ∈ (𝑅 RingIso 𝑆)) → (𝐹 ∘ 𝐺) ∈ (𝑅 RingIso 𝑇))
Theorem	ricsym 42492	Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
Theorem	rictr 42493	Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇)
Theorem	riccrng1 42494	Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ CRing) → 𝑆 ∈ CRing)
Theorem	riccrng 42495	A ring is commutative if and only if an isomorphic ring is commutative. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ CRing ↔ 𝑆 ∈ CRing))
Theorem	domnexpgn0cl 42496	In a domain, a (nonnegative) power of a nonzero element is nonzero. (Contributed by SN, 6-Jul-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ↑ = (.g‘(mulGrp‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝑁 ∈ ℕ0)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑁 ↑ 𝑋) ∈ (𝐵 ∖ { 0 }))
Theorem	drnginvrn0d 42497	A multiplicative inverse in a division ring is nonzero. (recne0d 11912 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ≠ 0 )
Theorem	drngmullcan 42498	Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11773 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ≠ 0 )    &   ⊢ (𝜑 → (𝑍 · 𝑋) = (𝑍 · 𝑌))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	drngmulrcan 42499	Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11774 analog). (Contributed by SN, 14-Aug-2024.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ≠ 0 )    &   ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	drnginvmuld 42500	Inverse of a nonzero product. (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘(𝑋 · 𝑌)) = ((𝐼‘𝑌) · (𝐼‘𝑋)))