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	renegneg 42401	A real number is equal to the negative of its negative. Compare negneg 11531. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴)
Theorem	readdcan2 42402	Commuted version of readdcan 11407 without ax-mulcom 11191. (Contributed by SN, 21-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	renegid2 42403	Commuted version of renegid 42363. (Contributed by SN, 4-May-2024.)
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0)
Theorem	remulneg2d 42404	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 42405	Proof of it0e0 12462 without ax-mulcom 11191. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11200 and renegid2 42403. (Contributed by SN, 30-Apr-2024.)
⊢ (i · 0) = 0
Theorem	sn-negex12 42406*	A combination of cnegex 11414 and cnegex2 11415, this proof takes cnre 11230 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 42407*	Proof of cnegex 11414 without ax-mulcom 11191. (Contributed by SN, 30-Apr-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0)
Theorem	sn-negex2 42408*	Proof of cnegex2 11415 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0)
Theorem	sn-addcand 42409	addcand 11436 without ax-mulcom 11191. Note how the proof is almost identical to addcan 11417. (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 42410	addrid 11413 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴)
Theorem	sn-addcan2d 42411	addcan2d 11437 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵))
Theorem	reixi 42412	ixi 11864 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ (i · i) = (0 −ℝ 1)
Theorem	rei4 42413	i4 14220 without ax-mulcom 11191. (Contributed by SN, 27-May-2024.)
⊢ ((i · i) · (i · i)) = 1
Theorem	sn-addid0 42414	A number that sums to itself is zero. Compare addid0 11654, readdridaddlidd 42256. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 = 0)
Theorem	sn-mul01 42415	mul01 11412 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
Theorem	sn-subeu 42416*	negeu 11470 without ax-mulcom 11191 and complex number version of resubeu 42367. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵)
Theorem	sn-subcl 42417	subcl 11479 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
Theorem	sn-subf 42418	subf 11482 without ax-mulcom 11191. (Contributed by SN, 5-May-2024.)
⊢ − :(ℂ × ℂ)⟶ℂ
Theorem	resubeqsub 42419	Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵))
Theorem	subresre 42420	Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.)
⊢ −ℝ = ( − ↾ (ℝ × ℝ))
Theorem	addinvcom 42421	A number commutes with its additive inverse. Compare remulinvcom 42422. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → (𝐴 + 𝐵) = 0)    ⇒   ⊢ (𝜑 → (𝐵 + 𝐴) = 0)
Theorem	remulinvcom 42422	A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11191. (Contributed by SN, 5-Feb-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → (𝐴 · 𝐵) = 1)    ⇒   ⊢ (𝜑 → (𝐵 · 𝐴) = 1)
Theorem	remullid 42423	Commuted version of ax-1rid 11197 without ax-mulcom 11191. (Contributed by SN, 5-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴)
Theorem	sn-1ticom 42424	Lemma for sn-mullid 42425 and sn-it1ei 42426. (Contributed by SN, 27-May-2024.)
⊢ (1 · i) = (i · 1)
Theorem	sn-mullid 42425	mullid 11232 without ax-mulcom 11191. (Contributed by SN, 27-May-2024.)
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴)
Theorem	sn-it1ei 42426	it1ei 42312 without ax-mulcom 11191. (See sn-mullid 42425 for commuted version). (Contributed by SN, 1-Jun-2024.)
⊢ (i · 1) = i
Theorem	ipiiie0 42427	The multiplicative inverse of i (per i4 14220) is also its additive inverse. (Contributed by SN, 30-Jun-2024.)
⊢ (i + (i · (i · i))) = 0
Theorem	remulcand 42428	Commuted version of remulcan2d 42255 without ax-mulcom 11191. (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))
Theorem	sn-0tie0 42429	Lemma for sn-mul02 42430. Commuted version of sn-it0e0 42405. (Contributed by SN, 30-Jun-2024.)
⊢ (0 · i) = 0
Theorem	sn-mul02 42430	mul02 11411 without ax-mulcom 11191. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11191 for an outline. (Contributed by SN, 30-Jun-2024.)
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0)
Theorem	sn-ltaddpos 42431	ltaddpos 11725 without ax-mulcom 11191. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴)))
Theorem	sn-ltaddneg 42432	ltaddneg 11449 without ax-mulcom 11191. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵))
Theorem	reposdif 42433	Comparison of two numbers whose difference is positive. Compare posdif 11728. (Contributed by SN, 13-Feb-2024.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴)))
Theorem	relt0neg1 42434	Comparison of a real and its negative to zero. Compare lt0neg1 11741. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴)))
Theorem	relt0neg2 42435	Comparison of a real and its negative to zero. Compare lt0neg2 11742. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0))
Theorem	sn-addlt0d 42436	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 42437	The sum of positive numbers is positive. Proof of addgt0d 11810 without ax-mulcom 11191. (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 42438	nnne0 12272 without ax-mulcom 11191. (Contributed by SN, 25-Jan-2025.)
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0)
Theorem	reelznn0nn 42439	elznn0nn 12600 restated using df-resub 42356. (Contributed by SN, 25-Jan-2025.)
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)))
Theorem	nn0addcom 42440	Addition is commutative for nonnegative integers. Proven without ax-mulcom 11191. (Contributed by SN, 1-Feb-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcomlem 42441	Lemma for zaddcom 42442. (Contributed by SN, 1-Feb-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	zaddcom 42442	Addition is commutative for integers. Proven without ax-mulcom 11191. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
Theorem	renegmulnnass 42443	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 42444	Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11191. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcomlem 42445	Lemma for zmulcom 42446. (Contributed by SN, 25-Jan-2025.)
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	zmulcom 42446	Multiplication is commutative for integers. Proven without ax-mulcom 11191. From this result and grpcominv1 42478, we can show that rationals commute under multiplication without using ax-mulcom 11191. (Contributed by SN, 25-Jan-2025.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
Theorem	mulgt0con1dlem 42447	Lemma for mulgt0con1d 42448. 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 42448	Counterpart to mulgt0con2d 42449, though not a lemma of anything. This is the first use of ax-pre-mulgt0 11204. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐴 < 0)
Theorem	mulgt0con2d 42449	Lemma for mulgt0b2d 42450 and contrapositive of mulgt0 11310. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    &   ⊢ (𝜑 → (𝐴 · 𝐵) < 0)    ⇒   ⊢ (𝜑 → 𝐵 < 0)
Theorem	mulgt0b2d 42450	Biconditional, deductive form of mulgt0 11310. The second factor is positive iff the product is. Note that the commuted form cannot be proven since resubdi 42386 does not have a commuted form. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → (0 < 𝐵 ↔ 0 < (𝐴 · 𝐵)))
Theorem	sn-ltmul2d 42451	ltmul2d 13091 without ax-mulcom 11191. (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 42452	ltmulgt11d 13084 without ax-mulcom 11191. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 0 < 𝐵)    ⇒   ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴)))
Theorem	sn-0lt1 42453	0lt1 11757 without ax-mulcom 11191. (Contributed by SN, 13-Feb-2024.)
⊢ 0 < 1
Theorem	sn-ltp1 42454	ltp1 12079 without ax-mulcom 11191. (Contributed by SN, 13-Feb-2024.)
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1))
Theorem	sn-mulgt1d 42455	mulgt1d 12176 without ax-mulcom 11191. (Contributed by SN, 26-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 1 < 𝐴)    &   ⊢ (𝜑 → 1 < 𝐵)    ⇒   ⊢ (𝜑 → 1 < (𝐴 · 𝐵))
Theorem	reneg1lt0 42456	Lemma for sn-inelr 42457. (Contributed by SN, 1-Jun-2024.)
⊢ (0 −ℝ 1) < 0
Theorem	sn-inelr 42457	inelr 12228 without ax-mulcom 11191. (Contributed by SN, 1-Jun-2024.)
⊢ ¬ i ∈ ℝ
Theorem	sn-itrere 42458	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 42459	Commuted version of sn-itrere 42458. (Contributed by SN, 27-Jun-2024.)
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0))
Theorem	cnreeu 42460	The reals in the expression given by cnre 11230 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 42461*	sup2 12196 with exactly the same proof except for using sn-ltp1 42454 instead of ltp1 12079, saving ax-mulcom 11191. (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 42462*	sup3 12197 without ax-mulcom 11191, proven trivially from sn-sup2 42461. (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 42463*	suprcld 12203 without ax-mulcom 11191, proven trivially from sn-sup3d 42462. (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 42464*	suprubd 12202 without ax-mulcom 11191, proven trivially from sn-suprcld 42463. (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 42465	Avoid axioms in base0 17231 by using the discouraged df-base 17227. 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 42466	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 42467	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 42468	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 42469	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 42470	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 42471	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 42472	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 42473	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 42474	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 42475	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 42476	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 42477	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 42478	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 42479	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 42480	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 42481	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 42482	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 42483	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 42484	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 42485	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑅 ∈ Ring)
Theorem	rimrcl2 42486	Reverse closure of a ring isomorphism. (Contributed by SN, 19-Feb-2025.)
⊢ (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝑆 ∈ Ring)
Theorem	rimcnv 42487	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 42488	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 42489	Ring isomorphism is symmetric. (Contributed by SN, 10-Jan-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → 𝑆 ≃𝑟 𝑅)
Theorem	rictr 42490	Ring isomorphism is transitive. (Contributed by SN, 17-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑆 ≃𝑟 𝑇) → 𝑅 ≃𝑟 𝑇)
Theorem	riccrng1 42491	Ring isomorphism preserves (multiplicative) commutativity. (Contributed by SN, 10-Jan-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ CRing) → 𝑆 ∈ CRing)
Theorem	riccrng 42492	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 42493	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 42494	A multiplicative inverse in a division ring is nonzero. (recne0d 12009 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 42495	Cancellation of a nonzero factor on the left for multiplication. (mulcanad 11870 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 42496	Cancellation of a nonzero factor on the right for multiplication. (mulcan2ad 11871 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 42497	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)))
Theorem	ricdrng1 42498	A ring isomorphism maps a division ring to a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ ((𝑅 ≃𝑟 𝑆 ∧ 𝑅 ∈ DivRing) → 𝑆 ∈ DivRing)
Theorem	ricdrng 42499	A ring is a division ring if and only if an isomorphic ring is a division ring. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ DivRing ↔ 𝑆 ∈ DivRing))
Theorem	ricfld 42500	A ring is a field if and only if an isomorphic ring is a field. (Contributed by SN, 18-Feb-2025.)
⊢ (𝑅 ≃𝑟 𝑆 → (𝑅 ∈ Field ↔ 𝑆 ∈ Field))