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Type | Label | Description |
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Statement | ||
Theorem | resubidaddlid 42401 | 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 42402 | 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 42403 | Equality of two left-additive identities. See resubidaddlid 42401. Uses ax-i2m1 11220. (Contributed by SN, 25-Dec-2023.) |
⊢ (1 −ℝ 1) = (0 −ℝ 0) | ||
Theorem | sn-00idlem1 42404 | Lemma for sn-00id 42407. (Contributed by SN, 25-Dec-2023.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · (0 −ℝ 0)) = (𝐴 −ℝ 𝐴)) | ||
Theorem | sn-00idlem2 42405 | Lemma for sn-00id 42407. (Contributed by SN, 25-Dec-2023.) |
⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1) | ||
Theorem | sn-00idlem3 42406 | Lemma for sn-00id 42407. (Contributed by SN, 25-Dec-2023.) |
⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0) | ||
Theorem | sn-00id 42407 | 00id 11433 proven without ax-mulcom 11216 but using ax-1ne0 11221. (Though note that the current version of 00id 11433 can be changed to avoid ax-icn 11211, ax-addcl 11212, ax-mulcl 11214, ax-i2m1 11220, ax-cnre 11225. Most of this is by using 0cnALT3 42272 instead of 0cn 11250). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.) |
⊢ (0 + 0) = 0 | ||
Theorem | re0m0e0 42408 | Real number version of 0m0e0 12383 proven without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (0 −ℝ 0) = 0 | ||
Theorem | readdlid 42409 | Real number version of addlid 11441. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (0 + 𝐴) = 𝐴) | ||
Theorem | sn-addlid 42410 | addlid 11441 without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℂ → (0 + 𝐴) = 𝐴) | ||
Theorem | remul02 42411 | Real number version of mul02 11436 proven without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) | ||
Theorem | sn-0ne2 42412 | 0ne2 12470 without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ 0 ≠ 2 | ||
Theorem | remul01 42413 | Real number version of mul01 11437 proven without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (𝐴 · 0) = 0) | ||
Theorem | resubid 42414 | Subtraction of a real number from itself (compare subid 11525). (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 𝐴) = 0) | ||
Theorem | readdrid 42415 | Real number version of addrid 11438 without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) | ||
Theorem | resubid1 42416 | Real number version of subid1 11526 without ax-mulcom 11216. (Contributed by SN, 23-Jan-2024.) |
⊢ (𝐴 ∈ ℝ → (𝐴 −ℝ 0) = 𝐴) | ||
Theorem | renegneg 42417 | A real number is equal to the negative of its negative. Compare negneg 11556. (Contributed by SN, 13-Feb-2024.) |
⊢ (𝐴 ∈ ℝ → (0 −ℝ (0 −ℝ 𝐴)) = 𝐴) | ||
Theorem | readdcan2 42418 | Commuted version of readdcan 11432 without ax-mulcom 11216. (Contributed by SN, 21-Feb-2024.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | renegid2 42419 | Commuted version of renegid 42379. (Contributed by SN, 4-May-2024.) |
⊢ (𝐴 ∈ ℝ → ((0 −ℝ 𝐴) + 𝐴) = 0) | ||
Theorem | remulneg2d 42420 | 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 42421 | Proof of it0e0 12485 without ax-mulcom 11216. Informally, a real number times 0 is 0, and ∃𝑟 ∈ ℝ𝑟 = i · 𝑠 by ax-cnre 11225 and renegid2 42419. (Contributed by SN, 30-Apr-2024.) |
⊢ (i · 0) = 0 | ||
Theorem | sn-negex12 42422* | A combination of cnegex 11439 and cnegex2 11440, this proof takes cnre 11255 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 42423* | Proof of cnegex 11439 without ax-mulcom 11216. (Contributed by SN, 30-Apr-2024.) |
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝐴 + 𝑏) = 0) | ||
Theorem | sn-negex2 42424* | Proof of cnegex2 11440 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ (𝐴 ∈ ℂ → ∃𝑏 ∈ ℂ (𝑏 + 𝐴) = 0) | ||
Theorem | sn-addcand 42425 | addcand 11461 without ax-mulcom 11216. Note how the proof is almost identical to addcan 11442. (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 42426 | addrid 11438 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | ||
Theorem | sn-addcan2d 42427 | addcan2d 11462 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) | ||
Theorem | reixi 42428 | ixi 11889 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ (i · i) = (0 −ℝ 1) | ||
Theorem | rei4 42429 | i4 14239 without ax-mulcom 11216. (Contributed by SN, 27-May-2024.) |
⊢ ((i · i) · (i · i)) = 1 | ||
Theorem | sn-addid0 42430 | A number that sums to itself is zero. Compare addid0 11679, readdridaddlidd 42277. (Contributed by SN, 5-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐴) = 𝐴) ⇒ ⊢ (𝜑 → 𝐴 = 0) | ||
Theorem | sn-mul01 42431 | mul01 11437 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | ||
Theorem | sn-subeu 42432* | negeu 11495 without ax-mulcom 11216 and complex number version of resubeu 42383. (Contributed by SN, 5-May-2024.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 𝐵) | ||
Theorem | sn-subcl 42433 | subcl 11504 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | ||
Theorem | sn-subf 42434 | subf 11507 without ax-mulcom 11216. (Contributed by SN, 5-May-2024.) |
⊢ − :(ℂ × ℂ)⟶ℂ | ||
Theorem | resubeqsub 42435 | Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵)) | ||
Theorem | subresre 42436 | Subtraction restricted to the reals. (Contributed by SN, 5-May-2024.) |
⊢ −ℝ = ( − ↾ (ℝ × ℝ)) | ||
Theorem | addinvcom 42437 | A number commutes with its additive inverse. Compare remulinvcom 42438. (Contributed by SN, 5-May-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → (𝐴 + 𝐵) = 0) ⇒ ⊢ (𝜑 → (𝐵 + 𝐴) = 0) | ||
Theorem | remulinvcom 42438 | A left multiplicative inverse is a right multiplicative inverse. Proven without ax-mulcom 11216. (Contributed by SN, 5-Feb-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → (𝐴 · 𝐵) = 1) ⇒ ⊢ (𝜑 → (𝐵 · 𝐴) = 1) | ||
Theorem | remullid 42439 | Commuted version of ax-1rid 11222 without ax-mulcom 11216. (Contributed by SN, 5-Feb-2024.) |
⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴) | ||
Theorem | sn-1ticom 42440 | Lemma for sn-mullid 42441 and sn-it1ei 42442. (Contributed by SN, 27-May-2024.) |
⊢ (1 · i) = (i · 1) | ||
Theorem | sn-mullid 42441 | mullid 11257 without ax-mulcom 11216. (Contributed by SN, 27-May-2024.) |
⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | ||
Theorem | sn-it1ei 42442 | it1ei 42329 without ax-mulcom 11216. (See sn-mullid 42441 for commuted version). (Contributed by SN, 1-Jun-2024.) |
⊢ (i · 1) = i | ||
Theorem | ipiiie0 42443 | The multiplicative inverse of i (per i4 14239) is also its additive inverse. (Contributed by SN, 30-Jun-2024.) |
⊢ (i + (i · (i · i))) = 0 | ||
Theorem | remulcand 42444 | Commuted version of remulcan2d 42276 without ax-mulcom 11216. (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 42445 | Lemma for sn-mul02 42446. Commuted version of sn-it0e0 42421. (Contributed by SN, 30-Jun-2024.) |
⊢ (0 · i) = 0 | ||
Theorem | sn-mul02 42446 | mul02 11436 without ax-mulcom 11216. See https://github.com/icecream17/Stuff/blob/main/math/0A%3D0.md 11216 for an outline. (Contributed by SN, 30-Jun-2024.) |
⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | ||
Theorem | sn-ltaddpos 42447 | ltaddpos 11750 without ax-mulcom 11216. (Contributed by SN, 13-Feb-2024.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < 𝐴 ↔ 𝐵 < (𝐵 + 𝐴))) | ||
Theorem | sn-ltaddneg 42448 | ltaddneg 11474 without ax-mulcom 11216. (Contributed by SN, 25-Jan-2025.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 0 ↔ (𝐵 + 𝐴) < 𝐵)) | ||
Theorem | reposdif 42449 | Comparison of two numbers whose difference is positive. Compare posdif 11753. (Contributed by SN, 13-Feb-2024.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 −ℝ 𝐴))) | ||
Theorem | relt0neg1 42450 | Comparison of a real and its negative to zero. Compare lt0neg1 11766. (Contributed by SN, 13-Feb-2024.) |
⊢ (𝐴 ∈ ℝ → (𝐴 < 0 ↔ 0 < (0 −ℝ 𝐴))) | ||
Theorem | relt0neg2 42451 | Comparison of a real and its negative to zero. Compare lt0neg2 11767. (Contributed by SN, 13-Feb-2024.) |
⊢ (𝐴 ∈ ℝ → (0 < 𝐴 ↔ (0 −ℝ 𝐴) < 0)) | ||
Theorem | sn-addlt0d 42452 | 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 42453 | The sum of positive numbers is positive. Proof of addgt0d 11835 without ax-mulcom 11216. (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 42454 | nnne0 12297 without ax-mulcom 11216. (Contributed by SN, 25-Jan-2025.) |
⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) | ||
Theorem | reelznn0nn 42455 | elznn0nn 12624 restated using df-resub 42372. (Contributed by SN, 25-Jan-2025.) |
⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ))) | ||
Theorem | nn0addcom 42456 | Addition is commutative for nonnegative integers. Proven without ax-mulcom 11216. (Contributed by SN, 1-Feb-2025.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | zaddcomlem 42457 | Lemma for zaddcom 42458. (Contributed by SN, 1-Feb-2025.) |
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | zaddcom 42458 | Addition is commutative for integers. Proven without ax-mulcom 11216. (Contributed by SN, 25-Jan-2025.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) | ||
Theorem | renegmulnnass 42459 | 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 42460 | Multiplication is commutative for nonnegative integers. Proven without ax-mulcom 11216. (Contributed by SN, 25-Jan-2025.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | zmulcomlem 42461 | Lemma for zmulcom 42462. (Contributed by SN, 25-Jan-2025.) |
⊢ (((𝐴 ∈ ℝ ∧ (0 −ℝ 𝐴) ∈ ℕ) ∧ 𝐵 ∈ ℕ0) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | zmulcom 42462 | Multiplication is commutative for integers. Proven without ax-mulcom 11216. From this result and grpcominv1 42494, we can show that rationals commute under multiplication without using ax-mulcom 11216. (Contributed by SN, 25-Jan-2025.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 · 𝐵) = (𝐵 · 𝐴)) | ||
Theorem | mulgt0con1dlem 42463 | Lemma for mulgt0con1d 42464. 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 42464 | Counterpart to mulgt0con2d 42465, though not a lemma of anything. This is the first use of ax-pre-mulgt0 11229. (Contributed by SN, 26-Jun-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐵) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ (𝜑 → 𝐴 < 0) | ||
Theorem | mulgt0con2d 42465 | Lemma for mulgt0b2d 42466 and contrapositive of mulgt0 11335. (Contributed by SN, 26-Jun-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐴) & ⊢ (𝜑 → (𝐴 · 𝐵) < 0) ⇒ ⊢ (𝜑 → 𝐵 < 0) | ||
Theorem | mulgt0b2d 42466 | Biconditional, deductive form of mulgt0 11335. The second factor is positive iff the product is. Note that the commuted form cannot be proven since resubdi 42402 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 42467 | ltmul2d 13116 without ax-mulcom 11216. (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 42468 | ltmulgt11d 13109 without ax-mulcom 11216. (Contributed by SN, 26-Jun-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 0 < 𝐵) ⇒ ⊢ (𝜑 → (1 < 𝐴 ↔ 𝐵 < (𝐵 · 𝐴))) | ||
Theorem | sn-0lt1 42469 | 0lt1 11782 without ax-mulcom 11216. (Contributed by SN, 13-Feb-2024.) |
⊢ 0 < 1 | ||
Theorem | sn-ltp1 42470 | ltp1 12104 without ax-mulcom 11216. (Contributed by SN, 13-Feb-2024.) |
⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | ||
Theorem | sn-mulgt1d 42471 | mulgt1d 12201 without ax-mulcom 11216. (Contributed by SN, 26-Jun-2024.) |
⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 1 < 𝐴) & ⊢ (𝜑 → 1 < 𝐵) ⇒ ⊢ (𝜑 → 1 < (𝐴 · 𝐵)) | ||
Theorem | reneg1lt0 42472 | Lemma for sn-inelr 42473. (Contributed by SN, 1-Jun-2024.) |
⊢ (0 −ℝ 1) < 0 | ||
Theorem | sn-inelr 42473 | inelr 12253 without ax-mulcom 11216. (Contributed by SN, 1-Jun-2024.) |
⊢ ¬ i ∈ ℝ | ||
Theorem | sn-itrere 42474 | 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 42475 | Commuted version of sn-itrere 42474. (Contributed by SN, 27-Jun-2024.) |
⊢ (𝑅 ∈ ℝ → ((𝑅 · i) ∈ ℝ ↔ 𝑅 = 0)) | ||
Theorem | cnreeu 42476 | The reals in the expression given by cnre 11255 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 42477* | sup2 12221 with exactly the same proof except for using sn-ltp1 42470 instead of ltp1 12104, saving ax-mulcom 11216. (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 42478* | sup3 12222 without ax-mulcom 11216, proven trivially from sn-sup2 42477. (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 42479* | suprcld 12228 without ax-mulcom 11216, proven trivially from sn-sup3d 42478. (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 42480* | suprubd 12227 without ax-mulcom 11216, proven trivially from sn-suprcld 42479. (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, < )) | ||
Theorem | sn-base0 42481 | Avoid axioms in base0 17249 by using the discouraged df-base 17245. 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 42482 | 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 42483 | 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 42484 | 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 42485 | 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 42486 | 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 42487 | 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 42488 | 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 42489 | 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 42490 | 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 42491 | 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 42492 | 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 42493 | 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 42494 | 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 42495 | 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 42496 | 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 42497 | 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 42498 | 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 42499 | 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 42500 | 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) |
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