HomeTheorem List (p. 204 of 494)
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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ringmneg1 20301 | Negation of a product in a ring. (mulneg1 11699 analog.) Compared with rngmneg1 20164, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌))) | ||
| Theorem | ringmneg2 20302 | Negation of a product in a ring. (mulneg2 11700 analog.) Compared with rngmneg2 20165, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) | ||
| Theorem | ringm2neg 20303 | Double negation of a product in a ring. (mul2neg 11702 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋) · (𝑁‘𝑌)) = (𝑋 · 𝑌)) | ||
| Theorem | ringsubdi 20304 | Ring multiplication distributes over subtraction. (subdi 11696 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) | ||
| Theorem | ringsubdir 20305 | Ring multiplication distributes over subtraction. (subdir 11697 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ − = (-g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍))) | ||
| Theorem | mulgass2 20306 | An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁 · 𝑋) × 𝑌) = (𝑁 · (𝑋 × 𝑌))) | ||
| Theorem | ring1 20307 | The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.) |
| ⊢ 𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ⇒ ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring) | ||
| Theorem | ringn0 20308 | Rings exist. (Contributed by AV, 29-Apr-2019.) |
| ⊢ Ring ≠ ∅ | ||
| Theorem | ringlghm 20309* | Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑋 · 𝑥)) ∈ (𝑅 GrpHom 𝑅)) | ||
| Theorem | ringrghm 20310* | Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑥 ∈ 𝐵 ↦ (𝑥 · 𝑋)) ∈ (𝑅 GrpHom 𝑅)) | ||
| Theorem | gsummulc1OLD 20311* | Obsolete version of gsummulc1 20313 as of 7-Mar-2025. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) | ||
| Theorem | gsummulc2OLD 20312* | Obsolete version of gsummulc2 20314 as of 7-Mar-2025. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
| Theorem | gsummulc1 20313* | A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) Remove unused hypothesis. (Revised by SN, 7-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑋 · 𝑌))) = ((𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) · 𝑌)) | ||
| Theorem | gsummulc2 20314* | A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 10-Jul-2019.) Remove unused hypothesis. (Revised by SN, 7-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑅 Σg (𝑘 ∈ 𝐴 ↦ (𝑌 · 𝑋))) = (𝑌 · (𝑅 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)))) | ||
| Theorem | gsummgp0 20315* | If one factor in a finite group sum of the multiplicative group of a commutative ring is 0, the whole "sum" (i.e. product) is 0. (Contributed by AV, 3-Jan-2019.) |
| ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ CRing) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅)) & ⊢ ((𝜑 ∧ 𝑛 = 𝑖) → 𝐴 = 𝐵) & ⊢ (𝜑 → ∃𝑖 ∈ 𝑁 𝐵 = 0 ) ⇒ ⊢ (𝜑 → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ 𝐴)) = 0 ) | ||
| Theorem | gsumdixp 20316* | Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.) (Revised by AV, 10-Jul-2019.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑋 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝑋) finSupp 0 ) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ 𝑌) finSupp 0 ) ⇒ ⊢ (𝜑 → ((𝑅 Σg (𝑥 ∈ 𝐼 ↦ 𝑋)) · (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑌))) = (𝑅 Σg (𝑥 ∈ 𝐼, 𝑦 ∈ 𝐽 ↦ (𝑋 · 𝑌)))) | ||
| Theorem | prdsmulrcl 20317 | A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.) (Proof shortened by AV, 30-Mar-2025.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = (.r‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) | ||
| Theorem | prdsringd 20318 | A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) ⇒ ⊢ (𝜑 → 𝑌 ∈ Ring) | ||
| Theorem | prdscrngd 20319 | A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶CRing) ⇒ ⊢ (𝜑 → 𝑌 ∈ CRing) | ||
| Theorem | prds1 20320 | Value of the ring unity in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Ring) ⇒ ⊢ (𝜑 → (1r ∘ 𝑅) = (1r‘𝑌)) | ||
| Theorem | pwsring 20321 | A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ Ring) | ||
| Theorem | pws1 20322 | Value of the ring unity in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 1 }) = (1r‘𝑌)) | ||
| Theorem | pwscrng 20323 | A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ CRing) | ||
| Theorem | pwsmgp 20324 | The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝑀 = (mulGrp‘𝑅) & ⊢ 𝑍 = (𝑀 ↑s 𝐼) & ⊢ 𝑁 = (mulGrp‘𝑌) & ⊢ 𝐵 = (Base‘𝑁) & ⊢ 𝐶 = (Base‘𝑍) & ⊢ + = (+g‘𝑁) & ⊢ ✚ = (+g‘𝑍) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐵 = 𝐶 ∧ + = ✚ )) | ||
| Theorem | pwspjmhmmgpd 20325* | The projection given by pwspjmhm 18843 is also a monoid homomorphism between the respective multiplicative groups. (Contributed by SN, 30-Jul-2024.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (mulGrp‘𝑌) & ⊢ 𝑇 = (mulGrp‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥‘𝐴)) ∈ (𝑀 MndHom 𝑇)) | ||
| Theorem | pwsexpg 20326 | Value of a group exponentiation in a structure power. Compare pwsmulg 19137. (Contributed by SN, 30-Jul-2024.) |
| ⊢ 𝑌 = (𝑅 ↑s 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑀 = (mulGrp‘𝑌) & ⊢ 𝑇 = (mulGrp‘𝑅) & ⊢ ∙ = (.g‘𝑀) & ⊢ · = (.g‘𝑇) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐴 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑁 ∙ 𝑋)‘𝐴) = (𝑁 · (𝑋‘𝐴))) | ||
| Theorem | imasring 20327* | The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) & ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑈 ∈ Ring ∧ (𝐹‘ 1 ) = (1r‘𝑈))) | ||
| Theorem | imasringf1 20328 | The image of a ring under an injection is a ring (imasmndf1 18789 analog). (Contributed by AV, 27-Feb-2025.) |
| ⊢ 𝑈 = (𝐹 “s 𝑅) & ⊢ 𝑉 = (Base‘𝑅) ⇒ ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Ring) → 𝑈 ∈ Ring) | ||
| Theorem | xpsringd 20329 | A product of two rings is a ring (xpsmnd 18790 analog). (Contributed by AV, 28-Feb-2025.) |
| ⊢ 𝑌 = (𝑆 ×s 𝑅) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → 𝑌 ∈ Ring) | ||
| Theorem | xpsring1d 20330 | The multiplicative identity element of a binary product of rings. (Contributed by AV, 16-Mar-2025.) |
| ⊢ 𝑌 = (𝑆 ×s 𝑅) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (1r‘𝑌) = ⟨(1r‘𝑆), (1r‘𝑅)⟩) | ||
| Theorem | qusring2 20331* | The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
| ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) & ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) & ⊢ + = (+g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → ∼ Er 𝑉) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞))) & ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) & ⊢ (𝜑 → 𝑅 ∈ Ring) ⇒ ⊢ (𝜑 → (𝑈 ∈ Ring ∧ [ 1 ] ∼ = (1r‘𝑈))) | ||
| Theorem | crngbinom 20332* | The binomial theorem for commutative rings (special case of csrgbinom 20229): (𝐴 + 𝐵)↑𝑁 is the sum from 𝑘 = 0 to 𝑁 of (𝑁C𝑘) · ((𝐴↑𝑘) · (𝐵↑(𝑁 − 𝑘)). (Contributed by AV, 24-Aug-2019.) |
| ⊢ 𝑆 = (Base‘𝑅) & ⊢ × = (.r‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ + = (+g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑅) & ⊢ ↑ = (.g‘𝐺) ⇒ ⊢ (((𝑅 ∈ CRing ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝑁 ↑ (𝐴 + 𝐵)) = (𝑅 Σg (𝑘 ∈ (0...𝑁) ↦ ((𝑁C𝑘) · (((𝑁 − 𝑘) ↑ 𝐴) × (𝑘 ↑ 𝐵)))))) | ||
| Syntax | coppr 20333 | The opposite ring operation. |
| class oppr | ||
| Definition | df-oppr 20334 | Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ oppr = (𝑓 ∈ V ↦ (𝑓 sSet ⟨(.r‘ndx), tpos (.r‘𝑓)⟩)) | ||
| Theorem | opprval 20335 | Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ 𝑂 = (𝑅 sSet ⟨(.r‘ndx), tpos · ⟩) | ||
| Theorem | opprmulfval 20336 | Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ ∙ = tpos · | ||
| Theorem | opprmul 20337 | Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ (𝑋 ∙ 𝑌) = (𝑌 · 𝑋) | ||
| Theorem | crngoppr 20338 | In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 20417). (Contributed by Mario Carneiro, 14-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑂 = (oppr‘𝑅) & ⊢ ∙ = (.r‘𝑂) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (𝑋 ∙ 𝑌)) | ||
| Theorem | opprlem 20339 | Lemma for opprbas 20341 and oppradd 20343. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐸 = Slot (𝐸‘ndx) & ⊢ (𝐸‘ndx) ≠ (.r‘ndx) ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | ||
| Theorem | opprlemOLD 20340 | Obsolete version of opprlem 20339 as of 6-Nov-2024. Lemma for opprbas 20341 and oppradd 20343. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 < 3 ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | ||
| Theorem | opprbas 20341 | Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
| Theorem | opprbasOLD 20342 | Obsolete version of opprbas 20341 as of 6-Nov-2024. Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ 𝐵 = (Base‘𝑂) | ||
| Theorem | oppradd 20343 | Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ + = (+g‘𝑂) | ||
| Theorem | oppraddOLD 20344 | Obsolete version of opprbas 20341 as of 6-Nov-2024. Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ + = (+g‘𝑅) ⇒ ⊢ + = (+g‘𝑂) | ||
| Theorem | opprrng 20345 | An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng → 𝑂 ∈ Rng) | ||
| Theorem | opprrngb 20346 | A class is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 20345. (Contributed by AV, 15-Feb-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Rng ↔ 𝑂 ∈ Rng) | ||
| Theorem | opprring 20347 | An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.) (Proof shortened by AV, 30-Mar-2025.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑂 ∈ Ring) | ||
| Theorem | opprringb 20348 | Bidirectional form of opprring 20347. (Contributed by Mario Carneiro, 6-Dec-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring ↔ 𝑂 ∈ Ring) | ||
| Theorem | oppr0 20349 | Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ 0 = (0g‘𝑂) | ||
| Theorem | oppr1 20350 | Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ 1 = (1r‘𝑂) | ||
| Theorem | opprneg 20351 | The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) |
| ⊢ 𝑂 = (oppr‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ 𝑁 = (invg‘𝑂) | ||
| Theorem | opprsubg 20352 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.) |
| ⊢ 𝑂 = (oppr‘𝑅) ⇒ ⊢ (SubGrp‘𝑅) = (SubGrp‘𝑂) | ||
| Theorem | mulgass3 20353 | An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.g‘𝑅) & ⊢ × = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌))) | ||
| Syntax | cdsr 20354 | Ring divisibility relation. |
| class ∥r | ||
| Syntax | cui 20355 | Units in a ring. |
| class Unit | ||
| Syntax | cir 20356 | Ring irreducibles. |
| class Irred | ||
| Definition | df-dvdsr 20357* | Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through (∥r‘(oppr‘𝑅)). (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ ∥r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (Base‘𝑤) ∧ ∃𝑧 ∈ (Base‘𝑤)(𝑧(.r‘𝑤)𝑥) = 𝑦)}) | ||
| Definition | df-unit 20358 | Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ Unit = (𝑤 ∈ V ↦ (◡((∥r‘𝑤) ∩ (∥r‘(oppr‘𝑤))) “ {(1r‘𝑤)})) | ||
| Definition | df-irred 20359* | Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ Irred = (𝑤 ∈ V ↦ ⦋((Base‘𝑤) ∖ (Unit‘𝑤)) / 𝑏⦌{𝑧 ∈ 𝑏 ∣ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(.r‘𝑤)𝑦) ≠ 𝑧}) | ||
| Theorem | reldvdsr 20360 | The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ Rel ∥ | ||
| Theorem | dvdsrval 20361* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ∥ = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑥) = 𝑦)} | ||
| Theorem | dvdsr 20362* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∥ 𝑌 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) | ||
| Theorem | dvdsr2 20363* | Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝑋 ∥ 𝑌 ↔ ∃𝑧 ∈ 𝐵 (𝑧 · 𝑋) = 𝑌)) | ||
| Theorem | dvdsrmul 20364 | A left-multiple of 𝑋 is divisible by 𝑋. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∥ (𝑌 · 𝑋)) | ||
| Theorem | dvdsrcl 20365 | Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑋 ∥ 𝑌 → 𝑋 ∈ 𝐵) | ||
| Theorem | dvdsrcl2 20366 | Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌) → 𝑌 ∈ 𝐵) | ||
| Theorem | dvdsrid 20367 | An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 𝑋) | ||
| Theorem | dvdsrtr 20368 | Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∥ 𝑍 ∧ 𝑍 ∥ 𝑋) → 𝑌 ∥ 𝑋) | ||
| Theorem | dvdsrmul1 20369 | The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∥ 𝑌) → (𝑋 · 𝑍) ∥ (𝑌 · 𝑍)) | ||
| Theorem | dvdsrneg 20370 | An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ (𝑁‘𝑋)) | ||
| Theorem | dvdsr01 20371 | In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 20694.) (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑋 ∥ 0 ) | ||
| Theorem | dvdsr02 20372 | Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) | ||
| Theorem | isunit 20373 | Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) & ⊢ 𝑆 = (oppr‘𝑅) & ⊢ 𝐸 = (∥r‘𝑆) ⇒ ⊢ (𝑋 ∈ 𝑈 ↔ (𝑋 ∥ 1 ∧ 𝑋𝐸 1 )) | ||
| Theorem | 1unit 20374 | The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 1 ∈ 𝑈) | ||
| Theorem | unitcl 20375 | A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ 𝐵) | ||
| Theorem | unitss 20376 | The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝑈 = (Unit‘𝑅) ⇒ ⊢ 𝑈 ⊆ 𝐵 | ||
| Theorem | opprunit 20377 | Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑆 = (oppr‘𝑅) ⇒ ⊢ 𝑈 = (Unit‘𝑆) | ||
| Theorem | crngunit 20378 | Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ (𝑅 ∈ CRing → (𝑋 ∈ 𝑈 ↔ 𝑋 ∥ 1 )) | ||
| Theorem | dvdsunit 20379 | A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ ∥ = (∥r‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑌 ∥ 𝑋 ∧ 𝑋 ∈ 𝑈) → 𝑌 ∈ 𝑈) | ||
| Theorem | unitmulcl 20380 | The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈) → (𝑋 · 𝑌) ∈ 𝑈) | ||
| Theorem | unitmulclb 20381 | Reversal of unitmulcl 20380 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) ∈ 𝑈 ↔ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈))) | ||
| Theorem | unitgrpbas 20382 | The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ 𝑈 = (Base‘𝐺) | ||
| Theorem | unitgrp 20383 | The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Grp) | ||
| Theorem | unitabl 20384 | The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) ⇒ ⊢ (𝑅 ∈ CRing → 𝐺 ∈ Abel) | ||
| Theorem | unitgrpid 20385 | The identity of the group of units of a ring is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 1 = (0g‘𝐺)) | ||
| Theorem | unitsubm 20386 | The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑀 = (mulGrp‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀)) | ||
| Syntax | cinvr 20387 | Extend class notation with multiplicative inverse. |
| class invr | ||
| Definition | df-invr 20388 | Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.) |
| ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | ||
| Theorem | invrfval 20389 | Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ 𝐼 = (invg‘𝐺) | ||
| Theorem | unitinvcl 20390 | The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝑈) | ||
| Theorem | unitinvinv 20391 | The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘(𝐼‘𝑋)) = 𝑋) | ||
| Theorem | ringinvcl 20392 | The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ 𝐵 = (Base‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) ∈ 𝐵) | ||
| Theorem | unitlinv 20393 | A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → ((𝐼‘𝑋) · 𝑋) = 1 ) | ||
| Theorem | unitrinv 20394 | A unit times its inverse is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝐼 = (invr‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑋 · (𝐼‘𝑋)) = 1 ) | ||
| Theorem | 1rinv 20395 | The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.) |
| ⊢ 𝐼 = (invr‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐼‘ 1 ) = 1 ) | ||
| Theorem | 0unit 20396 | The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝑅 ∈ Ring → ( 0 ∈ 𝑈 ↔ 1 = 0 )) | ||
| Theorem | unitnegcl 20397 | The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.) |
| ⊢ 𝑈 = (Unit‘𝑅) & ⊢ 𝑁 = (invg‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) | ||
| Theorem | ringunitnzdiv 20398 | In a unitary ring, a unit is not a zero divisor. (Contributed by AV, 7-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑋 ∈ (Unit‘𝑅)) ⇒ ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 )) | ||
| Theorem | ring1nzdiv 20399 | In a unitary ring, the ring unity is not a zero divisor. (Contributed by AV, 7-Mar-2025.) |
| ⊢ 𝐵 = (Base‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → (( 1 · 𝑌) = 0 ↔ 𝑌 = 0 )) | ||
| Syntax | cdvr 20400 | Extend class notation with ring division. |
| class /r | ||
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