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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | idlnegcl 38001 | An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑁 = (inv‘𝐺) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴 ∈ 𝐼) → (𝑁‘𝐴) ∈ 𝐼) | ||
| Theorem | idlsubcl 38002 | An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐷 = ( /𝑔 ‘𝐺) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴 ∈ 𝐼 ∧ 𝐵 ∈ 𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼) | ||
| Theorem | rngoidl 38003 | A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅)) | ||
| Theorem | 0idl 38004 | The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅)) | ||
| Theorem | 1idl 38005 | Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋)) | ||
| Theorem | 0rngo 38006 | In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) | ||
| Theorem | divrngidl 38007 | The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋}) | ||
| Theorem | intidl 38008 | The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∩ 𝐶 ∈ (Idl‘𝑅)) | ||
| Theorem | inidl 38009 | The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼 ∩ 𝐽) ∈ (Idl‘𝑅)) | ||
| Theorem | unichnidl 38010* | The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
| ⊢ ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖 ∈ 𝐶 ∀𝑗 ∈ 𝐶 (𝑖 ⊆ 𝑗 ∨ 𝑗 ⊆ 𝑖))) → ∪ 𝐶 ∈ (Idl‘𝑅)) | ||
| Theorem | keridl 38011 | The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑆) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (◡𝐹 “ {𝑍}) ∈ (Idl‘𝑅)) | ||
| Theorem | pridlval 38012* | The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (PrIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑖 → (𝑎 ⊆ 𝑖 ∨ 𝑏 ⊆ 𝑖)))}) | ||
| Theorem | ispridl 38013* | The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥𝐻𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))) | ||
| Theorem | pridlidl 38014 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅)) | ||
| Theorem | pridlnr 38015 | A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ≠ 𝑋) | ||
| Theorem | pridl 38016* | The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐻 = (2nd ‘𝑅) ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (Idl‘𝑅) ∧ 𝐵 ∈ (Idl‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐻𝑦) ∈ 𝑃)) → (𝐴 ⊆ 𝑃 ∨ 𝐵 ⊆ 𝑃)) | ||
| Theorem | ispridl2 38017* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38049 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) → 𝑃 ∈ (PrIdl‘𝑅)) | ||
| Theorem | maxidlval 38018* | The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (MaxIdl‘𝑅) = {𝑖 ∈ (Idl‘𝑅) ∣ (𝑖 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑖 ⊆ 𝑗 → (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))}) | ||
| Theorem | ismaxidl 38019* | The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀 ≠ 𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀 ⊆ 𝑗 → (𝑗 = 𝑀 ∨ 𝑗 = 𝑋))))) | ||
| Theorem | maxidlidl 38020 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | ||
| Theorem | maxidlnr 38021 | A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ≠ 𝑋) | ||
| Theorem | maxidlmax 38022 | A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀 ⊆ 𝐼)) → (𝐼 = 𝑀 ∨ 𝐼 = 𝑋)) | ||
| Theorem | maxidln1 38023 | One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ¬ 𝑈 ∈ 𝑀) | ||
| Theorem | maxidln0 38024 | A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑈 ≠ 𝑍) | ||
| Syntax | cprrng 38025 | Extend class notation with the class of prime rings. |
| class PrRing | ||
| Syntax | cdmn 38026 | Extend class notation with the class of domains. |
| class Dmn | ||
| Definition | df-prrngo 38027 | Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ PrRing = {𝑟 ∈ RingOps ∣ {(GId‘(1st ‘𝑟))} ∈ (PrIdl‘𝑟)} | ||
| Definition | df-dmn 38028 | Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ Dmn = (PrRing ∩ Com2) | ||
| Theorem | isprrngo 38029 | The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdl‘𝑅))) | ||
| Theorem | prrngorngo 38030 | A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | ||
| Theorem | smprngopr 38031 | A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝑅) = {{𝑍}, 𝑋}) → 𝑅 ∈ PrRing) | ||
| Theorem | divrngpr 38032 | A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) | ||
| Theorem | isdmn 38033 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | ||
| Theorem | isdmn2 38034 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) | ||
| Theorem | dmncrng 38035 | A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) | ||
| Theorem | dmnrngo 38036 | A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps) | ||
| Theorem | flddmn 38037 | A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) | ||
| Syntax | cigen 38038 | Extend class notation with the ideal generation function. |
| class IdlGen | ||
| Definition | df-igen 38039* | Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ IdlGen = (𝑟 ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st ‘𝑟) ↦ ∩ {𝑗 ∈ (Idl‘𝑟) ∣ 𝑠 ⊆ 𝑗}) | ||
| Theorem | igenval 38040* | The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑗}) | ||
| Theorem | igenss 38041 | A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆)) | ||
| Theorem | igenidl 38042 | The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅)) | ||
| Theorem | igenmin 38043 | The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼) | ||
| Theorem | igenidl2 38044 | The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑅 IdlGen 𝐼) = 𝐼) | ||
| Theorem | igenval2 38045* | The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))) | ||
| Theorem | prnc 38046* | A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) → (𝑅 IdlGen {𝐴}) = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑋 𝑥 = (𝑦𝐻𝐴)}) | ||
| Theorem | isfldidl 38047 | Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝐾) & ⊢ 𝐻 = (2nd ‘𝐾) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | ||
| Theorem | isfldidl2 38048 | Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝐾) & ⊢ 𝐻 = (2nd ‘𝐾) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 ≠ {𝑍} ∧ (Idl‘𝐾) = {{𝑍}, 𝑋})) | ||
| Theorem | ispridlc 38049* | The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))) | ||
| Theorem | pridlc 38050 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)) | ||
| Theorem | pridlc2 38051 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃) | ||
| Theorem | pridlc3 38052 | Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 ⇒ ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃)) | ||
| Theorem | isdmn3 38053* | The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) & ⊢ 𝑈 = (GId‘𝐻) ⇒ ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ 𝑈 ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))) | ||
| Theorem | dmnnzd 38054 | A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)) | ||
| Theorem | dmncan1 38055 | Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐴 ≠ 𝑍) → ((𝐴𝐻𝐵) = (𝐴𝐻𝐶) → 𝐵 = 𝐶)) | ||
| Theorem | dmncan2 38056 | Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ 𝐺 = (1st ‘𝑅) & ⊢ 𝐻 = (2nd ‘𝑅) & ⊢ 𝑋 = ran 𝐺 & ⊢ 𝑍 = (GId‘𝐺) ⇒ ⊢ (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝐶 ≠ 𝑍) → ((𝐴𝐻𝐶) = (𝐵𝐻𝐶) → 𝐴 = 𝐵)) | ||
The results in this section are mostly meant for being used by automatic proof building programs. As a result, they might appear less useful or meaningful than others to human beings. | ||
| Theorem | efald2 38057 | A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ 𝜑 → ⊥) ⇒ ⊢ 𝜑 | ||
| Theorem | notbinot1 38058 | Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ (¬ 𝜑 ↔ 𝜓) ↔ (𝜑 ↔ 𝜓)) | ||
| Theorem | bicontr 38059 | Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥) | ||
| Theorem | impor 38060 | An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒)) | ||
| Theorem | orfa 38061 | The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((𝜑 ∨ ⊥) ↔ 𝜑) | ||
| Theorem | notbinot2 38062 | Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓)) | ||
| Theorem | biimpor 38063 | A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ↔ 𝜓) ∨ 𝜒)) | ||
| Theorem | orfa1 38064 | Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ ⊥) → 𝜓) | ||
| Theorem | orfa2 38065 | Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ⊥) ⇒ ⊢ ((𝜑 ∨ 𝜓) → 𝜓) | ||
| Theorem | bifald 38066 | Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → (𝜓 ↔ ⊥)) | ||
| Theorem | orsild 38067 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
| Theorem | orsird 38068 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜒) | ||
| Theorem | cnf1dd 38069 | A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (𝜓 → ¬ 𝜒)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃)) | ||
| Theorem | cnf2dd 38070 | A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (𝜓 → ¬ 𝜃)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||
| Theorem | cnfn1dd 38071 | A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (𝜓 → 𝜒)) & ⊢ (𝜑 → (𝜓 → (¬ 𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜃)) | ||
| Theorem | cnfn2dd 38072 | A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (𝜓 → (𝜒 ∨ ¬ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 → 𝜒)) | ||
| Theorem | or32dd 38073 | A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃) ∨ 𝜏))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜏) ∨ 𝜃))) | ||
| Theorem | notornotel1 38074 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | notornotel2 38075 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
| Theorem | contrd 38076 | A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) & ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | an12i 38077 | An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.) |
| ⊢ (𝜑 ∧ (𝜓 ∧ 𝜒)) ⇒ ⊢ (𝜓 ∧ (𝜑 ∧ 𝜒)) | ||
| Theorem | exmid2 38078 | An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.) |
| ⊢ ((𝜓 ∧ 𝜑) → 𝜒) & ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜂) → 𝜒) | ||
| Theorem | selconj 38079 | An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.) |
| ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒)) ⇒ ⊢ ((𝜂 ∧ 𝜑) ↔ (𝜓 ∧ (𝜂 ∧ 𝜒))) | ||
| Theorem | truconj 38080 | Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.) |
| ⊢ (𝜑 ↔ (⊤ ∧ 𝜑)) | ||
| Theorem | orel 38081 | An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ ((𝜓 ∧ 𝜂) → 𝜃) & ⊢ ((𝜒 ∧ 𝜌) → 𝜃) & ⊢ (𝜑 → (𝜓 ∨ 𝜒)) ⇒ ⊢ ((𝜑 ∧ (𝜂 ∧ 𝜌)) → 𝜃) | ||
| Theorem | negel 38082 | An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜓 → 𝜒) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ⊥) | ||
| Theorem | botel 38083 | An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜑 → ⊥) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | tradd 38084 | Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ (⊤ ∧ 𝜓)) | ||
| Theorem | gm-sbtru 38085 | Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤) | ||
| Theorem | sbfal 38086 | Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥) | ||
| Theorem | sbcani 38087 | Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 ∧ 𝜓) ↔ (𝜒 ∧ 𝜂)) | ||
| Theorem | sbcori 38088 | Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 ∨ 𝜓) ↔ (𝜒 ∨ 𝜂)) | ||
| Theorem | sbcimi 38089 | Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜒) & ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝜂) ⇒ ⊢ ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ (𝜒 → 𝜂)) | ||
| Theorem | sbcni 38090 | Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ 𝜓) | ||
| Theorem | sbali 38091 | Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]∀𝑥𝜑 ↔ ∀𝑥𝜑) | ||
| Theorem | sbexi 38092 | Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]∃𝑥𝜑 ↔ ∃𝑥𝜑) | ||
| Theorem | sbcalf 38093* | Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.) |
| ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑) | ||
| Theorem | sbcexf 38094* | Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.) |
| ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑) | ||
| Theorem | sbcalfi 38095* | Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓) | ||
| Theorem | sbcexfi 38096* | Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ Ⅎ𝑦𝐴 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓) | ||
| Theorem | spsbcdi 38097 | A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ (𝜑 → ∀𝑥𝜒) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | alrimii 38098* | A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ Ⅎ𝑦𝜑 & ⊢ (𝜑 → 𝜓) & ⊢ ([𝑦 / 𝑥]𝜒 ↔ 𝜓) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ (𝜑 → ∀𝑥𝜒) | ||
| Theorem | spesbcdi 38099 | A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ (𝜑 → 𝜓) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜓) ⇒ ⊢ (𝜑 → ∃𝑥𝜒) | ||
| Theorem | exlimddvf 38100 | A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑥𝜓 & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
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