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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | unichnidl 38001* | 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 38002 | 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 38003* | 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 38004* | 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 38005 | A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅)) | ||
| Theorem | pridlnr 38006 | 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 38007* | 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 38008* | A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 38040 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 38009* | 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 38010* | 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 38011 | A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.) |
| ⊢ ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀 ∈ (Idl‘𝑅)) | ||
| Theorem | maxidlnr 38012 | 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 38013 | 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 38014 | 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 38015 | 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 38016 | Extend class notation with the class of prime rings. |
| class PrRing | ||
| Syntax | cdmn 38017 | Extend class notation with the class of domains. |
| class Dmn | ||
| Definition | df-prrngo 38018 | 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 38019 | 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 38020 | 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 38021 | A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ PrRing → 𝑅 ∈ RingOps) | ||
| Theorem | smprngopr 38022 | 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 38023 | A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ PrRing) | ||
| Theorem | isdmn 38024 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2)) | ||
| Theorem | isdmn2 38025 | The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps)) | ||
| Theorem | dmncrng 38026 | A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ CRingOps) | ||
| Theorem | dmnrngo 38027 | A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.) |
| ⊢ (𝑅 ∈ Dmn → 𝑅 ∈ RingOps) | ||
| Theorem | flddmn 38028 | A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.) |
| ⊢ (𝐾 ∈ Fld → 𝐾 ∈ Dmn) | ||
| Syntax | cigen 38029 | Extend class notation with the ideal generation function. |
| class IdlGen | ||
| Definition | df-igen 38030* | 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 38031* | 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 38032 | 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 38033 | 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 38034 | 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 38035 | 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 38036* | 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 38037* | 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 38038 | 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 38039 | 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 38040* | 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 38041 | 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 38042 | 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 38043 | 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 38044* | 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 38045 | 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 38046 | 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 38047 | 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 38048 | A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ 𝜑 → ⊥) ⇒ ⊢ 𝜑 | ||
| Theorem | notbinot1 38049 | Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ (¬ 𝜑 ↔ 𝜓) ↔ (𝜑 ↔ 𝜓)) | ||
| Theorem | bicontr 38050 | Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((¬ 𝜑 ↔ 𝜑) ↔ ⊥) | ||
| Theorem | impor 38051 | An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ ((¬ 𝜑 ∨ 𝜓) ∨ 𝜒)) | ||
| Theorem | orfa 38052 | The falsum ⊥ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ ((𝜑 ∨ ⊥) ↔ 𝜑) | ||
| Theorem | notbinot2 38053 | Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ 𝜓)) | ||
| Theorem | biimpor 38054 | A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((¬ 𝜑 ↔ 𝜓) ∨ 𝜒)) | ||
| Theorem | orfa1 38055 | Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → 𝜓) ⇒ ⊢ ((𝜑 ∨ ⊥) → 𝜓) | ||
| Theorem | orfa2 38056 | Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ⊥) ⇒ ⊢ ((𝜑 ∨ 𝜓) → 𝜓) | ||
| Theorem | bifald 38057 | 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 38058 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜓) | ||
| Theorem | orsird 38059 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → ¬ 𝜒) | ||
| Theorem | cnf1dd 38060 | 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 38061 | 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 38062 | 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 38063 | 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 38064 | 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 38065 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → ¬ (¬ 𝜓 ∨ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | notornotel2 38066 | A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → ¬ (𝜓 ∨ ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜒) | ||
| Theorem | contrd 38067 | A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.) |
| ⊢ (𝜑 → (¬ 𝜓 → 𝜒)) & ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | an12i 38068 | 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 38069 | An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.) |
| ⊢ ((𝜓 ∧ 𝜑) → 𝜒) & ⊢ ((¬ 𝜓 ∧ 𝜂) → 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜂) → 𝜒) | ||
| Theorem | selconj 38070 | 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 38071 | Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.) |
| ⊢ (𝜑 ↔ (⊤ ∧ 𝜑)) | ||
| Theorem | orel 38072 | 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 38073 | An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜓 → 𝜒) & ⊢ (𝜑 → ¬ 𝜒) ⇒ ⊢ ((𝜑 ∧ 𝜓) → ⊥) | ||
| Theorem | botel 38074 | An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜑 → ⊥) ⇒ ⊢ (𝜑 → 𝜓) | ||
| Theorem | tradd 38075 | Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (𝜑 ↔ (⊤ ∧ 𝜓)) | ||
| Theorem | gm-sbtru 38076 | Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤) | ||
| Theorem | sbfal 38077 | Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥) | ||
| Theorem | sbcani 38078 | 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 38079 | 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 38080 | 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 38081 | 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 38082 | 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 38083 | 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 38084* | 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 38085* | 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 38086* | 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 38087* | 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 38088 | 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 38089* | 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 38090 | 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 38091 | A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.) |
| ⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑥𝜓 & ⊢ ((𝜃 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
| Theorem | exlimddvfi 38092 | A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| ⊢ (𝜑 → ∃𝑥𝜃) & ⊢ Ⅎ𝑦𝜃 & ⊢ Ⅎ𝑦𝜓 & ⊢ ([𝑦 / 𝑥]𝜃 ↔ 𝜂) & ⊢ ((𝜂 ∧ 𝜓) → 𝜒) & ⊢ Ⅎ𝑦𝜒 ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | ||
| Theorem | sbceq1ddi 38093 | A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| ⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜓 → 𝜃) & ⊢ ([𝐴 / 𝑥]𝜒 ↔ 𝜃) & ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝜂) ⇒ ⊢ ((𝜑 ∧ 𝜓) → 𝜂) | ||
| Theorem | sbccom2lem 38094* | Lemma for sbccom2 38095. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
| Theorem | sbccom2 38095* | Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| ⊢ 𝐴 ∈ V ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
| Theorem | sbccom2f 38096* | Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [⦋𝐴 / 𝑥⦌𝐵 / 𝑦][𝐴 / 𝑥]𝜑) | ||
| Theorem | sbccom2fi 38097* | Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓) ⇒ ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐶 / 𝑦]𝜓) | ||
| Theorem | csbcom2fi 38098* | Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) |
| ⊢ 𝐴 ∈ V & ⊢ Ⅎ𝑦𝐴 & ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 & ⊢ ⦋𝐴 / 𝑥⦌𝐷 = 𝐸 ⇒ ⊢ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝐷 = ⦋𝐶 / 𝑦⦌𝐸 | ||
A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form. | ||
| Theorem | fald 38099 | Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
| ⊢ (𝜃 → ¬ ⊥) | ||
| Theorem | tsim1 38100 | A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.) |
| ⊢ (𝜃 → ((¬ 𝜑 ∨ 𝜓) ∨ ¬ (𝜑 → 𝜓))) | ||
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