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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ss2mcls 33101 | The closure is monotonic under subsets of the original set of expressions and the set of disjoint variable conditions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ mFS) & ⊢ (𝜑 → 𝐾 ⊆ 𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐸) & ⊢ (𝜑 → 𝑋 ⊆ 𝐾) & ⊢ (𝜑 → 𝑌 ⊆ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵)) | ||
Theorem | mclsax 33102* | The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ mFS) & ⊢ (𝜑 → 𝐾 ⊆ 𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐸) & ⊢ 𝐴 = (mAx‘𝑇) & ⊢ 𝐿 = (mSubst‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) & ⊢ (𝜑 → 〈𝑀, 𝑂, 𝑃〉 ∈ 𝐴) & ⊢ (𝜑 → 𝑆 ∈ ran 𝐿) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) ⇒ ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵)) | ||
Theorem | mclsind 33103* | Induction theorem for closure: any other set 𝑄 closed under the axioms and the hypotheses contains all the elements of the closure. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ mFS) & ⊢ (𝜑 → 𝐾 ⊆ 𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐸) & ⊢ 𝐴 = (mAx‘𝑇) & ⊢ 𝐿 = (mSubst‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) & ⊢ (𝜑 → 𝐵 ⊆ 𝑄) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄) & ⊢ ((𝜑 ∧ (〈𝑚, 𝑜, 𝑝〉 ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄) ⇒ ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄) | ||
Theorem | mppspstlem 33104* | Lemma for mppspst 33107. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) ⇒ ⊢ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃 | ||
Theorem | mppsval 33105* | Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) ⇒ ⊢ 𝐽 = {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} | ||
Theorem | elmpps 33106 | Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) ⇒ ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝐽 ↔ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))) | ||
Theorem | mppspst 33107 | A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) ⇒ ⊢ 𝐽 ⊆ 𝑃 | ||
Theorem | mthmval 33108 | A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ 𝑈 = (◡𝑅 “ (𝑅 “ 𝐽)) | ||
Theorem | elmthm 33109* | A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋)) | ||
Theorem | mthmi 33110 | A statement whose reduct is the reduct of a provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈) | ||
Theorem | mthmsta 33111 | A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑈 = (mThm‘𝑇) & ⊢ 𝑆 = (mPreSt‘𝑇) ⇒ ⊢ 𝑈 ⊆ 𝑆 | ||
Theorem | mppsthm 33112 | A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ 𝐽 ⊆ 𝑈 | ||
Theorem | mthmblem 33113 | Lemma for mthmb 33114. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈)) | ||
Theorem | mthmb 33114 | If two statements have the same reduct then one is a theorem iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) ⇒ ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈)) | ||
Theorem | mthmpps 33115 | Given a theorem, there is an explicitly definable witnessing provable pre-statement for the provability of the theorem. (However, this pre-statement requires infinitely many disjoint variable conditions, which is sometimes inconvenient.) (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝑈 = (mThm‘𝑇) & ⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝑉 = (mVars‘𝑇) & ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴})) & ⊢ 𝑀 = (𝐶 ∪ (𝐷 ∖ (𝑍 × 𝑍))) ⇒ ⊢ (𝑇 ∈ mFS → (〈𝐶, 𝐻, 𝐴〉 ∈ 𝑈 ↔ (〈𝑀, 𝐻, 𝐴〉 ∈ 𝐽 ∧ (𝑅‘〈𝑀, 𝐻, 𝐴〉) = (𝑅‘〈𝐶, 𝐻, 𝐴〉)))) | ||
Theorem | mclsppslem 33116* | The closure is closed under application of provable pre-statements. (Compare mclsax 33102.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ mFS) & ⊢ (𝜑 → 𝐾 ⊆ 𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐸) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝐿 = (mSubst‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) & ⊢ (𝜑 → 〈𝑀, 𝑂, 𝑃〉 ∈ 𝐽) & ⊢ (𝜑 → 𝑆 ∈ ran 𝐿) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) & ⊢ (𝜑 → 〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑇)) & ⊢ (𝜑 → 𝑠 ∈ ran 𝐿) & ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵))) & ⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀)) ⇒ ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵))) | ||
Theorem | mclspps 33117* | The closure is closed under application of provable pre-statements. (Compare mclsax 33102.) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐷 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCls‘𝑇) & ⊢ (𝜑 → 𝑇 ∈ mFS) & ⊢ (𝜑 → 𝐾 ⊆ 𝐷) & ⊢ (𝜑 → 𝐵 ⊆ 𝐸) & ⊢ 𝐽 = (mPPSt‘𝑇) & ⊢ 𝐿 = (mSubst‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) & ⊢ (𝜑 → 〈𝑀, 𝑂, 𝑃〉 ∈ 𝐽) & ⊢ (𝜑 → 𝑆 ∈ ran 𝐿) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵)) & ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏) ⇒ ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵)) | ||
Syntax | cm0s 33118 | Mapping expressions to statements. |
class m0St | ||
Syntax | cmsa 33119 | The set of syntax axioms. |
class mSA | ||
Syntax | cmwgfs 33120 | The set of weakly grammatical formal systems. |
class mWGFS | ||
Syntax | cmsy 33121 | The syntax typecode function. |
class mSyn | ||
Syntax | cmesy 33122 | The syntax typecode function for expressions. |
class mESyn | ||
Syntax | cmgfs 33123 | The set of grammatical formal systems. |
class mGFS | ||
Syntax | cmtree 33124 | The set of proof trees. |
class mTree | ||
Syntax | cmst 33125 | The set of syntax trees. |
class mST | ||
Syntax | cmsax 33126 | The indexing set for a syntax axiom. |
class mSAX | ||
Syntax | cmufs 33127 | The set of unambiguous formal sytems. |
class mUFS | ||
Definition | df-m0s 33128 | Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ m0St = (𝑎 ∈ V ↦ 〈∅, ∅, 𝑎〉) | ||
Definition | df-msa 33129* | Define the set of syntax axioms. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (◡(2nd ‘𝑎) ↾ (mVR‘𝑡)))}) | ||
Definition | df-mwgfs 33130* | Define the set of weakly grammatical formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((〈𝑑, ℎ, 𝑎〉 ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))} | ||
Definition | df-msyn 33131 | Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mSyn = Slot 6 | ||
Definition | df-mesyn 33132* | Define the syntax typecode function for expressions. (Contributed by Mario Carneiro, 12-Jun-2023.) |
⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒))) | ||
Definition | df-mgfs 33133* | Define the set of grammatical formal systems. (Contributed by Mario Carneiro, 12-Jun-2023.) |
⊢ mGFS = {𝑡 ∈ mWGFS ∣ ((mSyn‘𝑡):(mTC‘𝑡)⟶(mVT‘𝑡) ∧ ∀𝑐 ∈ (mVT‘𝑡)((mSyn‘𝑡)‘𝑐) = 𝑐 ∧ ∀𝑑∀ℎ∀𝑎(〈𝑑, ℎ, 𝑎〉 ∈ (mAx‘𝑡) → ∀𝑒 ∈ (ℎ ∪ {𝑎})((mESyn‘𝑡)‘𝑒) ∈ (mPPSt‘𝑡)))} | ||
Definition | df-mtree 33134* | Define the set of proof trees. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟〈(m0St‘𝑒), ∅〉 ∧ ∀𝑒 ∈ ℎ 𝑒𝑟〈((mStRed‘𝑡)‘〈𝑑, ℎ, 𝑒〉), ∅〉 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠‘𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ∪ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠‘𝑒)})) ⊆ 𝑟)))})) | ||
Definition | df-mst 33135 | Define the function mapping syntax expressions to syntax trees. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mST = (𝑡 ∈ V ↦ ((∅(mTree‘𝑡)∅) ↾ ((mEx‘𝑡) ↾ (mVT‘𝑡)))) | ||
Definition | df-msax 33136* | Define the indexing set for a syntax axiom's representation in a tree. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mSAX = (𝑡 ∈ V ↦ (𝑝 ∈ (mSA‘𝑡) ↦ ((mVH‘𝑡) “ ((mVars‘𝑡)‘𝑝)))) | ||
Definition | df-mufs 33137 | Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)} | ||
Syntax | cmuv 33138 | The universe of a model. |
class mUV | ||
Syntax | cmvl 33139 | The set of valuations. |
class mVL | ||
Syntax | cmvsb 33140 | Substitution for a valuation. |
class mVSubst | ||
Syntax | cmfsh 33141 | The freshness relation of a model. |
class mFresh | ||
Syntax | cmfr 33142 | The set of freshness relations. |
class mFRel | ||
Syntax | cmevl 33143 | The evaluation function of a model. |
class mEval | ||
Syntax | cmdl 33144 | The set of models. |
class mMdl | ||
Syntax | cusyn 33145 | The syntax function applied to elements of the model. |
class mUSyn | ||
Syntax | cgmdl 33146 | The set of models in a grammatical formal system. |
class mGMdl | ||
Syntax | cmitp 33147 | The interpretation function of the model. |
class mItp | ||
Syntax | cmfitp 33148 | The evaluation function derived from the interpretation. |
class mFromItp | ||
Definition | df-muv 33149 | Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mUV = Slot 7 | ||
Definition | df-mfsh 33150 | Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mFresh = Slot 8 | ||
Definition | df-mevl 33151 | Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mEval = Slot 9 | ||
Definition | df-mvl 33152* | Define the set of valuations. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)})) | ||
Definition | df-mvsb 33153* | Define substitution applied to a valuation. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVSubst = (𝑡 ∈ V ↦ {〈〈𝑠, 𝑚〉, 𝑥〉 ∣ ((𝑠 ∈ ran (mSubst‘𝑡) ∧ 𝑚 ∈ (mVL‘𝑡)) ∧ ∀𝑣 ∈ (mVR‘𝑡)𝑚dom (mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)) ∧ 𝑥 = (𝑣 ∈ (mVR‘𝑡) ↦ (𝑚(mEval‘𝑡)(𝑠‘((mVH‘𝑡)‘𝑣)))))}) | ||
Definition | df-mfrel 33154* | Define the set of freshness relations. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mFRel = (𝑡 ∈ V ↦ {𝑟 ∈ 𝒫 ((mUV‘𝑡) × (mUV‘𝑡)) ∣ (◡𝑟 = 𝑟 ∧ ∀𝑐 ∈ (mVT‘𝑡)∀𝑤 ∈ (𝒫 (mUV‘𝑡) ∩ Fin)∃𝑣 ∈ ((mUV‘𝑡) “ {𝑐})𝑤 ⊆ (𝑟 “ {𝑣}))}) | ||
Definition | df-mdl 33155* | Define the set of models of a formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mMdl = {𝑡 ∈ mFS ∣ [(mUV‘𝑡) / 𝑢][(mEx‘𝑡) / 𝑥][(mVL‘𝑡) / 𝑣][(mEval‘𝑡) / 𝑛][(mFresh‘𝑡) / 𝑓]((𝑢 ⊆ ((mTC‘𝑡) × V) ∧ 𝑓 ∈ (mFRel‘𝑡) ∧ 𝑛 ∈ (𝑢 ↑pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚 ∈ 𝑣 ((∀𝑒 ∈ 𝑥 (𝑛 “ {〈𝑚, 𝑒〉}) ⊆ (𝑢 “ {(1st ‘𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)〈𝑚, ((mVH‘𝑡)‘𝑦)〉𝑛(𝑚‘𝑦) ∧ ∀𝑑∀ℎ∀𝑎(〈𝑑, ℎ, 𝑎〉 ∈ (mAx‘𝑡) → ((∀𝑦∀𝑧(𝑦𝑑𝑧 → (𝑚‘𝑦)𝑓(𝑚‘𝑧)) ∧ ℎ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(〈𝑠, 𝑚〉(mVSubst‘𝑡)𝑦 → (𝑛 “ {〈𝑚, (𝑠‘𝑒)〉}) = (𝑛 “ {〈𝑦, 𝑒〉})) ∧ ∀𝑝 ∈ 𝑣 ∀𝑒 ∈ 𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {〈𝑚, 𝑒〉}) = (𝑛 “ {〈𝑝, 𝑒〉})) ∧ ∀𝑦 ∈ 𝑢 ∀𝑒 ∈ 𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {〈𝑚, 𝑒〉}) ⊆ (𝑓 “ {𝑦})))))} | ||
Definition | df-musyn 33156* | Define the syntax typecode function for the model universe. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ 〈((mSyn‘𝑡)‘(1st ‘𝑣)), (2nd ‘𝑣)〉)) | ||
Definition | df-gmdl 33157* | Define the set of models of a grammatical formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mGMdl = {𝑡 ∈ (mGFS ∩ mMdl) ∣ (∀𝑐 ∈ (mTC‘𝑡)((mUV‘𝑡) “ {𝑐}) ⊆ ((mUV‘𝑡) “ {((mSyn‘𝑡)‘𝑐)}) ∧ ∀𝑣 ∈ (mUV‘𝑐)∀𝑤 ∈ (mUV‘𝑐)(𝑣(mFresh‘𝑡)𝑤 ↔ 𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {〈𝑚, 𝑒〉}) = (((mEval‘𝑡) “ {〈𝑚, ((mESyn‘𝑡)‘𝑒)〉}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)})))} | ||
Definition | df-mitp 33158* | Define the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔 ∈ X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥∃𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎)))))) | ||
Definition | df-mfitp 33159* | Define a function that produces the evaluation function, given the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mFromItp = (𝑡 ∈ V ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st ‘𝑡)‘𝑎)}) ↑m X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)})) ↦ (℩𝑛 ∈ ((mUV‘𝑡) ↑pm ((mVL‘𝑡) × (mEx‘𝑡)))∀𝑚 ∈ (mVL‘𝑡)(∀𝑣 ∈ (mVR‘𝑡)〈𝑚, ((mVH‘𝑡)‘𝑣)〉𝑛(𝑚‘𝑣) ∧ ∀𝑒∀𝑎∀𝑔(𝑒(mST‘𝑡)〈𝑎, 𝑔〉 → 〈𝑚, 𝑒〉𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {〈𝑚, 𝑒〉}) = ((𝑛 “ {〈𝑚, ((mESyn‘𝑡)‘𝑒)〉}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)})))))) | ||
Syntax | citr 33160 | Integral subring of a ring. |
class IntgRing | ||
Syntax | ccpms 33161 | Completion of a metric space. |
class cplMetSp | ||
Syntax | chlb 33162 | Embeddings for a direct limit. |
class HomLimB | ||
Syntax | chlim 33163 | Direct limit structure. |
class HomLim | ||
Syntax | cpfl 33164 | Polynomial extension field. |
class polyFld | ||
Syntax | csf1 33165 | Splitting field for a single polynomial (auxiliary). |
class splitFld1 | ||
Syntax | csf 33166 | Splitting field for a finite set of polynomials. |
class splitFld | ||
Syntax | cpsl 33167 | Splitting field for a sequence of polynomials. |
class polySplitLim | ||
Definition | df-irng 33168* | Define the subring of elements of 𝑟 integral over 𝑠 in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ IntgRing = (𝑟 ∈ V, 𝑠 ∈ V ↦ ∪ 𝑓 ∈ (Monic1p‘(𝑟 ↾s 𝑠))(◡𝑓 “ {(0g‘𝑟)})) | ||
Definition | df-cplmet 33169* | A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ cplMetSp = (𝑤 ∈ V ↦ ⦋((𝑤 ↑s ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟⦌⦋(Base‘𝑟) / 𝑣⦌⦋{〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑔‘𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒⦌((𝑟 /s 𝑒) sSet {〈(dist‘ndx), {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑝 ∈ 𝑣 ∃𝑞 ∈ 𝑣 ((𝑥 = [𝑝]𝑒 ∧ 𝑦 = [𝑞]𝑒) ∧ (𝑝 ∘f (dist‘𝑟)𝑞) ⇝ 𝑧)}〉})) | ||
Definition | df-homlimb 33170* | The input to this function is a sequence (on ℕ) of homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined. This function returns the pair 〈𝑆, 𝐺〉 where 𝑆 is the terminal object and 𝐺 is a sequence of functions such that 𝐺(𝑛):𝑅(𝑛)⟶𝑆 and 𝐺(𝑛) = 𝐹(𝑛) ∘ 𝐺(𝑛 + 1). (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ HomLimB = (𝑓 ∈ V ↦ ⦋∪ 𝑛 ∈ ℕ ({𝑛} × dom (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ 〈((1st ‘𝑥) + 1), ((𝑓‘(1st ‘𝑥))‘(2nd ‘𝑥))〉) ⊆ 𝑠)} / 𝑒⦌〈(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [〈𝑛, 𝑥〉]𝑒))〉) | ||
Definition | df-homlim 33171* | The input to this function is a sequence (on ℕ) of structures 𝑅(𝑛) and homomorphisms 𝐹(𝑛):𝑅(𝑛)⟶𝑅(𝑛 + 1). The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ HomLim = (𝑟 ∈ V, 𝑓 ∈ V ↦ ⦋( HomLimB ‘𝑓) / 𝑒⦌⦋(1st ‘𝑒) / 𝑣⦌⦋(2nd ‘𝑒) / 𝑔⦌({〈(Base‘ndx), 𝑣〉, 〈(+g‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, ((𝑔‘𝑛)‘(𝑥(+g‘(𝑟‘𝑛))𝑦))〉)〉, 〈(.r‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, ((𝑔‘𝑛)‘(𝑥(.r‘(𝑟‘𝑛))𝑦))〉)〉} ∪ {〈(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ (◡(𝑔‘𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟‘𝑛))}〉, 〈(dist‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔‘𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔‘𝑛)‘𝑛) ↦ 〈〈((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)〉, (𝑥(dist‘(𝑟‘𝑛))𝑦)〉)〉, 〈(le‘ndx), ∪ 𝑛 ∈ ℕ (◡(𝑔‘𝑛) ∘ ((le‘(𝑟‘𝑛)) ∘ (𝑔‘𝑛)))〉})) | ||
Definition | df-plfl 33172* | Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑧 ∈ (Base‘𝑟) ↦ [(𝑧( ·𝑠 ‘𝑠)(1r‘𝑠))](𝑠 ~QG 𝑖)) / 𝑓⦌〈⦋(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet 〈(le‘ndx), ⦋(𝑧 ∈ (Base‘𝑡) ↦ (℩𝑞 ∈ 𝑧 (𝑟 deg1 𝑞) < (𝑟 deg1 𝑝))) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))〉), 𝑓〉) | ||
Definition | df-sfl1 33173* |
Temporary construction for the splitting field of a polynomial. The
inputs are a field 𝑟 and a polynomial 𝑝 that we
want to split,
along with a tuple 𝑗 in the same format as the output.
The output
is a tuple 〈𝑆, 𝐹〉 where 𝑆 is the splitting field
and 𝐹
is an injective homomorphism from the original field 𝑟.
The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋( mPoly ‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠 deg1 𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), 〈𝑠, 𝑓〉, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌〈(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))〉)), 𝑗)‘(card‘(1...(𝑟 deg1 𝑝)))))) | ||
Definition | df-sfl 33174* | Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple 〈𝑆, 𝐹〉 where 𝑆 is the totally ordered splitting field and 𝐹 is an injective homomorphism from the original field 𝑟. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {〈0, 〈𝑟, ( I ↾ (Base‘𝑟))〉〉}))‘(♯‘𝑝))))) | ||
Definition | df-psl 33175* | Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring 𝑟, a strict order on 𝑟, and a sequence 𝑝:ℕ⟶(𝒫 𝑟 ∩ Fin) of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ polySplitLim = (𝑟 ∈ V, 𝑝 ∈ ((𝒫 (Base‘𝑟) ∩ Fin) ↑m ℕ) ↦ ⦋(1st ∘ seq0((𝑔 ∈ V, 𝑞 ∈ V ↦ ⦋(1st ‘𝑔) / 𝑒⦌⦋(1st ‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2nd ‘𝑔)))) / 𝑓⦌〈𝑓, ((2nd ‘𝑔) ∘ (2nd ‘𝑓))〉), (𝑝 ∪ {〈0, 〈〈𝑟, ∅〉, ( I ↾ (Base‘𝑟))〉〉}))) / 𝑓⦌((1st ∘ (𝑓 shift 1)) HomLim (2nd ∘ 𝑓))) | ||
Syntax | czr 33176 | Integral elements of a ring. |
class ZRing | ||
Syntax | cgf 33177 | Galois finite field. |
class GF | ||
Syntax | cgfo 33178 | Galois limit field. |
class GF∞ | ||
Syntax | ceqp 33179 | Equivalence relation for df-qp 33190. |
class ~Qp | ||
Syntax | crqp 33180 | Equivalence relation representatives for df-qp 33190. |
class /Qp | ||
Syntax | cqp 33181 | The set of 𝑝-adic rational numbers. |
class Qp | ||
Syntax | czp 33182 | The set of 𝑝-adic integers. (Not to be confused with czn 20323.) |
class Zp | ||
Syntax | cqpa 33183 | Algebraic completion of the 𝑝-adic rational numbers. |
class _Qp | ||
Syntax | ccp 33184 | Metric completion of _Qp. |
class Cp | ||
Definition | df-zrng 33185 | Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟))) | ||
Definition | df-gf 33186* | Define the Galois finite field of order 𝑝↑𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) | ||
Definition | df-gfoo 33187* | Define the Galois field of order 𝑝↑+∞, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ GF∞ = (𝑝 ∈ ℙ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)}))) | ||
Definition | df-eqp 33188* | Define an equivalence relation on ℤ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum Σ𝑘 ≤ 𝑛𝑓(𝑘)(𝑝↑𝑘) is a multiple of 𝑝↑(𝑛 + 1) for every 𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ ~Qp = (𝑝 ∈ ℙ ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑m ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)}) | ||
Definition | df-rqp 33189* | There is a unique element of (ℤ ↑m (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑m ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑m ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑m (0...(𝑝 − 1))))))) | ||
Definition | df-qp 33190* | Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 10-Oct-2021.) |
⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑m (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ≥(◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘f + 𝑔)))〉, 〈(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))〉} ∪ {〈(le‘ndx), {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}〉}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ, < )))))) | ||
Definition | df-zp 33191 | Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ Zp = (ZRing ∘ Qp) | ||
Definition | df-qpa 33192* | Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟 deg1 𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))}))) | ||
Definition | df-cp 33193 | Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ Cp = ( cplMetSp ∘ _Qp) | ||
I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from https://us.metamath.org/other/milpgame/milpgame.html.) | ||
Theorem | problem1 33194 | Practice problem 1. Clues: 5p4e9 11874 3p2e5 11867 eqtri 2761 oveq1i 7180. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ ((3 + 2) + 4) = 9 | ||
Theorem | problem2 33195 | Practice problem 2. Clues: oveq12i 7182 adddiri 10732 add4i 10942 mulcli 10726 recni 10733 2re 11790 3eqtri 2765 10re 12198 5re 11803 1re 10719 4re 11800 eqcomi 2747 5p4e9 11874 oveq1i 7180 df-3 11780. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.) |
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) | ||
Theorem | problem3 33196 | Practice problem 3. Clues: eqcomi 2747 eqtri 2761 subaddrii 11053 recni 10733 4re 11800 3re 11796 1re 10719 df-4 11781 addcomi 10909. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ (𝐴 + 3) = 4 ⇒ ⊢ 𝐴 = 1 | ||
Theorem | problem4 33197 | Practice problem 4. Clues: pm3.2i 474 eqcomi 2747 eqtri 2761 subaddrii 11053 recni 10733 7re 11809 6re 11806 ax-1cn 10673 df-7 11784 ax-mp 5 oveq1i 7180 3cn 11797 2cn 11791 df-3 11780 mulid2i 10724 subdiri 11168 mp3an 1462 mulcli 10726 subadd23 10976 oveq2i 7181 oveq12i 7182 3t2e6 11882 mulcomi 10727 subcli 11040 biimpri 231 subadd2i 11052. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 3 & ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7 ⇒ ⊢ (𝐴 = 1 ∧ 𝐵 = 2) | ||
Theorem | problem5 33198 | Practice problem 5. Clues: 3brtr3i 5059 mpbi 233 breqtri 5055 ltaddsubi 11279 remulcli 10735 2re 11790 3re 11796 9re 11815 eqcomi 2747 mvlladdi 10982 3cn 6cn 11807 eqtr3i 2763 6p3e9 11876 addcomi 10909 ltdiv1ii 11647 6re 11806 nngt0i 11755 2nn 11789 divcan3i 11464 recni 10733 2cn 11791 2ne0 11820 mpbir 234 eqtri 2761 mulcomi 10727 3t2e6 11882 divmuli 11472. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℝ & ⊢ ((2 · 𝐴) + 3) < 9 ⇒ ⊢ 𝐴 < 3 | ||
Theorem | quad3 33199 | Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) |
⊢ 𝑋 ∈ ℂ & ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ⇒ ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴))) | ||
Theorem | climuzcnv 33200* | Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.) |
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑)))) |
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