Home | Metamath
Proof Explorer Theorem List (p. 327 of 449) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28623) |
Hilbert Space Explorer
(28624-30146) |
Users' Mathboxes
(30147-44804) |
Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-gzun 32601 | The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxUn = ∃𝑔1o∀𝑔2o(∃𝑔1o((2o∈𝑔1o)∧𝑔(1o∈𝑔∅)) →𝑔 (2o∈𝑔1o)) | ||
Definition | df-gzreg 32602 | The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxReg = (∃𝑔1o(1o∈𝑔∅) →𝑔 ∃𝑔1o((1o∈𝑔∅)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ¬𝑔(2o∈𝑔∅)))) | ||
Definition | df-gzinf 32603 | The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxInf = ∃𝑔1o((∅∈𝑔1o)∧𝑔∀𝑔2o((2o∈𝑔1o) →𝑔 ∃𝑔∅((2o∈𝑔∅)∧𝑔(∅∈𝑔1o)))) | ||
Definition | df-gzf 32604* | Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ ZF = {𝑚 ∣ ((Tr 𝑚 ∧ 𝑚⊧AxExt ∧ 𝑚⊧AxPow) ∧ (𝑚⊧AxUn ∧ 𝑚⊧AxReg ∧ 𝑚⊧AxInf) ∧ ∀𝑢 ∈ (Fmla‘ω)𝑚⊧(AxRep‘𝑢))} | ||
This is a formalization of Appendix C of the Metamath book, which describes the mathematical representation of a formal system, of which set.mm (this file) is one. | ||
Syntax | cmcn 32605 | The set of constants. |
class mCN | ||
Syntax | cmvar 32606 | The set of variables. |
class mVR | ||
Syntax | cmty 32607 | The type function. |
class mType | ||
Syntax | cmvt 32608 | The set of variable typecodes. |
class mVT | ||
Syntax | cmtc 32609 | The set of typecodes. |
class mTC | ||
Syntax | cmax 32610 | The set of axioms. |
class mAx | ||
Syntax | cmrex 32611 | The set of raw expressions. |
class mREx | ||
Syntax | cmex 32612 | The set of expressions. |
class mEx | ||
Syntax | cmdv 32613 | The set of distinct variables. |
class mDV | ||
Syntax | cmvrs 32614 | The variables in an expression. |
class mVars | ||
Syntax | cmrsub 32615 | The set of raw substitutions. |
class mRSubst | ||
Syntax | cmsub 32616 | The set of substitutions. |
class mSubst | ||
Syntax | cmvh 32617 | The set of variable hypotheses. |
class mVH | ||
Syntax | cmpst 32618 | The set of pre-statements. |
class mPreSt | ||
Syntax | cmsr 32619 | The reduct of a pre-statement. |
class mStRed | ||
Syntax | cmsta 32620 | The set of statements. |
class mStat | ||
Syntax | cmfs 32621 | The set of formal systems. |
class mFS | ||
Syntax | cmcls 32622 | The closure of a set of statements. |
class mCls | ||
Syntax | cmpps 32623 | The set of provable pre-statements. |
class mPPSt | ||
Syntax | cmthm 32624 | The set of theorems. |
class mThm | ||
Definition | df-mcn 32625 | Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCN = Slot 1 | ||
Definition | df-mvar 32626 | Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVR = Slot 2 | ||
Definition | df-mty 32627 | Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mType = Slot 3 | ||
Definition | df-mtc 32628 | Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mTC = Slot 4 | ||
Definition | df-mmax 32629 | Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mAx = Slot 5 | ||
Definition | df-mvt 32630 | Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVT = (𝑡 ∈ V ↦ ran (mType‘𝑡)) | ||
Definition | df-mrex 32631 | Define the set of "raw expressions", which are expressions without a typecode attached. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mREx = (𝑡 ∈ V ↦ Word ((mCN‘𝑡) ∪ (mVR‘𝑡))) | ||
Definition | df-mex 32632 | Define the set of expressions, which are strings of constants and variables headed by a typecode constant. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mEx = (𝑡 ∈ V ↦ ((mTC‘𝑡) × (mREx‘𝑡))) | ||
Definition | df-mdv 32633 | Define the set of distinct variable conditions, which are pairs of distinct variables. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mDV = (𝑡 ∈ V ↦ (((mVR‘𝑡) × (mVR‘𝑡)) ∖ I )) | ||
Definition | df-mvrs 32634* | Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVars = (𝑡 ∈ V ↦ (𝑒 ∈ (mEx‘𝑡) ↦ (ran (2nd ‘𝑒) ∩ (mVR‘𝑡)))) | ||
Definition | df-mrsub 32635* | Define a substitution of raw expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) | ||
Definition | df-msub 32636* | Define a substitution of expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mEx‘𝑡) ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑡)‘𝑓)‘(2nd ‘𝑒))〉))) | ||
Definition | df-mvh 32637* | Define the mapping from variables to their variable hypothesis. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVH = (𝑡 ∈ V ↦ (𝑣 ∈ (mVR‘𝑡) ↦ 〈((mType‘𝑡)‘𝑣), 〈“𝑣”〉〉)) | ||
Definition | df-mpst 32638* | Define the set of all pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡))) | ||
Definition | df-msr 32639* | Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStRed = (𝑡 ∈ V ↦ (𝑠 ∈ (mPreSt‘𝑡) ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌〈((1st ‘(1st ‘𝑠)) ∩ ⦋∪ ((mVars‘𝑡) “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎〉)) | ||
Definition | df-msta 32640 | Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | ||
Definition | df-mfs 32641* | Define the set of all formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mFS = {𝑡 ∣ ((((mCN‘𝑡) ∩ (mVR‘𝑡)) = ∅ ∧ (mType‘𝑡):(mVR‘𝑡)⟶(mTC‘𝑡)) ∧ ((mAx‘𝑡) ⊆ (mStat‘𝑡) ∧ ∀𝑣 ∈ (mVT‘𝑡) ¬ (◡(mType‘𝑡) “ {𝑣}) ∈ Fin))} | ||
Definition | df-mcls 32642* | Define the closure of a set of statements relative to a set of disjointness constraints. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(〈𝑚, 𝑜, 𝑝〉 ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})) | ||
Definition | df-mpps 32643* | Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mPPSt = (𝑡 ∈ V ↦ {〈〈𝑑, ℎ〉, 𝑎〉 ∣ (〈𝑑, ℎ, 𝑎〉 ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))}) | ||
Definition | df-mthm 32644 | Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | ||
Theorem | mvtval 32645 | The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVT‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ 𝑉 = ran 𝑌 | ||
Theorem | mrexval 32646 | The set of "raw expressions", which are expressions without a typecode, that is, just sequences of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → 𝑅 = Word (𝐶 ∪ 𝑉)) | ||
Theorem | mexval 32647 | The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐾 = (mTC‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) ⇒ ⊢ 𝐸 = (𝐾 × 𝑅) | ||
Theorem | mexval2 32648 | The set of expressions, which are pairs whose first element is a typecode, and whose second element is a list of constants and variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐾 = (mTC‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) ⇒ ⊢ 𝐸 = (𝐾 × Word (𝐶 ∪ 𝑉)) | ||
Theorem | mdvval 32649 | The set of disjoint variable conditions, which are pairs of distinct variables. (This definition differs from appendix C, which uses unordered pairs instead. We use ordered pairs, but all sets of disjoint variable conditions of interest will be symmetric, so it does not matter.) (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐷 = (mDV‘𝑇) ⇒ ⊢ 𝐷 = ((𝑉 × 𝑉) ∖ I ) | ||
Theorem | mvrsval 32650 | The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) = (ran (2nd ‘𝑋) ∩ 𝑉)) | ||
Theorem | mvrsfpw 32651 | The set of variables in an expression is a finite subset of 𝑉. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑊 = (mVars‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝐸 → (𝑊‘𝑋) ∈ (𝒫 𝑉 ∩ Fin)) | ||
Theorem | mrsubffval 32652* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) ⇒ ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), 〈“𝑣”〉)) ∘ 𝑒))))) | ||
Theorem | mrsubfval 32653* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) | ||
Theorem | mrsubval 32654* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋))) | ||
Theorem | mrsubcv 32655 | The value of a substituted singleton. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ (𝐶 ∪ 𝑉)) → ((𝑆‘𝐹)‘〈“𝑋”〉) = if(𝑋 ∈ 𝐴, (𝐹‘𝑋), 〈“𝑋”〉)) | ||
Theorem | mrsubvr 32656 | The value of a substituted variable. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐴) → ((𝑆‘𝐹)‘〈“𝑋”〉) = (𝐹‘𝑋)) | ||
Theorem | mrsubff 32657 | A substitution is a function from 𝑅 to 𝑅. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) | ||
Theorem | mrsubrn 32658 | Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) | ||
Theorem | mrsubff1 32659 | When restricted to complete mappings, the substitution-producing function is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅)) | ||
Theorem | mrsubff1o 32660 | When restricted to complete mappings, the substitution-producing function is bijective to the set of all substitutions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆) | ||
Theorem | mrsub0 32661 | The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) | ||
Theorem | mrsubf 32662 | A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) ⇒ ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝑅⟶𝑅) | ||
Theorem | mrsubccat 32663 | Substitution distributes over concatenation. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) ⇒ ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹‘𝑋) ++ (𝐹‘𝑌))) | ||
Theorem | mrsubcn 32664 | A substitution does not change the value of constant substrings. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐶 = (mCN‘𝑇) ⇒ ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ (𝐶 ∖ 𝑉)) → (𝐹‘〈“𝑋”〉) = 〈“𝑋”〉) | ||
Theorem | elmrsubrn 32665* | Characterization of the substitutions as functions from expressions to expressions that distribute under concatenation and map constants to themselves. (The constant part uses (𝐶 ∖ 𝑉) because we don't know that 𝐶 and 𝑉 are disjoint until we get to ismfs 32694.) (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐶 = (mCN‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → (𝐹 ∈ ran 𝑆 ↔ (𝐹:𝑅⟶𝑅 ∧ ∀𝑐 ∈ (𝐶 ∖ 𝑉)(𝐹‘〈“𝑐”〉) = 〈“𝑐”〉 ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹‘𝑥) ++ (𝐹‘𝑦))))) | ||
Theorem | mrsubco 32666 | The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆) | ||
Theorem | mrsubvrs 32667* | The set of variables in a substitution is the union, indexed by the variables in the original expression, of the variables in the substitution to that variable. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) ⇒ ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑋 ∈ 𝑅) → (ran (𝐹‘𝑋) ∩ 𝑉) = ∪ 𝑥 ∈ (ran 𝑋 ∩ 𝑉)(ran (𝐹‘〈“𝑥”〉) ∩ 𝑉)) | ||
Theorem | msubffval 32668* | A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑂 = (mRSubst‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))〉))) | ||
Theorem | msubfval 32669* | A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑂 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))〉)) | ||
Theorem | msubval 32670 | A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑂 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) | ||
Theorem | msubrsub 32671 | A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑂 = (mRSubst‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋))) | ||
Theorem | msubty 32672 | The type of a substituted expression is the same as the original type. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) ⇒ ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1st ‘((𝑆‘𝐹)‘𝑋)) = (1st ‘𝑋)) | ||
Theorem | elmsubrn 32673* | Characterization of substitution in terms of raw substitution, without reference to the generating functions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑂 = (mRSubst‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) ⇒ ⊢ ran 𝑆 = ran (𝑓 ∈ ran 𝑂 ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (𝑓‘(2nd ‘𝑒))〉)) | ||
Theorem | msubrn 32674 | Although it is defined for partial mappings of variables, every partial substitution is a substitution on some complete mapping of the variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) ⇒ ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) | ||
Theorem | msubff 32675 | A substitution is a function from 𝐸 to 𝐸. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑅 = (mREx‘𝑇) & ⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) | ||
Theorem | msubco 32676 | The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mSubst‘𝑇) ⇒ ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆) | ||
Theorem | msubf 32677 | A substitution is a function. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mSubst‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) ⇒ ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:𝐸⟶𝐸) | ||
Theorem | mvhfval 32678* | Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) ⇒ ⊢ 𝐻 = (𝑣 ∈ 𝑉 ↦ 〈(𝑌‘𝑣), 〈“𝑣”〉〉) | ||
Theorem | mvhval 32679 | Value of the function mapping variables to their corresponding variable expressions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) & ⊢ 𝐻 = (mVH‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝐻‘𝑋) = 〈(𝑌‘𝑋), 〈“𝑋”〉〉) | ||
Theorem | mpstval 32680* | A pre-statement is an ordered triple, whose first member is a symmetric set of disjoint variable conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) | ||
Theorem | elmpst 32681 | Property of being a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mDV‘𝑇) & ⊢ 𝐸 = (mEx‘𝑇) & ⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸)) | ||
Theorem | msrfval 32682* | Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVars‘𝑇) & ⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑅 = (mStRed‘𝑇) ⇒ ⊢ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd ‘(1st ‘𝑠)) / ℎ⦌⦋(2nd ‘𝑠) / 𝑎⦌〈((1st ‘(1st ‘𝑠)) ∩ ⦋∪ (𝑉 “ (ℎ ∪ {𝑎})) / 𝑧⦌(𝑧 × 𝑧)), ℎ, 𝑎〉) | ||
Theorem | msrval 32683 | Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVars‘𝑇) & ⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴})) ⇒ ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝑅‘〈𝐷, 𝐻, 𝐴〉) = 〈(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴〉) | ||
Theorem | mpstssv 32684 | A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ 𝑃 ⊆ ((V × V) × V) | ||
Theorem | mpst123 32685 | Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝑃 → 𝑋 = 〈(1st ‘(1st ‘𝑋)), (2nd ‘(1st ‘𝑋)), (2nd ‘𝑋)〉) | ||
Theorem | mpstrcl 32686 | The elements of a pre-statement are sets. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V)) | ||
Theorem | msrf 32687 | The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑅 = (mStRed‘𝑇) ⇒ ⊢ 𝑅:𝑃⟶𝑃 | ||
Theorem | msrrcl 32688 | If 𝑋 and 𝑌 have the same reduct, then one is a pre-statement iff the other is. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑅 = (mStRed‘𝑇) ⇒ ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑃 ↔ 𝑌 ∈ 𝑃)) | ||
Theorem | mstaval 32689 | Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 = ran 𝑅 | ||
Theorem | msrid 32690 | The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ (𝑋 ∈ 𝑆 → (𝑅‘𝑋) = 𝑋) | ||
Theorem | msrfo 32691 | The reduct of a pre-statement is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) & ⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ 𝑅:𝑃–onto→𝑆 | ||
Theorem | mstapst 32692 | A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 ⊆ 𝑃 | ||
Theorem | elmsta 32693 | Property of being a statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) & ⊢ 𝑉 = (mVars‘𝑇) & ⊢ 𝑍 = ∪ (𝑉 “ (𝐻 ∪ {𝐴})) ⇒ ⊢ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑆 ↔ (〈𝐷, 𝐻, 𝐴〉 ∈ 𝑃 ∧ 𝐷 ⊆ (𝑍 × 𝑍))) | ||
Theorem | ismfs 32694* | A formal system is a tuple 〈mCN, mVR, mType, mVT, mTC, mAx〉 such that: mCN and mVR are disjoint; mType is a function from mVR to mVT; mVT is a subset of mTC; mAx is a set of statements; and for each variable typecode, there are infinitely many variables of that type. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) & ⊢ 𝐹 = (mVT‘𝑇) & ⊢ 𝐾 = (mTC‘𝑇) & ⊢ 𝐴 = (mAx‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ (𝑇 ∈ 𝑊 → (𝑇 ∈ mFS ↔ (((𝐶 ∩ 𝑉) = ∅ ∧ 𝑌:𝑉⟶𝐾) ∧ (𝐴 ⊆ 𝑆 ∧ ∀𝑣 ∈ 𝐹 ¬ (◡𝑌 “ {𝑣}) ∈ Fin)))) | ||
Theorem | mfsdisj 32695 | The constants and variables of a formal system are disjoint. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐶 = (mCN‘𝑇) & ⊢ 𝑉 = (mVR‘𝑇) ⇒ ⊢ (𝑇 ∈ mFS → (𝐶 ∩ 𝑉) = ∅) | ||
Theorem | mtyf2 32696 | The type function maps variables to typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐾 = (mTC‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐾) | ||
Theorem | mtyf 32697 | The type function maps variables to variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVR‘𝑇) & ⊢ 𝐹 = (mVT‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ (𝑇 ∈ mFS → 𝑌:𝑉⟶𝐹) | ||
Theorem | mvtss 32698 | The set of variable typecodes is a subset of all typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐹 = (mVT‘𝑇) & ⊢ 𝐾 = (mTC‘𝑇) ⇒ ⊢ (𝑇 ∈ mFS → 𝐹 ⊆ 𝐾) | ||
Theorem | maxsta 32699 | An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐴 = (mAx‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆) | ||
Theorem | mvtinf 32700 | Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝐹 = (mVT‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ ((𝑇 ∈ mFS ∧ 𝑋 ∈ 𝐹) → ¬ (◡𝑌 “ {𝑋}) ∈ Fin) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |