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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-gzrep 32701 | The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxRep = (𝑢 ∈ (Fmla‘ω) ↦ (∀𝑔3o∃𝑔1o∀𝑔2o(∀𝑔1o𝑢 →𝑔 (2o=𝑔1o)) →𝑔 ∀𝑔1o∀𝑔2o((2o∈𝑔1o) ↔𝑔 ∃𝑔3o((3o∈𝑔∅)∧𝑔∀𝑔1o𝑢)))) | ||
Definition | df-gzpow 32702 | The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxPow = ∃𝑔1o∀𝑔2o(∀𝑔1o((1o∈𝑔2o) ↔𝑔 (1o∈𝑔∅)) →𝑔 (2o∈𝑔1o)) | ||
Definition | df-gzun 32703 | 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 32704 | 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 32705 | 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 32706* | 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 32707 | The set of constants. |
class mCN | ||
Syntax | cmvar 32708 | The set of variables. |
class mVR | ||
Syntax | cmty 32709 | The type function. |
class mType | ||
Syntax | cmvt 32710 | The set of variable typecodes. |
class mVT | ||
Syntax | cmtc 32711 | The set of typecodes. |
class mTC | ||
Syntax | cmax 32712 | The set of axioms. |
class mAx | ||
Syntax | cmrex 32713 | The set of raw expressions. |
class mREx | ||
Syntax | cmex 32714 | The set of expressions. |
class mEx | ||
Syntax | cmdv 32715 | The set of distinct variables. |
class mDV | ||
Syntax | cmvrs 32716 | The variables in an expression. |
class mVars | ||
Syntax | cmrsub 32717 | The set of raw substitutions. |
class mRSubst | ||
Syntax | cmsub 32718 | The set of substitutions. |
class mSubst | ||
Syntax | cmvh 32719 | The set of variable hypotheses. |
class mVH | ||
Syntax | cmpst 32720 | The set of pre-statements. |
class mPreSt | ||
Syntax | cmsr 32721 | The reduct of a pre-statement. |
class mStRed | ||
Syntax | cmsta 32722 | The set of statements. |
class mStat | ||
Syntax | cmfs 32723 | The set of formal systems. |
class mFS | ||
Syntax | cmcls 32724 | The closure of a set of statements. |
class mCls | ||
Syntax | cmpps 32725 | The set of provable pre-statements. |
class mPPSt | ||
Syntax | cmthm 32726 | The set of theorems. |
class mThm | ||
Definition | df-mcn 32727 | Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCN = Slot 1 | ||
Definition | df-mvar 32728 | Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVR = Slot 2 | ||
Definition | df-mty 32729 | Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mType = Slot 3 | ||
Definition | df-mtc 32730 | Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mTC = Slot 4 | ||
Definition | df-mmax 32731 | Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mAx = Slot 5 | ||
Definition | df-mvt 32732 | 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 32733 | 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 32734 | 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 32735 | 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 32736* | 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 32737* | 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 32738* | 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 32739* | 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 32740* | 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 32741* | 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 32742 | Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | ||
Definition | df-mfs 32743* | 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 32744* | 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 32745* | 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 32746 | Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | ||
Theorem | mvtval 32747 | The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVT‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ 𝑉 = ran 𝑌 | ||
Theorem | mrexval 32748 | 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 32749 | 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 32750 | 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 32751 | 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 32752 | 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 32753 | 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 32754* | 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 32755* | 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 32756* | 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 32757 | 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 32758 | 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 32759 | 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 32760 | 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 32761 | 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 32762 | 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 32763 | The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) | ||
Theorem | mrsubf 32764 | 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 32765 | 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 32766 | 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 32767* | 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 32796.) (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 32768 | 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 32769* | 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 32770* | 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 32771* | 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 32772 | 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 32773 | 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 32774 | 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 32775* | 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 32776 | 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 32777 | 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 32778 | 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 32779 | 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 32780* | 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 32781 | 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 32782* | 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 32783 | 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 32784* | 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 32785 | 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 32786 | A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ 𝑃 ⊆ ((V × V) × V) | ||
Theorem | mpst123 32787 | 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 32788 | 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 32789 | 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 32790 | 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 32791 | Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 = ran 𝑅 | ||
Theorem | msrid 32792 | 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 32793 | 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 32794 | A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 ⊆ 𝑃 | ||
Theorem | elmsta 32795 | 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 32796* | 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 32797 | 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 32798 | 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 32799 | 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 32800 | 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) |
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