Home | Metamath
Proof Explorer Theorem List (p. 307 of 424) | < 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
(1-27159) |
Hilbert Space Explorer
(27160-28684) |
Users' Mathboxes
(28685-42360) |
Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cgzg 30601 | The Axiom of Regularity. |
class AxReg | ||
Syntax | cgzi 30602 | The Axiom of Infinity. |
class AxInf | ||
Syntax | cgzf 30603 | The set of models of ZF. |
class ZF | ||
Definition | df-gzext 30604 | The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.) |
⊢ AxExt = (∀𝑔2𝑜((2𝑜∈𝑔∅) ↔𝑔 (2𝑜∈𝑔1𝑜)) →𝑔 (∅=𝑔1𝑜)) | ||
Definition | df-gzrep 30605 | 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 30606 | 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 30607 | 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 30608 | 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 30609 | 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 30610* | 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 30611 | The set of constants. |
class mCN | ||
Syntax | cmvar 30612 | The set of variables. |
class mVR | ||
Syntax | cmty 30613 | The type function. |
class mType | ||
Syntax | cmvt 30614 | The set of variable typecodes. |
class mVT | ||
Syntax | cmtc 30615 | The set of typecodes. |
class mTC | ||
Syntax | cmax 30616 | The set of axioms. |
class mAx | ||
Syntax | cmrex 30617 | The set of raw expressions. |
class mREx | ||
Syntax | cmex 30618 | The set of expressions. |
class mEx | ||
Syntax | cmdv 30619 | The set of distinct variables. |
class mDV | ||
Syntax | cmvrs 30620 | The variables in an expression. |
class mVars | ||
Syntax | cmrsub 30621 | The set of raw substitutions. |
class mRSubst | ||
Syntax | cmsub 30622 | The set of substitutions. |
class mSubst | ||
Syntax | cmvh 30623 | The set of variable hypotheses. |
class mVH | ||
Syntax | cmpst 30624 | The set of pre-statements. |
class mPreSt | ||
Syntax | cmsr 30625 | The reduct of a pre-statement. |
class mStRed | ||
Syntax | cmsta 30626 | The set of statements. |
class mStat | ||
Syntax | cmfs 30627 | The set of formal systems. |
class mFS | ||
Syntax | cmcls 30628 | The closure of a set of statements. |
class mCls | ||
Syntax | cmpps 30629 | The set of provable pre-statements. |
class mPPSt | ||
Syntax | cmthm 30630 | The set of theorems. |
class mThm | ||
Definition | df-mcn 30631 | Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mCN = Slot 1 | ||
Definition | df-mvar 30632 | Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mVR = Slot 2 | ||
Definition | df-mty 30633 | Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mType = Slot 3 | ||
Definition | df-mtc 30634 | Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mTC = Slot 4 | ||
Definition | df-mmax 30635 | Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mAx = Slot 5 | ||
Definition | df-mvt 30636 | 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 30637 | 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 30638 | 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 30639 | 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 30640* | 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 30641* | 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 30642* | 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 30643* | 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 30644* | 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 30645* | 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 30646 | Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mStat = (𝑡 ∈ V ↦ ran (mStRed‘𝑡)) | ||
Definition | df-mfs 30647* | 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 30648* | 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 30649* | 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 30650 | Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
⊢ mThm = (𝑡 ∈ V ↦ (◡(mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡)))) | ||
Theorem | mvtval 30651 | The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑉 = (mVT‘𝑇) & ⊢ 𝑌 = (mType‘𝑇) ⇒ ⊢ 𝑉 = ran 𝑌 | ||
Theorem | mrexval 30652 | 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 30653 | 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 30654 | 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 30655 | 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 dv 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 30656 | 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 30657 | 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 30658* | 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 30659* | 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 30660* | 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 30661 | 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 30662 | 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 30663 | 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 30664 | 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 30665 | 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 30666 | 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 30667 | The value of the substituted empty string. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑆 = (mRSubst‘𝑇) ⇒ ⊢ (𝐹 ∈ ran 𝑆 → (𝐹‘∅) = ∅) | ||
Theorem | mrsubf 30668 | 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 30669 | 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 30670 | 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 30671* | 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 30700.) (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 30672 | 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 30673* | 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 30674* | 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 30675* | 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 30676 | 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 30677 | 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 30678 | 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 30679* | 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 30680 | 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 30681 | 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 30682 | 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 30683 | 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 30684* | 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 30685 | 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 30686* | A pre-statement is an ordered triple, whose first member is a symmetric set of dv 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 30687 | 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 30688* | 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 30689 | 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 30690 | A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) ⇒ ⊢ 𝑃 ⊆ ((V × V) × V) | ||
Theorem | mpst123 30691 | 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 30692 | 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 30693 | 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 30694 | 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 30695 | Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑅 = (mStRed‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 = ran 𝑅 | ||
Theorem | msrid 30696 | 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 30697 | 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 30698 | A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.) |
⊢ 𝑃 = (mPreSt‘𝑇) & ⊢ 𝑆 = (mStat‘𝑇) ⇒ ⊢ 𝑆 ⊆ 𝑃 | ||
Theorem | elmsta 30699 | 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 30700* | 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)))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |