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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | satfun 34701 | The satisfaction predicate as function over wff codes in the model π and the binary relation πΈ on π. (Contributed by AV, 29-Oct-2023.) |
β’ ((π β π β§ πΈ β π) β ((π Sat πΈ)βΟ):(FmlaβΟ)βΆπ« (π βm Ο)) | ||
Theorem | satfvel 34702 | An element of the value of the satisfaction predicate as function over wff codes in the model π and the binary relation πΈ on π at the code π for a wff using β , βΌ , β is a valuation π:ΟβΆπ of the variables (v0 = (πββ ), v1 = (πβ1o), etc.) so that π is true under the assignment π. (Contributed by AV, 29-Oct-2023.) |
β’ (((π β π β§ πΈ β π) β§ π β (FmlaβΟ) β§ π β (((π Sat πΈ)βΟ)βπ)) β π:ΟβΆπ) | ||
Theorem | satfv0fvfmla0 34703* | The value of the satisfaction predicate as function over a wff code at β . (Contributed by AV, 2-Nov-2023.) |
β’ π = (π Sat πΈ) β β’ ((π β π β§ πΈ β π β§ π β (Fmlaββ )) β ((πββ )βπ) = {π β (π βm Ο) β£ (πβ(1st β(2nd βπ)))πΈ(πβ(2nd β(2nd βπ)))}) | ||
Theorem | satefv 34704 | The simplified satisfaction predicate as function over wff codes in the model π at the code π. (Contributed by AV, 30-Oct-2023.) |
β’ ((π β π β§ π β π) β (π Satβ π) = (((π Sat ( E β© (π Γ π)))βΟ)βπ)) | ||
Theorem | sate0 34705 | The simplified satisfaction predicate for any wff code over an empty model. (Contributed by AV, 6-Oct-2023.) (Revised by AV, 5-Nov-2023.) |
β’ (π β π β (β Satβ π) = (((β Sat β )βΟ)βπ)) | ||
Theorem | satef 34706 | The simplified satisfaction predicate as function over wff codes over an empty model. (Contributed by AV, 30-Oct-2023.) |
β’ ((π β π β§ π β (FmlaβΟ) β§ π β (π Satβ π)) β π:ΟβΆπ) | ||
Theorem | sate0fv0 34707 | A simplified satisfaction predicate as function over wff codes over an empty model is an empty set. (Contributed by AV, 31-Oct-2023.) |
β’ (π β (FmlaβΟ) β (π β (β Satβ π) β π = β )) | ||
Theorem | satefvfmla0 34708* | The simplified satisfaction predicate for wff codes of height 0. (Contributed by AV, 4-Nov-2023.) |
β’ ((π β π β§ π β (Fmlaββ )) β (π Satβ π) = {π β (π βm Ο) β£ (πβ(1st β(2nd βπ))) β (πβ(2nd β(2nd βπ)))}) | ||
Theorem | sategoelfvb 34709 | Characterization of a valuation π of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.) |
β’ πΈ = (π Satβ (π΄βππ΅)) β β’ ((π β π β§ (π΄ β Ο β§ π΅ β Ο)) β (π β πΈ β (π β (π βm Ο) β§ (πβπ΄) β (πβπ΅)))) | ||
Theorem | sategoelfv 34710 | Condition of a valuation π of a simplified satisfaction predicate for a Godel-set of membership: The sets in model π corresponding to the variables π΄ and π΅ under the assignment of π are in a membership relation in π. (Contributed by AV, 5-Nov-2023.) |
β’ πΈ = (π Satβ (π΄βππ΅)) β β’ ((π β π β§ (π΄ β Ο β§ π΅ β Ο) β§ π β πΈ) β (πβπ΄) β (πβπ΅)) | ||
Theorem | ex-sategoelel 34711* | Example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.) |
β’ πΈ = (π Satβ (π΄βππ΅)) & β’ π = (π₯ β Ο β¦ if(π₯ = π΄, π, if(π₯ = π΅, π« π, β ))) β β’ (((π β WUni β§ π β π) β§ (π΄ β Ο β§ π΅ β Ο β§ π΄ β π΅)) β π β πΈ) | ||
Theorem | ex-sategoel 34712* | Instance of sategoelfv 34710 for the example of a valuation of a simplified satisfaction predicate for a Godel-set of membership. (Contributed by AV, 5-Nov-2023.) |
β’ πΈ = (π Satβ (π΄βππ΅)) & β’ π = (π₯ β Ο β¦ if(π₯ = π΄, π, if(π₯ = π΅, π« π, β ))) β β’ (((π β WUni β§ π β π) β§ (π΄ β Ο β§ π΅ β Ο β§ π΄ β π΅)) β (πβπ΄) β (πβπ΅)) | ||
Theorem | satfv1fvfmla1 34713* | The value of the satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.) |
β’ π = ((πΌβππ½)βΌπ(πΎβππΏ)) β β’ (((π β π β§ πΈ β π) β§ (πΌ β Ο β§ π½ β Ο) β§ (πΎ β Ο β§ πΏ β Ο)) β (((π Sat πΈ)β1o)βπ) = {π β (π βm Ο) β£ (Β¬ (πβπΌ)πΈ(πβπ½) β¨ Β¬ (πβπΎ)πΈ(πβπΏ))}) | ||
Theorem | 2goelgoanfmla1 34714 | Two Godel-sets of membership combined with a Godel-set for NAND is a Godel formula of height 1. (Contributed by AV, 17-Nov-2023.) |
β’ π = ((πΌβππ½)βΌπ(πΎβππΏ)) β β’ (((πΌ β Ο β§ π½ β Ο) β§ (πΎ β Ο β§ πΏ β Ο)) β π β (Fmlaβ1o)) | ||
Theorem | satefvfmla1 34715* | The simplified satisfaction predicate at two Godel-sets of membership combined with a Godel-set for NAND. (Contributed by AV, 17-Nov-2023.) |
β’ π = ((πΌβππ½)βΌπ(πΎβππΏ)) β β’ ((π β π β§ (πΌ β Ο β§ π½ β Ο) β§ (πΎ β Ο β§ πΏ β Ο)) β (π Satβ π) = {π β (π βm Ο) β£ (Β¬ (πβπΌ) β (πβπ½) β¨ Β¬ (πβπΎ) β (πβπΏ))}) | ||
Theorem | ex-sategoelelomsuc 34716* | Example of a valuation of a simplified satisfaction predicate over the ordinal numbers as model for a Godel-set of membership using the properties of a successor: (πβ2o) = π β suc π = (πβ2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2oβπ1o) should not be confused with 2o β 1o, which is false. (Contributed by AV, 19-Nov-2023.) |
β’ π = (π₯ β Ο β¦ if(π₯ = 2o, π, suc π)) β β’ (π β Ο β π β (Ο Satβ (2oβπ1o))) | ||
Theorem | ex-sategoelel12 34717 | Example of a valuation of a simplified satisfaction predicate over a proper pair (of ordinal numbers) as model for a Godel-set of membership using the properties of a successor: (πβ2o) = 1o β 2o = (πβ2o). Remark: the indices 1o and 2o are intentionally reversed to distinguish them from elements of the model: (2oβπ1o) should not be confused with 2o β 1o, which is false. (Contributed by AV, 19-Nov-2023.) |
β’ π = (π₯ β Ο β¦ if(π₯ = 2o, 1o, 2o)) β β’ π β ({1o, 2o} Satβ (2oβπ1o)) | ||
Theorem | prv 34718 | The "proves" relation on a set. A wff encoded as π is true in a model π iff for every valuation π β (π βm Ο), the interpretation of the wff using the membership relation on π is true. (Contributed by AV, 5-Nov-2023.) |
β’ ((π β π β§ π β π) β (πβ§π β (π Satβ π) = (π βm Ο))) | ||
Theorem | elnanelprv 34719 | The wff (π΄ β π΅ βΌ π΅ β π΄) encoded as ((π΄βππ΅) βΌπ(π΅βππ΄)) is true in any model π. This is the model theoretic proof of elnanel 9606. (Contributed by AV, 5-Nov-2023.) |
β’ ((π β π β§ π΄ β Ο β§ π΅ β Ο) β πβ§((π΄βππ΅)βΌπ(π΅βππ΄))) | ||
Theorem | prv0 34720 | Every wff encoded as π is true in an "empty model" (π = β ). Since β§ is defined in terms of the interpretations making the given formula true, it is not defined on the "empty model", since there are no interpretations. In particular, the empty set on the LHS of β§ should not be interpreted as the empty model, because βπ₯π₯ = π₯ is not satisfied on the empty model. (Contributed by AV, 19-Nov-2023.) |
β’ (π β (FmlaβΟ) β β β§π) | ||
Theorem | prv1n 34721 | No wff encoded as a Godel-set of membership is true in a model with only one element. (Contributed by AV, 19-Nov-2023.) |
β’ ((πΌ β Ο β§ π½ β Ο β§ π β π) β Β¬ {π}β§(πΌβππ½)) | ||
Syntax | cgon 34722 | The Godel-set of negation. (Note that this is not a wff.) |
class Β¬ππ | ||
Syntax | cgoa 34723 | The Godel-set of conjunction. |
class β§π | ||
Syntax | cgoi 34724 | The Godel-set of implication. |
class βπ | ||
Syntax | cgoo 34725 | The Godel-set of disjunction. |
class β¨π | ||
Syntax | cgob 34726 | The Godel-set of equivalence. |
class βπ | ||
Syntax | cgoq 34727 | The Godel-set of equality. |
class =π | ||
Syntax | cgox 34728 | The Godel-set of existential quantification. (Note that this is not a wff.) |
class βπππ | ||
Definition | df-gonot 34729 | Define the Godel-set of negation. Here the argument π is also a Godel-set corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ Β¬ππ = (πβΌππ) | ||
Definition | df-goan 34730* | Define the Godel-set of conjunction. Here the arguments π and π are also Godel-sets corresponding to smaller formulas. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ β§π = (π’ β V, π£ β V β¦ Β¬π(π’βΌππ£)) | ||
Definition | df-goim 34731* | Define the Godel-set of implication. Here the arguments π and π are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ βπ = (π’ β V, π£ β V β¦ (π’βΌπΒ¬ππ£)) | ||
Definition | df-goor 34732* | Define the Godel-set of disjunction. Here the arguments π and π are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ β¨π = (π’ β V, π£ β V β¦ (Β¬ππ’ βπ π£)) | ||
Definition | df-gobi 34733* | Define the Godel-set of equivalence. Here the arguments π and π are also Godel-sets corresponding to smaller formulas. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ βπ = (π’ β V, π£ β V β¦ ((π’ βπ π£)β§π(π£ βπ π’))) | ||
Definition | df-goeq 34734* | Define the Godel-set of equality. Here the arguments π₯ = β¨π, πβ© correspond to vN and vP , so (β =π1o) actually means v0 = v1 , not 0 = 1. Here we use the trick mentioned in ax-ext 2702 to introduce equality as a defined notion in terms of βπ. The expression suc (π’ βͺ π£) = max (π’, π£) + 1 here is a convenient way of getting a dummy variable distinct from π’ and π£. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ =π = (π’ β Ο, π£ β Ο β¦ β¦suc (π’ βͺ π£) / π€β¦βππ€((π€βππ’) βπ (π€βππ£))) | ||
Definition | df-goex 34735 | Define the Godel-set of existential quantification. Here π β Ο corresponds to vN , and π represents another formula, and this expression is [βπ₯π] = βπππ where π₯ is the π-th variable, π = [π] is the code for π. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ βπππ = Β¬πβππΒ¬ππ | ||
Syntax | cgze 34736 | The Axiom of Extensionality. |
class AxExt | ||
Syntax | cgzr 34737 | The Axiom Scheme of Replacement. |
class AxRep | ||
Syntax | cgzp 34738 | The Axiom of Power Sets. |
class AxPow | ||
Syntax | cgzu 34739 | The Axiom of Unions. |
class AxUn | ||
Syntax | cgzg 34740 | The Axiom of Regularity. |
class AxReg | ||
Syntax | cgzi 34741 | The Axiom of Infinity. |
class AxInf | ||
Syntax | cgzf 34742 | The set of models of ZF. |
class ZF | ||
Definition | df-gzext 34743 | The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ AxExt = (βπ2o((2oβπβ ) βπ (2oβπ1o)) βπ (β =π1o)) | ||
Definition | df-gzrep 34744 | 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.) |
β’ AxRep = (π’ β (FmlaβΟ) β¦ (βπ3oβπ1oβπ2o(βπ1oπ’ βπ (2o=π1o)) βπ βπ1oβπ2o((2oβπ1o) βπ βπ3o((3oβπβ )β§πβπ1oπ’)))) | ||
Definition | df-gzpow 34745 | The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ AxPow = βπ1oβπ2o(βπ1o((1oβπ2o) βπ (1oβπβ )) βπ (2oβπ1o)) | ||
Definition | df-gzun 34746 | The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ AxUn = βπ1oβπ2o(βπ1o((2oβπ1o)β§π(1oβπβ )) βπ (2oβπ1o)) | ||
Definition | df-gzreg 34747 | The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ AxReg = (βπ1o(1oβπβ ) βπ βπ1o((1oβπβ )β§πβπ2o((2oβπ1o) βπ Β¬π(2oβπβ )))) | ||
Definition | df-gzinf 34748 | The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ AxInf = βπ1o((β βπ1o)β§πβπ2o((2oβπ1o) βπ βπβ ((2oβπβ )β§π(β βπ1o)))) | ||
Definition | df-gzf 34749* | Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.) |
β’ ZF = {π β£ ((Tr π β§ πβ§AxExt β§ πβ§AxPow) β§ (πβ§AxUn β§ πβ§AxReg β§ πβ§AxInf) β§ βπ’ β (FmlaβΟ)πβ§(AxRepβπ’))} | ||
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 34750 | The set of constants. |
class mCN | ||
Syntax | cmvar 34751 | The set of variables. |
class mVR | ||
Syntax | cmty 34752 | The type function. |
class mType | ||
Syntax | cmvt 34753 | The set of variable typecodes. |
class mVT | ||
Syntax | cmtc 34754 | The set of typecodes. |
class mTC | ||
Syntax | cmax 34755 | The set of axioms. |
class mAx | ||
Syntax | cmrex 34756 | The set of raw expressions. |
class mREx | ||
Syntax | cmex 34757 | The set of expressions. |
class mEx | ||
Syntax | cmdv 34758 | The set of distinct variables. |
class mDV | ||
Syntax | cmvrs 34759 | The variables in an expression. |
class mVars | ||
Syntax | cmrsub 34760 | The set of raw substitutions. |
class mRSubst | ||
Syntax | cmsub 34761 | The set of substitutions. |
class mSubst | ||
Syntax | cmvh 34762 | The set of variable hypotheses. |
class mVH | ||
Syntax | cmpst 34763 | The set of pre-statements. |
class mPreSt | ||
Syntax | cmsr 34764 | The reduct of a pre-statement. |
class mStRed | ||
Syntax | cmsta 34765 | The set of statements. |
class mStat | ||
Syntax | cmfs 34766 | The set of formal systems. |
class mFS | ||
Syntax | cmcls 34767 | The closure of a set of statements. |
class mCls | ||
Syntax | cmpps 34768 | The set of provable pre-statements. |
class mPPSt | ||
Syntax | cmthm 34769 | The set of theorems. |
class mThm | ||
Definition | df-mcn 34770 | Define the set of constants in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mCN = Slot 1 | ||
Definition | df-mvar 34771 | Define the set of variables in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mVR = Slot 2 | ||
Definition | df-mty 34772 | Define the type function in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mType = Slot 3 | ||
Definition | df-mtc 34773 | Define the set of typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mTC = Slot 4 | ||
Definition | df-mmax 34774 | Define the set of axioms in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mAx = Slot 5 | ||
Definition | df-mvt 34775 | Define the set of variable typecodes in a Metamath formal system. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mVT = (π‘ β V β¦ ran (mTypeβπ‘)) | ||
Definition | df-mrex 34776 | Define the set of "raw expressions", which are expressions without a typecode attached. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mREx = (π‘ β V β¦ Word ((mCNβπ‘) βͺ (mVRβπ‘))) | ||
Definition | df-mex 34777 | Define the set of expressions, which are strings of constants and variables headed by a typecode constant. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mEx = (π‘ β V β¦ ((mTCβπ‘) Γ (mRExβπ‘))) | ||
Definition | df-mdv 34778 | Define the set of distinct variable conditions, which are pairs of distinct variables. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mDV = (π‘ β V β¦ (((mVRβπ‘) Γ (mVRβπ‘)) β I )) | ||
Definition | df-mvrs 34779* | Define the set of variables in an expression. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mVars = (π‘ β V β¦ (π β (mExβπ‘) β¦ (ran (2nd βπ) β© (mVRβπ‘)))) | ||
Definition | df-mrsub 34780* | Define a substitution of raw expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mRSubst = (π‘ β V β¦ (π β ((mRExβπ‘) βpm (mVRβπ‘)) β¦ (π β (mRExβπ‘) β¦ ((freeMndβ((mCNβπ‘) βͺ (mVRβπ‘))) Ξ£g ((π£ β ((mCNβπ‘) βͺ (mVRβπ‘)) β¦ if(π£ β dom π, (πβπ£), β¨βπ£ββ©)) β π))))) | ||
Definition | df-msub 34781* | Define a substitution of expressions given a mapping from variables to expressions. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mSubst = (π‘ β V β¦ (π β ((mRExβπ‘) βpm (mVRβπ‘)) β¦ (π β (mExβπ‘) β¦ β¨(1st βπ), (((mRSubstβπ‘)βπ)β(2nd βπ))β©))) | ||
Definition | df-mvh 34782* | Define the mapping from variables to their variable hypothesis. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mVH = (π‘ β V β¦ (π£ β (mVRβπ‘) β¦ β¨((mTypeβπ‘)βπ£), β¨βπ£ββ©β©)) | ||
Definition | df-mpst 34783* | Define the set of all pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mPreSt = (π‘ β V β¦ (({π β π« (mDVβπ‘) β£ β‘π = π} Γ (π« (mExβπ‘) β© Fin)) Γ (mExβπ‘))) | ||
Definition | df-msr 34784* | Define the reduct of a pre-statement. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mStRed = (π‘ β V β¦ (π β (mPreStβπ‘) β¦ β¦(2nd β(1st βπ )) / ββ¦β¦(2nd βπ ) / πβ¦β¨((1st β(1st βπ )) β© β¦βͺ ((mVarsβπ‘) β (β βͺ {π})) / π§β¦(π§ Γ π§)), β, πβ©)) | ||
Definition | df-msta 34785 | Define the set of all statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mStat = (π‘ β V β¦ ran (mStRedβπ‘)) | ||
Definition | df-mfs 34786* | Define the set of all formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mFS = {π‘ β£ ((((mCNβπ‘) β© (mVRβπ‘)) = β β§ (mTypeβπ‘):(mVRβπ‘)βΆ(mTCβπ‘)) β§ ((mAxβπ‘) β (mStatβπ‘) β§ βπ£ β (mVTβπ‘) Β¬ (β‘(mTypeβπ‘) β {π£}) β Fin))} | ||
Definition | df-mcls 34787* | Define the closure of a set of statements relative to a set of disjointness constraints. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mCls = (π‘ β V β¦ (π β π« (mDVβπ‘), β β π« (mExβπ‘) β¦ β© {π β£ ((β βͺ ran (mVHβπ‘)) β π β§ βπβπβπ(β¨π, π, πβ© β (mAxβπ‘) β βπ β ran (mSubstβπ‘)(((π β (π βͺ ran (mVHβπ‘))) β π β§ βπ₯βπ¦(π₯ππ¦ β (((mVarsβπ‘)β(π β((mVHβπ‘)βπ₯))) Γ ((mVarsβπ‘)β(π β((mVHβπ‘)βπ¦)))) β π)) β (π βπ) β π)))})) | ||
Definition | df-mpps 34788* | Define the set of provable pre-statements. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mPPSt = (π‘ β V β¦ {β¨β¨π, ββ©, πβ© β£ (β¨π, β, πβ© β (mPreStβπ‘) β§ π β (π(mClsβπ‘)β))}) | ||
Definition | df-mthm 34789 | Define the set of theorems. (Contributed by Mario Carneiro, 14-Jul-2016.) |
β’ mThm = (π‘ β V β¦ (β‘(mStRedβπ‘) β ((mStRedβπ‘) β (mPPStβπ‘)))) | ||
Theorem | mvtval 34790 | The set of variable typecodes. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ π = (mVTβπ) & β’ π = (mTypeβπ) β β’ π = ran π | ||
Theorem | mrexval 34791 | 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.) |
β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) & β’ π = (mRExβπ) β β’ (π β π β π = Word (πΆ βͺ π)) | ||
Theorem | mexval 34792 | 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.) |
β’ πΎ = (mTCβπ) & β’ πΈ = (mExβπ) & β’ π = (mRExβπ) β β’ πΈ = (πΎ Γ π ) | ||
Theorem | mexval2 34793 | 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.) |
β’ πΎ = (mTCβπ) & β’ πΈ = (mExβπ) & β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) β β’ πΈ = (πΎ Γ Word (πΆ βͺ π)) | ||
Theorem | mdvval 34794 | 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.) |
β’ π = (mVRβπ) & β’ π· = (mDVβπ) β β’ π· = ((π Γ π) β I ) | ||
Theorem | mvrsval 34795 | The set of variables in an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ π = (mVRβπ) & β’ πΈ = (mExβπ) & β’ π = (mVarsβπ) β β’ (π β πΈ β (πβπ) = (ran (2nd βπ) β© π)) | ||
Theorem | mvrsfpw 34796 | The set of variables in an expression is a finite subset of π. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ π = (mVRβπ) & β’ πΈ = (mExβπ) & β’ π = (mVarsβπ) β β’ (π β πΈ β (πβπ) β (π« π β© Fin)) | ||
Theorem | mrsubffval 34797* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) & β’ π = (mRExβπ) & β’ π = (mRSubstβπ) & β’ πΊ = (freeMndβ(πΆ βͺ π)) β β’ (π β π β π = (π β (π βpm π) β¦ (π β π β¦ (πΊ Ξ£g ((π£ β (πΆ βͺ π) β¦ if(π£ β dom π, (πβπ£), β¨βπ£ββ©)) β π))))) | ||
Theorem | mrsubfval 34798* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) & β’ π = (mRExβπ) & β’ π = (mRSubstβπ) & β’ πΊ = (freeMndβ(πΆ βͺ π)) β β’ ((πΉ:π΄βΆπ β§ π΄ β π) β (πβπΉ) = (π β π β¦ (πΊ Ξ£g ((π£ β (πΆ βͺ π) β¦ if(π£ β π΄, (πΉβπ£), β¨βπ£ββ©)) β π)))) | ||
Theorem | mrsubval 34799* | The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) & β’ π = (mRExβπ) & β’ π = (mRSubstβπ) & β’ πΊ = (freeMndβ(πΆ βͺ π)) β β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β π ) β ((πβπΉ)βπ) = (πΊ Ξ£g ((π£ β (πΆ βͺ π) β¦ if(π£ β π΄, (πΉβπ£), β¨βπ£ββ©)) β π))) | ||
Theorem | mrsubcv 34800 | The value of a substituted singleton. (Contributed by Mario Carneiro, 18-Jul-2016.) |
β’ πΆ = (mCNβπ) & β’ π = (mVRβπ) & β’ π = (mRExβπ) & β’ π = (mRSubstβπ) β β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β (πΆ βͺ π)) β ((πβπΉ)ββ¨βπββ©) = if(π β π΄, (πΉβπ), β¨βπββ©)) |
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