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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmfsdisj 30701 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 → (𝐶𝑉) = ∅)

Theoremmtyf2 30702 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:𝑉𝐾)

Theoremmtyf 30703 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:𝑉𝐹)

Theoremmvtss 30704 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 → 𝐹𝐾)

Theoremmaxsta 30705 An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐴 = (mAx‘𝑇)    &   𝑆 = (mStat‘𝑇)       (𝑇 ∈ mFS → 𝐴𝑆)

Theoremmvtinf 30706 Each variable typecode has infinitely many variables. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐹 = (mVT‘𝑇)    &   𝑌 = (mType‘𝑇)       ((𝑇 ∈ mFS ∧ 𝑋𝐹) → ¬ (𝑌 “ {𝑋}) ∈ Fin)

Theoremmsubff1 30707 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 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)       (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1→(𝐸𝑚 𝐸))

Theoremmsubff1o 30708 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 = (mSubst‘𝑇)       (𝑇 ∈ mFS → (𝑆 ↾ (𝑅𝑚 𝑉)):(𝑅𝑚 𝑉)–1-1-onto→ran 𝑆)

Theoremmvhf 30709 The function mapping variables to variable expressions is a function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐻 = (mVH‘𝑇)       (𝑇 ∈ mFS → 𝐻:𝑉𝐸)

Theoremmvhf1 30710 The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑉 = (mVR‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐻 = (mVH‘𝑇)       (𝑇 ∈ mFS → 𝐻:𝑉1-1𝐸)

Theoremmsubvrs 30711* 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 = (mSubst‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝑉 = (mVars‘𝑇)    &   𝐻 = (mVH‘𝑇)       ((𝑇 ∈ mFS ∧ 𝐹 ∈ ran 𝑆𝑋𝐸) → (𝑉‘(𝐹𝑋)) = 𝑥 ∈ (𝑉𝑋)(𝑉‘(𝐹‘(𝐻𝑥))))

Theoremmclsrcl 30712 Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)       (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))

Theoremmclsssvlem 30713* Lemma for mclsssv 30715. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝐴 = (mAx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝑉 = (mVars‘𝑇)       (𝜑 {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))} ⊆ 𝐸)

Theoremmclsval 30714* The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝐴 = (mAx‘𝑇)    &   𝑆 = (mSubst‘𝑇)    &   𝑉 = (mVars‘𝑇)       (𝜑 → (𝐾𝐶𝐵) = {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻𝑥))) × (𝑉‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑐)))})

Theoremmclsssv 30715 The closure of a set of expressions is a set of expressions. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)       (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸)

Theoremssmclslem 30716 Lemma for ssmcls 30718. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)       (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵))

Theoremvhmcls 30717 All variable hypotheses are in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐻 = (mVH‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐻𝑋) ∈ (𝐾𝐶𝐵))

Theoremssmcls 30718 The original expressions are also in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)       (𝜑𝐵 ⊆ (𝐾𝐶𝐵))

Theoremss2mcls 30719 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))

Theoremmclsax 30720* The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐷 = (mDV‘𝑇)    &   𝐸 = (mEx‘𝑇)    &   𝐶 = (mCls‘𝑇)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐴 = (mAx‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)       (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))

Theoremmclsind 30721* 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐴 = (mAx‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑𝐵𝑄)    &   ((𝜑𝑣𝑉) → (𝐻𝑣) ∈ 𝑄)    &   ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻𝑥))) × (𝑊‘(𝑠‘(𝐻𝑦)))) ⊆ 𝐾)) → (𝑠𝑝) ∈ 𝑄)       (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)

Theoremmppspstlem 30722* Lemma for mppspst 30725. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝐽 = (mPPSt‘𝑇)    &   𝐶 = (mCls‘𝑇)       {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃

Theoremmppsval 30723* 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, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}

Theoremelmpps 30724 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⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Theoremmppspst 30725 A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑃 = (mPreSt‘𝑇)    &   𝐽 = (mPPSt‘𝑇)       𝐽𝑃

Theoremmthmval 30726 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 “ (𝑅𝐽))

Theoremelmthm 30727* 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)       (𝑋𝑈 ↔ ∃𝑥𝐽 (𝑅𝑥) = (𝑅𝑋))

Theoremmthmi 30728 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)       ((𝑋𝐽 ∧ (𝑅𝑋) = (𝑅𝑌)) → 𝑌𝑈)

Theoremmthmsta 30729 A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑈 = (mThm‘𝑇)    &   𝑆 = (mPreSt‘𝑇)       𝑈𝑆

Theoremmppsthm 30730 A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝐽 = (mPPSt‘𝑇)    &   𝑈 = (mThm‘𝑇)       𝐽𝑈

Theoremmthmblem 30731 Lemma for mthmb 30732. (Contributed by Mario Carneiro, 18-Jul-2016.)
𝑅 = (mStRed‘𝑇)    &   𝑈 = (mThm‘𝑇)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Theoremmthmb 30732 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)       ((𝑅𝑋) = (𝑅𝑌) → (𝑋𝑈𝑌𝑈))

Theoremmthmpps 30733 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 dv 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⟩))))

Theoremmclsppslem 30734* The closure is closed under application of provable pre-statements. (Compare mclsax 30720.) 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)    &   (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))    &   (𝜑𝑠 ∈ ran 𝐿)    &   (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (𝑆 “ (𝐾𝐶𝐵)))    &   (𝜑 → ∀𝑧𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻𝑧))) × (𝑊‘(𝑠‘(𝐻𝑤)))) ⊆ 𝑀))       (𝜑 → (𝑠𝑝) ∈ (𝑆 “ (𝐾𝐶𝐵)))

Theoremmclspps 30735* The closure is closed under application of provable pre-statements. (Compare mclsax 30720.) 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)    &   (𝜑𝑇 ∈ mFS)    &   (𝜑𝐾𝐷)    &   (𝜑𝐵𝐸)    &   𝐽 = (mPPSt‘𝑇)    &   𝐿 = (mSubst‘𝑇)    &   𝑉 = (mVR‘𝑇)    &   𝐻 = (mVH‘𝑇)    &   𝑊 = (mVars‘𝑇)    &   (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   (𝜑𝑆 ∈ ran 𝐿)    &   ((𝜑𝑥𝑂) → (𝑆𝑥) ∈ (𝐾𝐶𝐵))    &   ((𝜑𝑣𝑉) → (𝑆‘(𝐻𝑣)) ∈ (𝐾𝐶𝐵))    &   ((𝜑 ∧ (𝑥𝑀𝑦𝑎 ∈ (𝑊‘(𝑆‘(𝐻𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻𝑦))))) → 𝑎𝐾𝑏)       (𝜑 → (𝑆𝑃) ∈ (𝐾𝐶𝐵))

21.5.13  Grammatical formal systems

Syntaxcm0s 30736 Mapping expressions to statements.
class m0St

Syntaxcmsa 30737 The set of syntax axioms.
class mSA

Syntaxcmwgfs 30738 The set of weakly grammatical formal systems.
class mWGFS

Syntaxcmsy 30739 The syntax typecode function.
class mSyn

Syntaxcmesy 30740 The syntax typecode function for expressions.
class mESyn

Syntaxcmgfs 30741 The set of grammatical formal systems.
class mGFS

Syntaxcmtree 30742 The set of proof trees.
class mTree

Syntaxcmst 30743 The set of syntax trees.
class mST

Syntaxcmsax 30744 The indexing set for a syntax axiom.
class mSAX

Syntaxcmufs 30745 The set of unambiguous formal sytems.
class mUFS

Definitiondf-m0s 30746 Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)

Definitiondf-msa 30747* Define the set of syntax axioms. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡) ∧ Fun ((2nd𝑎) ↾ (mVR‘𝑡)))})

Definitiondf-mwgfs 30748* Define the set of weakly grammatical formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑𝑎((⟨𝑑, , 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}

Definitiondf-msyn 30749 Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
mSyn = Slot 6

Definitiondf-mtree 30750* Define the set of proof trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟⟨(m0St‘𝑒), ∅⟩ ∧ ∀𝑒 𝑒𝑟⟨((mStRed‘𝑡)‘⟨𝑑, , 𝑒⟩), ∅⟩ ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠𝑒)})) ⊆ 𝑟)))}))

Definitiondf-mst 30751 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))))

Definitiondf-msax 30752* 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))))

Definitiondf-mufs 30753 Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}

21.5.14  Models of formal systems

Syntaxcmuv 30754 The universe of a model.
class mUV

Syntaxcmvl 30755 The set of valuations.
class mVL

Syntaxcmvsb 30756 Substitution for a valuation.
class mVSubst

Syntaxcmfsh 30757 The freshness relation of a model.
class mFresh

Syntaxcmfr 30758 The set of freshness relations.
class mFRel

Syntaxcmevl 30759 The evaluation function of a model.
class mEval

Syntaxcmdl 30760 The set of models.
class mMdl

Syntaxcusyn 30761 The syntax function applied to elements of the model.
class mUSyn

Syntaxcgmdl 30762 The set of models in a grammatical formal system.
class mGMdl

Syntaxcmitp 30763 The interpretation function of the model.
class mItp

Syntaxcmfitp 30764 The evaluation function derived from the interpretation.
class mFromItp

Definitiondf-muv 30765 Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUV = Slot 7

Definitiondf-mfsh 30766 Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mFresh = Slot 8

Definitiondf-mevl 30767 Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mEval = Slot 9

Definitiondf-mvl 30768* Define the set of valuations. (Contributed by Mario Carneiro, 14-Jul-2016.)
mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))

Definitiondf-mvsb 30769* 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)))))})

Definitiondf-mfrel 30770* 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}))})

Definitiondf-mdl 30771* 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 ∈ (𝑢pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚𝑣 ((∀𝑒𝑥 (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑢 “ {(1st𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑦)⟩𝑛(𝑚𝑦) ∧ ∀𝑑𝑎(⟨𝑑, , 𝑎⟩ ∈ (mAx‘𝑡) → ((∀𝑦𝑧(𝑦𝑑𝑧 → (𝑚𝑦)𝑓(𝑚𝑧)) ∧ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(⟨𝑠, 𝑚⟩(mVSubst‘𝑡)𝑦 → (𝑛 “ {⟨𝑚, (𝑠𝑒)⟩}) = (𝑛 “ {⟨𝑦, 𝑒⟩})) ∧ ∀𝑝𝑣𝑒𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {⟨𝑚, 𝑒⟩}) = (𝑛 “ {⟨𝑝, 𝑒⟩})) ∧ ∀𝑦𝑢𝑒𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑓 “ {𝑦})))))}

Definitiondf-musyn 30772* Define the syntax typecode function for the model universe. (Contributed by Mario Carneiro, 14-Jul-2016.)
mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st𝑣)), (2nd𝑣)⟩))

Definitiondf-gmdl 30773* 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)𝑤𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {⟨𝑚, 𝑒⟩}) = (((mEval‘𝑡) “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st𝑒)})))}

Definitiondf-mitp 30774* Define the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎))))))

Definitiondf-mfitp 30775* 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 ↦ (𝑓X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st𝑡)‘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(𝑚𝑣) ∧ ∀𝑒𝑎𝑔(𝑒(mST‘𝑡)⟨𝑎, 𝑔⟩ → ⟨𝑚, 𝑒𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {⟨𝑚, 𝑒⟩}) = ((𝑛 “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st𝑒)}))))))

21.5.15  Splitting fields

Syntaxcitr 30776 Integral subring of a ring.
class IntgRing

Syntaxccpms 30777 Completion of a metric space.
class cplMetSp

Syntaxchlb 30778 Embeddings for a direct limit.
class HomLimB

Syntaxchlim 30779 Direct limit structure.
class HomLim

Syntaxcpfl 30780 Polynomial extension field.
class polyFld

Syntaxcsf1 30781 Splitting field for a single polynomial (auxiliary).
class splitFld1

Syntaxcsf 30782 Splitting field for a finite set of polynomials.
class splitFld

Syntaxcpsl 30783 Splitting field for a sequence of polynomials.
class polySplitLim

Definitiondf-irng 30784* 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‘(𝑟s 𝑠))(𝑓 “ {(0g𝑟)}))

Definitiondf-cplmet 30785* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
cplMetSp = (𝑤 ∈ V ↦ ((𝑤s ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟(Base‘𝑟) / 𝑣{⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑔𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒((𝑟 /s 𝑒) sSet {⟨(dist‘ndx), {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝑣𝑞𝑣 ((𝑥 = [𝑝]𝑒𝑦 = [𝑞]𝑒) ∧ (𝑝𝑓 (dist‘𝑟)𝑞) ⇝ 𝑧)}⟩}))

Definitiondf-homlimb 30786* The input to this function is a sequence (on ) 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 (𝑓𝑛)) / 𝑣 {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥𝑣 ↦ ⟨((1st𝑥) + 1), ((𝑓‘(1st𝑥))‘(2nd𝑥))⟩) ⊆ 𝑠)} / 𝑒⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)

Definitiondf-homlim 30787* The input to this function is a sequence (on ) 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(1st𝑒) / 𝑣(2nd𝑒) / 𝑔({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔𝑛), 𝑦 ∈ dom (𝑔𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, ((𝑔𝑛)‘(𝑥(+g‘(𝑟𝑛))𝑦))⟩)⟩, ⟨(.r‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔𝑛), 𝑦 ∈ dom (𝑔𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, ((𝑔𝑛)‘(𝑥(.r‘(𝑟𝑛))𝑦))⟩)⟩} ∪ {⟨(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ ((𝑔𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟𝑛))}⟩, ⟨(dist‘ndx), 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔𝑛)‘𝑛) ↦ ⟨⟨((𝑔𝑛)‘𝑥), ((𝑔𝑛)‘𝑦)⟩, (𝑥(dist‘(𝑟𝑛))𝑦)⟩)⟩, ⟨(le‘ndx), 𝑛 ∈ ℕ ((𝑔𝑛) ∘ ((le‘(𝑟𝑛)) ∘ (𝑔𝑛)))⟩}))

Definitiondf-plfl 30788* 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 ↦ (Poly1𝑟) / 𝑠((RSpan‘𝑠)‘{𝑝}) / 𝑖(𝑧 ∈ (Base‘𝑟) ↦ [(𝑧( ·𝑠𝑠)(1r𝑠))](𝑠 ~QG 𝑖)) / 𝑓(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡((𝑡 toNrmGrp (𝑛 ∈ (AbsVal‘𝑡)(𝑛𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), (𝑧 ∈ (Base‘𝑡) ↦ (𝑞𝑧 (𝑟 deg1 𝑞) < (𝑟 deg1 𝑝))) / 𝑔(𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)

Definitiondf-sfl1 30789* 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 ∈ (Poly1𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ( mPoly ‘𝑠) / 𝑚{𝑔 ∈ ((Monic1p𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r𝑚)(𝑝𝑓) ∧ 1 < (𝑠 deg1 𝑔))} / 𝑏if(((𝑝𝑓) = (0g𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, (glb‘𝑏) / (𝑠 polyFld ) / 𝑡⟨(1st𝑡), (𝑓 ∘ (2nd𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟 deg1 𝑝))))))

Definitiondf-sfl 30790* 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 ↦ (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))))

Definitiondf-psl 30791* 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 ↦ (1st𝑔) / 𝑒(1st𝑒) / 𝑠(𝑠 splitFld ran (𝑥𝑞 ↦ (𝑥 ∘ (2nd𝑔)))) / 𝑓𝑓, ((2nd𝑔) ∘ (2nd𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓((1st ∘ (𝑓 shift 1)) HomLim (2nd𝑓)))

Syntaxczr 30792 Integral elements of a ring.
class ZRing

Syntaxcgf 30793 Galois finite field.
class GF

Syntaxcgfo 30794 Galois limit field.
class GF

Syntaxceqp 30795 Equivalence relation for df-qp 30807.
class ~Qp

Syntaxcrqp 30796 Equivalence relation representatives for df-qp 30807.
class /Qp

Syntaxcqp 30797 The set of 𝑝-adic rational numbers.
class Qp

SyntaxcqpOLD 30798 The set of 𝑝-adic rational numbers. (New usage is discouraged.)
class QpOLD

Syntaxczp 30799 The set of 𝑝-adic integers. (Not to be confused with czn 19670.)
class Zp

Syntaxcqpa 30800 Algebraic completion of the 𝑝-adic rational numbers.
class _Qp

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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