P. 335 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 33401-33500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mpstssv 33401	A pre-statement is an ordered triple. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    ⇒   ⊢ 𝑃 ⊆ ((V × V) × V)
Theorem	mpst123 33402	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 33403	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 33404	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 33405	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 33406	Value of the set of statements. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ 𝑆 = ran 𝑅
Theorem	msrid 33407	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 33408	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 33409	A statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ 𝑆 ⊆ 𝑃
Theorem	elmsta 33410	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 33411*	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 33412	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 33413	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 33414	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 33415	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 33416	An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)
Theorem	mvtinf 33417	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)
Theorem	msubff1 33418	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 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝐸 ↑m 𝐸))
Theorem	msubff1o 33419	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 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆)
Theorem	mvhf 33420	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:𝑉⟶𝐸)
Theorem	mvhf1 33421	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:𝑉–1-1→𝐸)
Theorem	msubvrs 33422*	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 𝑆 ∧ 𝑋 ∈ 𝐸) → (𝑉‘(𝐹‘𝑋)) = ∪ 𝑥 ∈ (𝑉‘𝑋)(𝑉‘(𝐹‘(𝐻‘𝑥))))
Theorem	mclsrcl 33423	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 ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))
Theorem	mclsssvlem 33424*	Lemma for mclsssv 33426. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    ⇒   ⊢ (𝜑 → ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))} ⊆ 𝐸)
Theorem	mclsval 33425*	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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVars‘𝑇)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) = ∩ {𝑐 ∣ ((𝐵 ∪ ran 𝐻) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 → ∀𝑠 ∈ ran 𝑆(((𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑉‘(𝑠‘(𝐻‘𝑥))) × (𝑉‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑐)))})
Theorem	mclsssv 33426	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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝐸)
Theorem	ssmclslem 33427	Lemma for ssmcls 33429. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    ⇒   ⊢ (𝜑 → (𝐵 ∪ ran 𝐻) ⊆ (𝐾𝐶𝐵))
Theorem	vhmcls 33428	All variable hypotheses are in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐻‘𝑋) ∈ (𝐾𝐶𝐵))
Theorem	ssmcls 33429	The original expressions are also in the closure. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    ⇒   ⊢ (𝜑 → 𝐵 ⊆ (𝐾𝐶𝐵))
Theorem	ss2mcls 33430	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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐶𝑌) ⊆ (𝐾𝐶𝐵))
Theorem	mclsax 33431*	The closure is closed under axiom application. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐷 = (mDV‘𝑇)    &   ⊢ 𝐸 = (mEx‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    ⇒   ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
Theorem	mclsind 33432*	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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝑄)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐻‘𝑣) ∈ 𝑄)    &   ⊢ ((𝜑 ∧ (⟨𝑚, 𝑜, 𝑝⟩ ∈ 𝐴 ∧ 𝑠 ∈ ran 𝐿 ∧ (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ 𝑄) ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → ((𝑊‘(𝑠‘(𝐻‘𝑥))) × (𝑊‘(𝑠‘(𝐻‘𝑦)))) ⊆ 𝐾)) → (𝑠‘𝑝) ∈ 𝑄)    ⇒   ⊢ (𝜑 → (𝐾𝐶𝐵) ⊆ 𝑄)
Theorem	mppspstlem 33433*	Lemma for mppspst 33436. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐶 = (mCls‘𝑇)    ⇒   ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃
Theorem	mppsval 33434*	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, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
Theorem	elmpps 33435	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⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
Theorem	mppspst 33436	A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑃 = (mPreSt‘𝑇)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    ⇒   ⊢ 𝐽 ⊆ 𝑃
Theorem	mthmval 33437	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 “ (𝑅 “ 𝐽))
Theorem	elmthm 33438*	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)    ⇒   ⊢ (𝑋 ∈ 𝑈 ↔ ∃𝑥 ∈ 𝐽 (𝑅‘𝑥) = (𝑅‘𝑋))
Theorem	mthmi 33439	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)    ⇒   ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑅‘𝑋) = (𝑅‘𝑌)) → 𝑌 ∈ 𝑈)
Theorem	mthmsta 33440	A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑈 = (mThm‘𝑇)    &   ⊢ 𝑆 = (mPreSt‘𝑇)    ⇒   ⊢ 𝑈 ⊆ 𝑆
Theorem	mppsthm 33441	A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ 𝐽 ⊆ 𝑈
Theorem	mthmblem 33442	Lemma for mthmb 33443. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))
Theorem	mthmb 33443	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)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 ↔ 𝑌 ∈ 𝑈))
Theorem	mthmpps 33444	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 disjoint variable 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⟩))))
Theorem	mclsppslem 33445*	The closure is closed under application of provable pre-statements. (Compare mclsax 33431.) 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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    &   ⊢ (𝜑 → ⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑇))    &   ⊢ (𝜑 → 𝑠 ∈ ran 𝐿)    &   ⊢ (𝜑 → (𝑠 “ (𝑜 ∪ ran 𝐻)) ⊆ (◡𝑆 “ (𝐾𝐶𝐵)))    &   ⊢ (𝜑 → ∀𝑧∀𝑤(𝑧𝑚𝑤 → ((𝑊‘(𝑠‘(𝐻‘𝑧))) × (𝑊‘(𝑠‘(𝐻‘𝑤)))) ⊆ 𝑀))    ⇒   ⊢ (𝜑 → (𝑠‘𝑝) ∈ (◡𝑆 “ (𝐾𝐶𝐵)))
Theorem	mclspps 33446*	The closure is closed under application of provable pre-statements. (Compare mclsax 33431.) 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)    &   ⊢ (𝜑 → 𝑇 ∈ mFS)    &   ⊢ (𝜑 → 𝐾 ⊆ 𝐷)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐸)    &   ⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝐿 = (mSubst‘𝑇)    &   ⊢ 𝑉 = (mVR‘𝑇)    &   ⊢ 𝐻 = (mVH‘𝑇)    &   ⊢ 𝑊 = (mVars‘𝑇)    &   ⊢ (𝜑 → ⟨𝑀, 𝑂, 𝑃⟩ ∈ 𝐽)    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐿)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑂) → (𝑆‘𝑥) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝑆‘(𝐻‘𝑣)) ∈ (𝐾𝐶𝐵))    &   ⊢ ((𝜑 ∧ (𝑥𝑀𝑦 ∧ 𝑎 ∈ (𝑊‘(𝑆‘(𝐻‘𝑥))) ∧ 𝑏 ∈ (𝑊‘(𝑆‘(𝐻‘𝑦))))) → 𝑎𝐾𝑏)    ⇒   ⊢ (𝜑 → (𝑆‘𝑃) ∈ (𝐾𝐶𝐵))
20.6.14  Grammatical formal systems
Syntax	cm0s 33447	Mapping expressions to statements.
class m0St
Syntax	cmsa 33448	The set of syntax axioms.
class mSA
Syntax	cmwgfs 33449	The set of weakly grammatical formal systems.
class mWGFS
Syntax	cmsy 33450	The syntax typecode function.
class mSyn
Syntax	cmesy 33451	The syntax typecode function for expressions.
class mESyn
Syntax	cmgfs 33452	The set of grammatical formal systems.
class mGFS
Syntax	cmtree 33453	The set of proof trees.
class mTree
Syntax	cmst 33454	The set of syntax trees.
class mST
Syntax	cmsax 33455	The indexing set for a syntax axiom.
class mSAX
Syntax	cmufs 33456	The set of unambiguous formal sytems.
class mUFS
Definition	df-m0s 33457	Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
Definition	df-msa 33458*	Define the set of syntax axioms. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mSA = (𝑡 ∈ V ↦ {𝑎 ∈ (mEx‘𝑡) ∣ ((m0St‘𝑎) ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡) ∧ Fun (◡(2nd ‘𝑎) ↾ (mVR‘𝑡)))})
Definition	df-mwgfs 33459*	Define the set of weakly grammatical formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mWGFS = {𝑡 ∈ mFS ∣ ∀𝑑∀ℎ∀𝑎((⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) ∧ (1st ‘𝑎) ∈ (mVT‘𝑡)) → ∃𝑠 ∈ ran (mSubst‘𝑡)𝑎 ∈ (𝑠 “ (mSA‘𝑡)))}
Definition	df-msyn 33460	Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mSyn = Slot 6
Definition	df-mesyn 33461*	Define the syntax typecode function for expressions. (Contributed by Mario Carneiro, 12-Jun-2023.)
⊢ mESyn = (𝑡 ∈ V ↦ (𝑐 ∈ (mTC‘𝑡), 𝑒 ∈ (mREx‘𝑡) ↦ (((mSyn‘𝑡)‘𝑐)m0St𝑒)))
Definition	df-mgfs 33462*	Define the set of grammatical formal systems. (Contributed by Mario Carneiro, 12-Jun-2023.)
⊢ mGFS = {𝑡 ∈ mWGFS ∣ ((mSyn‘𝑡):(mTC‘𝑡)⟶(mVT‘𝑡) ∧ ∀𝑐 ∈ (mVT‘𝑡)((mSyn‘𝑡)‘𝑐) = 𝑐 ∧ ∀𝑑∀ℎ∀𝑎(⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) → ∀𝑒 ∈ (ℎ ∪ {𝑎})((mESyn‘𝑡)‘𝑒) ∈ (mPPSt‘𝑡)))}
Definition	df-mtree 33463*	Define the set of proof trees. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mTree = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑟 ∣ (∀𝑒 ∈ ran (mVH‘𝑡)𝑒𝑟⟨(m0St‘𝑒), ∅⟩ ∧ ∀𝑒 ∈ ℎ 𝑒𝑟⟨((mStRed‘𝑡)‘⟨𝑑, ℎ, 𝑒⟩), ∅⟩ ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑) → ({(𝑠‘𝑝)} × X𝑒 ∈ (𝑜 ∪ ((mVH‘𝑡) “ ∪ ((mVars‘𝑡) “ (𝑜 ∪ {𝑝}))))(𝑟 “ {(𝑠‘𝑒)})) ⊆ 𝑟)))}))
Definition	df-mst 33464	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))))
Definition	df-msax 33465*	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))))
Definition	df-mufs 33466	Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
20.6.15  Models of formal systems
Syntax	cmuv 33467	The universe of a model.
class mUV
Syntax	cmvl 33468	The set of valuations.
class mVL
Syntax	cmvsb 33469	Substitution for a valuation.
class mVSubst
Syntax	cmfsh 33470	The freshness relation of a model.
class mFresh
Syntax	cmfr 33471	The set of freshness relations.
class mFRel
Syntax	cmevl 33472	The evaluation function of a model.
class mEval
Syntax	cmdl 33473	The set of models.
class mMdl
Syntax	cusyn 33474	The syntax function applied to elements of the model.
class mUSyn
Syntax	cgmdl 33475	The set of models in a grammatical formal system.
class mGMdl
Syntax	cmitp 33476	The interpretation function of the model.
class mItp
Syntax	cmfitp 33477	The evaluation function derived from the interpretation.
class mFromItp
Definition	df-muv 33478	Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUV = Slot 7
Definition	df-mfsh 33479	Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFresh = Slot ;19
Definition	df-mevl 33480	Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mEval = Slot ;20
Definition	df-mvl 33481*	Define the set of valuations. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mVL = (𝑡 ∈ V ↦ X𝑣 ∈ (mVR‘𝑡)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑣)}))
Definition	df-mvsb 33482*	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)))))})
Definition	df-mfrel 33483*	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}))})
Definition	df-mdl 33484*	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 ∈ (𝑢 ↑pm (𝑣 × (mEx‘𝑡)))) ∧ ∀𝑚 ∈ 𝑣 ((∀𝑒 ∈ 𝑥 (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑢 “ {(1st ‘𝑒)}) ∧ ∀𝑦 ∈ (mVR‘𝑡)⟨𝑚, ((mVH‘𝑡)‘𝑦)⟩𝑛(𝑚‘𝑦) ∧ ∀𝑑∀ℎ∀𝑎(⟨𝑑, ℎ, 𝑎⟩ ∈ (mAx‘𝑡) → ((∀𝑦∀𝑧(𝑦𝑑𝑧 → (𝑚‘𝑦)𝑓(𝑚‘𝑧)) ∧ ℎ ⊆ (dom 𝑛 “ {𝑚})) → 𝑚dom 𝑛 𝑎))) ∧ (∀𝑠 ∈ ran (mSubst‘𝑡)∀𝑒 ∈ (mEx‘𝑡)∀𝑦(⟨𝑠, 𝑚⟩(mVSubst‘𝑡)𝑦 → (𝑛 “ {⟨𝑚, (𝑠‘𝑒)⟩}) = (𝑛 “ {⟨𝑦, 𝑒⟩})) ∧ ∀𝑝 ∈ 𝑣 ∀𝑒 ∈ 𝑥 ((𝑚 ↾ ((mVars‘𝑡)‘𝑒)) = (𝑝 ↾ ((mVars‘𝑡)‘𝑒)) → (𝑛 “ {⟨𝑚, 𝑒⟩}) = (𝑛 “ {⟨𝑝, 𝑒⟩})) ∧ ∀𝑦 ∈ 𝑢 ∀𝑒 ∈ 𝑥 ((𝑚 “ ((mVars‘𝑡)‘𝑒)) ⊆ (𝑓 “ {𝑦}) → (𝑛 “ {⟨𝑚, 𝑒⟩}) ⊆ (𝑓 “ {𝑦})))))}
Definition	df-musyn 33485*	Define the syntax typecode function for the model universe. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUSyn = (𝑡 ∈ V ↦ (𝑣 ∈ (mUV‘𝑡) ↦ ⟨((mSyn‘𝑡)‘(1st ‘𝑣)), (2nd ‘𝑣)⟩))
Definition	df-gmdl 33486*	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)𝑤 ↔ 𝑣(mFresh‘𝑡)((mUSyn‘𝑡)‘𝑤)) ∧ ∀𝑚 ∈ (mVL‘𝑡)∀𝑒 ∈ (mEx‘𝑡)((mEval‘𝑡) “ {⟨𝑚, 𝑒⟩}) = (((mEval‘𝑡) “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)})))}
Definition	df-mitp 33487*	Define the interpretation function for a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mItp = (𝑡 ∈ V ↦ (𝑎 ∈ (mSA‘𝑡) ↦ (𝑔 ∈ X𝑖 ∈ ((mVars‘𝑡)‘𝑎)((mUV‘𝑡) “ {((mType‘𝑡)‘𝑖)}) ↦ (℩𝑥∃𝑚 ∈ (mVL‘𝑡)(𝑔 = (𝑚 ↾ ((mVars‘𝑡)‘𝑎)) ∧ 𝑥 = (𝑚(mEval‘𝑡)𝑎))))))
Definition	df-mfitp 33488*	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 ↦ (𝑓 ∈ X𝑎 ∈ (mSA‘𝑡)(((mUV‘𝑡) “ {((1st ‘𝑡)‘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(𝑚‘𝑣) ∧ ∀𝑒∀𝑎∀𝑔(𝑒(mST‘𝑡)⟨𝑎, 𝑔⟩ → ⟨𝑚, 𝑒⟩𝑛(𝑓‘(𝑖 ∈ ((mVars‘𝑡)‘𝑎) ↦ (𝑚𝑛(𝑔‘((mVH‘𝑡)‘𝑖)))))) ∧ ∀𝑒 ∈ (mEx‘𝑡)(𝑛 “ {⟨𝑚, 𝑒⟩}) = ((𝑛 “ {⟨𝑚, ((mESyn‘𝑡)‘𝑒)⟩}) ∩ ((mUV‘𝑡) “ {(1st ‘𝑒)}))))))
20.6.16  Splitting fields
Syntax	citr 33489	Integral subring of a ring.
class IntgRing
Syntax	ccpms 33490	Completion of a metric space.
class cplMetSp
Syntax	chlb 33491	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 33492	Direct limit structure.
class HomLim
Syntax	cpfl 33493	Polynomial extension field.
class polyFld
Syntax	csf1 33494	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 33495	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 33496	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-irng 33497*	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‘(𝑟 ↾s 𝑠))(◡𝑓 “ {(0g‘𝑟)}))
Definition	df-cplmet 33498*	A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ cplMetSp = (𝑤 ∈ V ↦ ⦋((𝑤 ↑s ℕ) ↾s (Cau‘(dist‘𝑤))) / 𝑟⦌⦋(Base‘𝑟) / 𝑣⦌⦋{⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑔‘𝑗)(ball‘(dist‘𝑤))𝑥))} / 𝑒⦌((𝑟 /s 𝑒) sSet {⟨(dist‘ndx), {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝 ∈ 𝑣 ∃𝑞 ∈ 𝑣 ((𝑥 = [𝑝]𝑒 ∧ 𝑦 = [𝑞]𝑒) ∧ (𝑝 ∘f (dist‘𝑟)𝑞) ⇝ 𝑧)}⟩}))
Definition	df-homlimb 33499*	The input to this function is a sequence (on ℕ) 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 (𝑓‘𝑛)) / 𝑣⦌⦋∩ {𝑠 ∣ (𝑠 Er 𝑣 ∧ (𝑥 ∈ 𝑣 ↦ ⟨((1st ‘𝑥) + 1), ((𝑓‘(1st ‘𝑥))‘(2nd ‘𝑥))⟩) ⊆ 𝑠)} / 𝑒⦌⟨(𝑣 / 𝑒), (𝑛 ∈ ℕ ↦ (𝑥 ∈ dom (𝑓‘𝑛) ↦ [⟨𝑛, 𝑥⟩]𝑒))⟩)
Definition	df-homlim 33500*	The input to this function is a sequence (on ℕ) 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⦌⦋(1st ‘𝑒) / 𝑣⦌⦋(2nd ‘𝑒) / 𝑔⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(+g‘(𝑟‘𝑛))𝑦))⟩)⟩, ⟨(.r‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom (𝑔‘𝑛), 𝑦 ∈ dom (𝑔‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, ((𝑔‘𝑛)‘(𝑥(.r‘(𝑟‘𝑛))𝑦))⟩)⟩} ∪ {⟨(TopOpen‘ndx), {𝑠 ∈ 𝒫 𝑣 ∣ ∀𝑛 ∈ ℕ (◡(𝑔‘𝑛) “ 𝑠) ∈ (TopOpen‘(𝑟‘𝑛))}⟩, ⟨(dist‘ndx), ∪ 𝑛 ∈ ℕ ran (𝑥 ∈ dom ((𝑔‘𝑛)‘𝑛), 𝑦 ∈ dom ((𝑔‘𝑛)‘𝑛) ↦ ⟨⟨((𝑔‘𝑛)‘𝑥), ((𝑔‘𝑛)‘𝑦)⟩, (𝑥(dist‘(𝑟‘𝑛))𝑦)⟩)⟩, ⟨(le‘ndx), ∪ 𝑛 ∈ ℕ (◡(𝑔‘𝑛) ∘ ((le‘(𝑟‘𝑛)) ∘ (𝑔‘𝑛)))⟩}))