P. 365 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	mfsdisj 36401	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 36402	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 36403	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 36404	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 36405	An axiom is a statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐴 = (mAx‘𝑇)    &   ⊢ 𝑆 = (mStat‘𝑇)    ⇒   ⊢ (𝑇 ∈ mFS → 𝐴 ⊆ 𝑆)
Theorem	mvtinf 36406	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 36407	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 36408	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 36409	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 36410	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 36411*	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 36412	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 36413*	Lemma for mclsssv 36415. (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 36414*	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 36415	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 36416	Lemma for ssmcls 36418. (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 36417	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 36418	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 36419	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 36420*	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 36421*	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 36422*	Lemma for mppspst 36425. (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 36423*	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 36424	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 36425	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 36426	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 36427*	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 36428	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 36429	A theorem is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑈 = (mThm‘𝑇)    &   ⊢ 𝑆 = (mPreSt‘𝑇)    ⇒   ⊢ 𝑈 ⊆ 𝑆
Theorem	mppsthm 36430	A provable pre-statement is a theorem. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝐽 = (mPPSt‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ 𝐽 ⊆ 𝑈
Theorem	mthmblem 36431	Lemma for mthmb 36432. (Contributed by Mario Carneiro, 18-Jul-2016.)
⊢ 𝑅 = (mStRed‘𝑇)    &   ⊢ 𝑈 = (mThm‘𝑇)    ⇒   ⊢ ((𝑅‘𝑋) = (𝑅‘𝑌) → (𝑋 ∈ 𝑈 → 𝑌 ∈ 𝑈))
Theorem	mthmb 36432	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 36433	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 36434*	The closure is closed under application of provable pre-statements. (Compare mclsax 36420.) 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 36435*	The closure is closed under application of provable pre-statements. (Compare mclsax 36420.) 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))
21.7.14  Grammatical formal systems
Syntax	cm0s 36436	Mapping expressions to statements.
class m0St
Syntax	cmsa 36437	The set of syntax axioms.
class mSA
Syntax	cmwgfs 36438	The set of weakly grammatical formal systems.
class mWGFS
Syntax	cmsy 36439	The syntax typecode function.
class mSyn
Syntax	cmesy 36440	The syntax typecode function for expressions.
class mESyn
Syntax	cmgfs 36441	The set of grammatical formal systems.
class mGFS
Syntax	cmtree 36442	The set of proof trees.
class mTree
Syntax	cmst 36443	The set of syntax trees.
class mST
Syntax	cmsax 36444	The indexing set for a syntax axiom.
class mSAX
Syntax	cmufs 36445	The set of unambiguous formal systems.
class mUFS
Definition	df-m0s 36446	Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
Definition	df-msa 36447*	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 36448*	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 36449	Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mSyn = Slot 6
Definition	df-mesyn 36450*	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 36451*	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 36452*	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 36453	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 36454*	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 36455	Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
21.7.15  Models of formal systems
Syntax	cmuv 36456	The universe of a model.
class mUV
Syntax	cmvl 36457	The set of valuations.
class mVL
Syntax	cmvsb 36458	Substitution for a valuation.
class mVSubst
Syntax	cmfsh 36459	The freshness relation of a model.
class mFresh
Syntax	cmfr 36460	The set of freshness relations.
class mFRel
Syntax	cmevl 36461	The evaluation function of a model.
class mEval
Syntax	cmdl 36462	The set of models.
class mMdl
Syntax	cusyn 36463	The syntax function applied to elements of the model.
class mUSyn
Syntax	cgmdl 36464	The set of models in a grammatical formal system.
class mGMdl
Syntax	cmitp 36465	The interpretation function of the model.
class mItp
Syntax	cmfitp 36466	The evaluation function derived from the interpretation.
class mFromItp
Definition	df-muv 36467	Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUV = Slot 7
Definition	df-mfsh 36468	Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFresh = Slot ;19
Definition	df-mevl 36469	Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mEval = Slot ;20
Definition	df-mvl 36470*	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 36471*	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 36472*	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 36473*	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 36474*	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 36475*	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 36476*	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 36477*	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)}))))))
21.7.16  Splitting fields
Syntax	ccpms 36478	Completion of a metric space.
class cplMetSp
Syntax	chlb 36479	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 36480	Direct limit structure.
class HomLim
Syntax	cpfl 36481	Polynomial extension field.
class polyFld
Syntax	csf1 36482	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 36483	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 36484	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-cplmet 36485*	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 36486*	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 36487*	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)))⟩}))
Definition	df-plfl 36488*	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.) (Revised by Thierry Arnoux and Steven Nguyen, 21-Jun-2025.)
⊢ polyFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑐 ∈ (Base‘𝑟) ↦ [(𝑐( ·𝑠 ‘𝑠)(1r‘𝑠))](𝑠 ~QG 𝑖)) / 𝑓⦌⟨⦋(𝑠 /s (𝑠 ~QG 𝑖)) / 𝑡⦌((𝑡 toNrmGrp (℩𝑛 ∈ (AbsVal‘𝑡)(𝑛 ∘ 𝑓) = (norm‘𝑟))) sSet ⟨(le‘ndx), ⦋(𝑧 ∈ (Base‘𝑡) ↦ (℩𝑞 ∈ 𝑧 (𝑞(rem1p‘𝑟)𝑝) = 𝑞)) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
Theorem	rexxfr3d 36489*	Transfer existential quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by SN, 20-Jun-2025.)
⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	rexxfr3dALT 36490*	Longer proof of rexxfr3d 36489 using ax-11 2193 instead of ax-12 2212, without the disjoint variable condition 𝐴𝑥𝑦. (Contributed by SN, 19-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ ∃𝑦 ∈ 𝐵 𝑥 = 𝑋))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒))
Theorem	rspssbasd 36491	The span of a set of ring elements is a set of ring elements. (Contributed by SN, 19-Jun-2025.)
⊢ 𝐾 = (RSpan‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐺 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐾‘𝐺) ⊆ 𝐵)
Theorem	ellcsrspsn 36492*	Membership in a left coset in a quotient of a ring by the span of a singleton (that is, by the ideal generated by an element). This characterization comes from eqglact 19396 and elrspsn 21530. (Contributed by SN, 19-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ ∼ = (𝑅 ~QG 𝐼)    &   ⊢ 𝑈 = (𝑅 /s ∼ )    &   ⊢ 𝐼 = ((RSpan‘𝑅)‘{𝑀})    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑈))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (𝑋 = [𝑥] ∼ ∧ 𝑋 = {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = (𝑥 + (𝑦 · 𝑀))}))
Theorem	ply1divalg3 36493*	Uniqueness of polynomial remainder: convert the subtraction in ply1divalg2 26462 to addition. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ + = (+g‘𝑃)    &   ⊢ ∙ = (.r‘𝑃)    &   ⊢ 𝐶 = (Unic1p‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝐵 (𝐷‘(𝐹 + (𝑞 ∙ 𝐺))) < (𝐷‘𝐺))
Theorem	r1peuqusdeg1 36494*	Uniqueness of polynomial remainder in terms of a quotient structure in the sense of the right hand side of r1pid2 26485. (Contributed by SN, 21-Jun-2025.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 𝐼 = ((RSpan‘𝑃)‘{𝐹})    &   ⊢ 𝑇 = (𝑃 /s (𝑃 ~QG 𝐼))    &   ⊢ 𝑄 = (Base‘𝑇)    &   ⊢ 𝑁 = (Unic1p‘𝑅)    &   ⊢ 𝐷 = (deg1‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑁)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑄)    ⇒   ⊢ (𝜑 → ∃!𝑞 ∈ 𝑍 (𝐷‘𝑞) < (𝐷‘𝐹))
Definition	df-sfl1 36495*	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 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))
Definition	df-sfl 36496*	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 ↦ (℩𝑥∃𝑓(𝑓 Isom < , (lt‘𝑟)((1...(♯‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(♯‘𝑝)))))
Definition	df-psl 36497*	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 ↦ ⦋(1st ‘𝑔) / 𝑒⦌⦋(1st ‘𝑒) / 𝑠⦌⦋(𝑠 splitFld ran (𝑥 ∈ 𝑞 ↦ (𝑥 ∘ (2nd ‘𝑔)))) / 𝑓⦌⟨𝑓, ((2nd ‘𝑔) ∘ (2nd ‘𝑓))⟩), (𝑝 ∪ {⟨0, ⟨⟨𝑟, ∅⟩, ( I ↾ (Base‘𝑟))⟩⟩}))) / 𝑓⦌((1st ∘ (𝑓 shift 1)) HomLim (2nd ∘ 𝑓)))
21.7.17  p-adic number fields
Syntax	czr 36498	Integral elements of a ring.
class ZRing
Syntax	cgf 36499	Galois finite field.
class GF
Syntax	cgfo 36500	Galois limit field.
class GF∞