P. 307 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30601-30700   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cgmdl 30601	The set of models in a grammatical formal system.
class mGMdl
Syntax	cmitp 30602	The interpretation function of the model.
class mItp
Syntax	cmfitp 30603	The evaluation function derived from the interpretation.
class mFromItp
Definition	df-muv 30604	Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUV = Slot 7
Definition	df-mfsh 30605	Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFresh = Slot 8
Definition	df-mevl 30606	Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mEval = Slot 9
Definition	df-mvl 30607*	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 30608*	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 30609*	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 30610*	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 30611*	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 30612*	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 30613*	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 30614*	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)}))))))
20.5.15  Splitting fields
Syntax	citr 30615	Integral subring of a ring.
class IntgRing
Syntax	ccpms 30616	Completion of a metric space.
class cplMetSp
Syntax	chlb 30617	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 30618	Direct limit structure.
class HomLim
Syntax	cpfl 30619	Polynomial extension field.
class polyFld
Syntax	csf1 30620	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 30621	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 30622	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-irng 30623*	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 30624*	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 30625*	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 30626*	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 30627*	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 ↦ ⦋(Poly1‘𝑟) / 𝑠⦌⦋((RSpan‘𝑠)‘{𝑝}) / 𝑖⦌⦋(𝑧 ∈ (Base‘𝑟) ↦ [(𝑧( ·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 (𝑟 deg1 𝑞) < (𝑟 deg1 𝑝))) / 𝑔⦌(◡𝑔 ∘ ((le‘𝑠) ∘ 𝑔))⟩), 𝑓⟩)
Definition	df-sfl1 30628*	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 ↦ ⦋( mPoly ‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠 deg1 𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟 deg1 𝑝))))))
Definition	df-sfl 30629*	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 30630*	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)))
20.5.16  p-adic number fields
Syntax	czr 30631	Integral elements of a ring.
class ZRing
Syntax	cgf 30632	Galois finite field.
class GF
Syntax	cgfo 30633	Galois limit field.
class GF∞
Syntax	ceqp 30634	Equivalence relation for df-qp 30645.
class ~Qp
Syntax	crqp 30635	Equivalence relation representatives for df-qp 30645.
class /Qp
Syntax	cqp 30636	The set of 𝑝-adic rational numbers.
class Qp
Syntax	czp 30637	The set of 𝑝-adic integers. (Not to be confused with czn 19579.)
class Zp
Syntax	cqpa 30638	Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
Syntax	ccp 30639	Metric completion of _Qp.
class Cp
Definition	df-zrng 30640	Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ZRing = (𝑟 ∈ V ↦ (𝑟 IntgRing ran (ℤRHom‘𝑟)))
Definition	df-gf 30641*	Define the Galois finite field of order 𝑝↑𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ GF = (𝑝 ∈ ℙ, 𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(1st ‘(𝑟 splitFld {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)})))
Definition	df-gfoo 30642*	Define the Galois field of order 𝑝↑+∞, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ GF∞ = (𝑝 ∈ ℙ ↦ ⦋(ℤ/nℤ‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {⦋(Poly1‘𝑟) / 𝑠⦌⦋(var1‘𝑟) / 𝑥⦌(((𝑝↑𝑛)(.g‘(mulGrp‘𝑠))𝑥)(-g‘𝑠)𝑥)})))
Definition	df-eqp 30643*	Define an equivalence relation on ℤ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum Σ𝑘 ≤ 𝑛𝑓(𝑘)(𝑝↑𝑘) is a multiple of 𝑝↑(𝑛 + 1) for every 𝑛. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ ~Qp = (𝑝 ∈ ℙ ↦ {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ (ℤ ↑𝑚 ℤ) ∧ ∀𝑛 ∈ ℤ Σ𝑘 ∈ (ℤ≥‘-𝑛)(((𝑓‘-𝑘) − (𝑔‘-𝑘)) / (𝑝↑(𝑘 + (𝑛 + 1)))) ∈ ℤ)})
Definition	df-rqp 30644*	There is a unique element of (ℤ ↑𝑚 (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑𝑚 ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑𝑚 ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑𝑚 (0...(𝑝 − 1)))))))
Definition	df-qp 30645*	Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑𝑚 (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ≥(◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘𝑓 + 𝑔)))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-sup((◡𝑓 “ (ℤ ∖ {0})), ℝ, ◡ < ))))))
Definition	df-zp 30646	Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Zp = (ZRing ∘ Qp)
Definition	df-qpa 30647*	Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟 deg1 𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
Definition	df-cp 30648	Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Cp = ( cplMetSp ∘ _Qp)
20.6  Mathbox for Filip Cernatescu I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from http://us.metamath.org/other/milpgame/milpgame.html.)
Theorem	problem1 30649	Practice problem 1. Clues: 5p4e9 10925 3p2e5 10918 eqtri 2536 oveq1i 6441. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ ((3 + 2) + 4) = 9
Theorem	problem2 30650	Practice problem 2. Clues: oveq12i 6443 adddiri 9810 add4i 10014 mulcli 9804 recni 9811 2re 10848 3eqtri 2540 10re 11260 5re 10857 1re 9798 4re 10855 eqcomi 2523 5p4e9 10925 oveq1i 6441 df-3 10838. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.)
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9)
Theorem	problem2OLD 30651	Practice problem 2. Clues: oveq12i 6443 adddiri 9810 add4i 10014 mulcli 9804 recni 9811 2re 10848 3eqtri 2540 10re 11260 5re 10857 1re 9798 4re 10855 eqcomi 2523 5p4e9 10925 oveq1i 6441 df-3 10838. (Contributed by Filip Cernatescu, 16-Mar-2019.) Obsolete version of problem2 30650 as of 9-Sep-2021. (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((2 · 10) + 5) + ((1 · 10) + 4)) = ((3 · 10) + 9)
Theorem	problem3 30652	Practice problem 3. Clues: eqcomi 2523 eqtri 2536 subaddrii 10124 recni 9811 4re 10855 3re 10852 1re 9798 df-4 10839 addcomi 9981. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 30653	Practice problem 4. Clues: pm3.2i 469 eqcomi 2523 eqtri 2536 subaddrii 10124 recni 9811 7re 10861 6re 10859 ax-1cn 9753 df-7 10842 ax-mp 5 oveq1i 6441 3cn 10853 2cn 10849 df-3 10838 mulid2i 9802 subdiri 10233 mp3an 1415 mulcli 9804 subadd23 10047 oveq2i 6442 oveq12i 6443 3t2e6 10937 mulcomi 9805 subcli 10111 biimpri 216 subadd2i 10123. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 3    &   ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7    ⇒   ⊢ (𝐴 = 1 ∧ 𝐵 = 2)
Theorem	problem5 30654	Practice problem 5. Clues: 3brtr3i 4510 mpbi 218 breqtri 4506 ltaddsubi 10341 remulcli 9813 2re 10848 3re 10852 9re 10865 eqcomi 2523 mvlladdi 10053 3cn 6cn 10860 eqtr3i 2538 6p3e9 10928 addcomi 9981 ltdiv1ii 10706 6re 10859 nngt0i 10812 2nn 10943 divcan3i 10523 recni 9811 2cn 10849 2ne0 10871 mpbir 219 eqtri 2536 mulcomi 9805 3t2e6 10937 divmuli 10531. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 30655	Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
⊢ 𝑋 ∈ ℂ    &   ⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0    ⇒   ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))
20.7  Mathbox for Paul Chapman
20.7.1  Real and complex numbers (cont.)
Theorem	climuzcnv 30656*	Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑))))
Theorem	sinccvglem 30657*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))    &   ⊢ (𝜑 → 𝐹 ⇝ 0)    &   ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))    &   ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)
Theorem	sinccvg 30658*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1)
Theorem	circum 30659*	The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ 𝐴 = ((2 · π) / 𝑛)    &   ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))))    &   ⊢ 𝑅 ∈ ℝ    ⇒   ⊢ 𝑃 ⇝ ((2 · π) · 𝑅)
20.7.2  Miscellaneous theorems
Theorem	elfzm12 30660	Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁)))
Theorem	nn0seqcvg 30661*	A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ 𝐹:ℕ0⟶ℕ0    &   ⊢ 𝑁 = (𝐹‘0)    &   ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘)))    ⇒   ⊢ (𝐹‘𝑁) = 0
Theorem	lediv2aALT 30662	Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	abs2sqlei 30663	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))
Theorem	abs2sqlti 30664	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))
Theorem	abs2sqle 30665	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)))
Theorem	abs2sqlt 30666	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)))
Theorem	abs2difi 30667	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 30668	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))
20.8  Mathbox for Scott Fenton
20.8.1  ZFC Axioms in primitive form
Theorem	axextprim 30669	ax-ext 2494 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))
Theorem	axrepprim 30670	ax-rep 4597 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥)))
Theorem	axunprim 30671	ax-un 6728 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpowprim 30672	ax-pow 4668 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦)
Theorem	axregprim 30673	ax-reg 8260 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axinfprim 30674	ax-inf 8298 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥))))
Theorem	axacprim 30675	ax-ac 9044 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
20.8.2  Untangled classes
Theorem	untelirr 30676*	We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 30777). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
Theorem	untuni 30677*	The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)
Theorem	untsucf 30678*	If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	unt0 30679	The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥
Theorem	untint 30680*	If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	efrunt 30681*	If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
Theorem	untangtr 30682*	A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
20.8.3  Extra propositional calculus theorems
Theorem	3orel1 30683	Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜓 ∨ 𝜒)))
Theorem	3orel2 30684	Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜒)))
Theorem	3orel3 30685	Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.)
⊢ (¬ 𝜒 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜓)))
Theorem	3pm3.2ni 30686	Triple negated disjunction introduction. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ¬ 𝜑    &   ⊢ ¬ 𝜓    &   ⊢ ¬ 𝜒    ⇒   ⊢ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒)
Theorem	3jaodd 30687	Double deduction form of 3jaoi 1382. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂)))
Theorem	3orit 30688	Closed form of 3ori 1379. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	biimpexp 30689	A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))
Theorem	3orel13 30690	Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.)
⊢ ((¬ 𝜑 ∧ ¬ 𝜒) → ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜓))
20.8.4  Misc. Useful Theorems
Theorem	nepss 30691	Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))
Theorem	3ccased 30692	Triple disjunction form of ccased 984. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓))    ⇒   ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓))
Theorem	dfso3 30693*	Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	brtpid1 30694	A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵
Theorem	brtpid2 30695	A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵
Theorem	brtpid3 30696	A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵
Theorem	ceqsrexv2 30697*	Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))
Theorem	iota5f 30698*	A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	ceqsralv2 30699*	Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓))
20.8.5  Properties of real and complex numbers
Theorem	sqdivzi 30700	Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))