P. 350 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34901-35000   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cusyn 34901	The syntax function applied to elements of the model.
class mUSyn
Syntax	cgmdl 34902	The set of models in a grammatical formal system.
class mGMdl
Syntax	cmitp 34903	The interpretation function of the model.
class mItp
Syntax	cmfitp 34904	The evaluation function derived from the interpretation.
class mFromItp
Definition	df-muv 34905	Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUV = Slot 7
Definition	df-mfsh 34906	Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFresh = Slot ;19
Definition	df-mevl 34907	Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mEval = Slot ;20
Definition	df-mvl 34908*	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 34909*	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 34910*	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 34911*	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 34912*	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 34913*	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 34914*	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 34915*	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)}))))))
21.6.16  Splitting fields
Syntax	ccpms 34916	Completion of a metric space.
class cplMetSp
Syntax	chlb 34917	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 34918	Direct limit structure.
class HomLim
Syntax	cpfl 34919	Polynomial extension field.
class polyFld
Syntax	csf1 34920	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 34921	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 34922	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-cplmet 34923*	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 34924*	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 34925*	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 34926*	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 34927*	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 34928*	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 34929*	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)))
21.6.17  p-adic number fields
Syntax	czr 34930	Integral elements of a ring.
class ZRing
Syntax	cgf 34931	Galois finite field.
class GF
Syntax	cgfo 34932	Galois limit field.
class GF∞
Syntax	ceqp 34933	Equivalence relation for df-qp 34944.
class ~Qp
Syntax	crqp 34934	Equivalence relation representatives for df-qp 34944.
class /Qp
Syntax	cqp 34935	The set of 𝑝-adic rational numbers.
class Qp
Syntax	czp 34936	The set of 𝑝-adic integers. (Not to be confused with czn 21271.)
class Zp
Syntax	cqpa 34937	Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
Syntax	ccp 34938	Metric completion of _Qp.
class Cp
Definition	df-zrng 34939	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 34940*	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 34941*	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 34942*	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 34943*	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 34944*	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.) (Revised by AV, 10-Oct-2021.)
⊢ 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↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
Definition	df-zp 34945	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 34946*	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 34947	Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Cp = ( cplMetSp ∘ _Qp)
21.7  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 https://us.metamath.org/other/milpgame/milpgame.html.)
Theorem	problem1 34948	Practice problem 1. Clues: 5p4e9 12374 3p2e5 12367 eqtri 2758 oveq1i 7421. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ ((3 + 2) + 4) = 9
Theorem	problem2 34949	Practice problem 2. Clues: oveq12i 7423 adddiri 11231 add4i 11442 mulcli 11225 recni 11232 2re 12290 3eqtri 2762 10re 12700 5re 12303 1re 11218 4re 12300 eqcomi 2739 5p4e9 12374 oveq1i 7421 df-3 12280. (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	problem3 34950	Practice problem 3. Clues: eqcomi 2739 eqtri 2758 subaddrii 11553 recni 11232 4re 12300 3re 12296 1re 11218 df-4 12281 addcomi 11409. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 34951	Practice problem 4. Clues: pm3.2i 469 eqcomi 2739 eqtri 2758 subaddrii 11553 recni 11232 7re 12309 6re 12306 ax-1cn 11170 df-7 12284 ax-mp 5 oveq1i 7421 3cn 12297 2cn 12291 df-3 12280 mullidi 11223 subdiri 11668 mp3an 1459 mulcli 11225 subadd23 11476 oveq2i 7422 oveq12i 7423 3t2e6 12382 mulcomi 11226 subcli 11540 biimpri 227 subadd2i 11552. (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 34952	Practice problem 5. Clues: 3brtr3i 5176 mpbi 229 breqtri 5172 ltaddsubi 11779 remulcli 11234 2re 12290 3re 12296 9re 12315 eqcomi 2739 mvlladdi 11482 3cn 6cn 12307 eqtr3i 2760 6p3e9 12376 addcomi 11409 ltdiv1ii 12147 6re 12306 nngt0i 12255 2nn 12289 divcan3i 11964 recni 11232 2cn 12291 2ne0 12320 mpbir 230 eqtri 2758 mulcomi 11226 3t2e6 12382 divmuli 11972. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 34953	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)))
21.8  Mathbox for Paul Chapman
21.8.1  Real and complex numbers (cont.)
Theorem	climuzcnv 34954*	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 34955*	((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 34956*	((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 34957*	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)
21.8.2  Miscellaneous theorems
Theorem	elfzm12 34958	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 34959*	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 34960	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 34961	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 34962	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 34963	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 34964	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 34965	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 34966	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))
21.9  Mathbox for Adrian Ducourtial
21.9.1  Propositional calculus
Theorem	currybi 34967	Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 35739 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)
⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)
21.9.2  Clone theory
Syntax	ccloneop 34968	Syntax for the function of the class of operations on a set.
class CloneOp
Definition	df-cloneop 34969*	Define the function that sends a set to the class of clone-theoretic operations on the set. For convenience, we take an operation on 𝑎 to be a function on finite sequences of elements of 𝑎 (rather than tuples) with values in 𝑎. Following line 6 of [Szendrei] p. 11, the arity 𝑛 of an operation (here, the length of the sequences at which the operation is defined) is always finite and non-zero, whence 𝑛 is taken to be a non-zero finite ordinal. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
Syntax	cprj 34970	Syntax for the function of projections on sets.
class prj
Definition	df-prj 34971*	Define the function that, for a set 𝑎, arity 𝑛, and index 𝑖, returns the 𝑖-th 𝑛-ary projection on 𝑎. This is the 𝑛-ary operation on 𝑎 that, for any sequence of 𝑛 elements of 𝑎, returns the element having index 𝑖. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
Syntax	csuppos 34972	Syntax for the function of superpositions.
class suppos
Definition	df-suppos 34973*	Define the function that, when given an 𝑛-ary operation 𝑓 and 𝑛 many 𝑚-ary operations (𝑔‘∅), ..., (𝑔‘∪ 𝑛), returns the superposition of 𝑓 with the (𝑔‘𝑖), itself another 𝑚-ary operation on 𝑎. Given 𝑥 (a sequence of 𝑚 arguments in 𝑎), the superposition effectively applies each of the (𝑔‘𝑖) to 𝑥, then applies 𝑓 to the resulting sequence of 𝑛 function values. This can be seen as a generalized version of function composition; see paragraph 3 of [Szendrei] p. 11. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (𝑓 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛)), 𝑔 ∈ ((𝑎 ↑m (𝑎 ↑m 𝑚)) ↑m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
21.10  Mathbox for Scott Fenton
21.10.1  ZFC Axioms in primitive form
Theorem	axextprim 34974	ax-ext 2701 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 34975	ax-rep 5284 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 34976	ax-un 7727 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 34977	ax-pow 5362 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 34978	ax-reg 9589 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 34979	ax-inf 9635 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 34980	ax-ac 10456 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))))))
21.10.2  Untangled classes
Theorem	untelirr 34981*	We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 35068). 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 34982*	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 34983*	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 34984	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 34985*	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 34986*	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 34987*	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))
21.10.3  Extra propositional calculus theorems
Theorem	3jaodd 34988	Double deduction form of 3jaoi 1425. (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 34989	Closed form of 3ori 1422. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	biimpexp 34990	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)))
21.10.4  Misc. Useful Theorems
Theorem	nepss 34991	Two classes are unequal 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 34992	Triple disjunction form of ccased 1035. (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 34993*	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 34994	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵
Theorem	brtpid2 34995	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵
Theorem	brtpid3 34996	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵
Theorem	iota5f 34997*	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	jath 34998	Closed form of ja 186. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.)
⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒)))
Theorem	xpab 34999*	Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
Theorem	nnuni 35000	The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)