P. 360 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35901-36000   *Has distinct variable group(s)

Type	Label	Description
Statement
Syntax	cmwgfs 35901	The set of weakly grammatical formal systems.
class mWGFS
Syntax	cmsy 35902	The syntax typecode function.
class mSyn
Syntax	cmesy 35903	The syntax typecode function for expressions.
class mESyn
Syntax	cmgfs 35904	The set of grammatical formal systems.
class mGFS
Syntax	cmtree 35905	The set of proof trees.
class mTree
Syntax	cmst 35906	The set of syntax trees.
class mST
Syntax	cmsax 35907	The indexing set for a syntax axiom.
class mSAX
Syntax	cmufs 35908	The set of unambiguous formal systems.
class mUFS
Definition	df-m0s 35909	Define a function mapping expressions to statements. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ m0St = (𝑎 ∈ V ↦ ⟨∅, ∅, 𝑎⟩)
Definition	df-msa 35910*	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 35911*	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 35912	Define the syntax typecode function. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mSyn = Slot 6
Definition	df-mesyn 35913*	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 35914*	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 35915*	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 35916	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 35917*	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 35918	Define the set of unambiguous formal systems. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUFS = {𝑡 ∈ mGFS ∣ Fun (mST‘𝑡)}
21.6.15  Models of formal systems
Syntax	cmuv 35919	The universe of a model.
class mUV
Syntax	cmvl 35920	The set of valuations.
class mVL
Syntax	cmvsb 35921	Substitution for a valuation.
class mVSubst
Syntax	cmfsh 35922	The freshness relation of a model.
class mFresh
Syntax	cmfr 35923	The set of freshness relations.
class mFRel
Syntax	cmevl 35924	The evaluation function of a model.
class mEval
Syntax	cmdl 35925	The set of models.
class mMdl
Syntax	cusyn 35926	The syntax function applied to elements of the model.
class mUSyn
Syntax	cgmdl 35927	The set of models in a grammatical formal system.
class mGMdl
Syntax	cmitp 35928	The interpretation function of the model.
class mItp
Syntax	cmfitp 35929	The evaluation function derived from the interpretation.
class mFromItp
Definition	df-muv 35930	Define the universe of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mUV = Slot 7
Definition	df-mfsh 35931	Define the freshness relation of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mFresh = Slot ;19
Definition	df-mevl 35932	Define the evaluation function of a model. (Contributed by Mario Carneiro, 14-Jul-2016.)
⊢ mEval = Slot ;20
Definition	df-mvl 35933*	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 35934*	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 35935*	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 35936*	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 35937*	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 35938*	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 35939*	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 35940*	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 35941	Completion of a metric space.
class cplMetSp
Syntax	chlb 35942	Embeddings for a direct limit.
class HomLimB
Syntax	chlim 35943	Direct limit structure.
class HomLim
Syntax	cpfl 35944	Polynomial extension field.
class polyFld
Syntax	csf1 35945	Splitting field for a single polynomial (auxiliary).
class splitFld1
Syntax	csf 35946	Splitting field for a finite set of polynomials.
class splitFld
Syntax	cpsl 35947	Splitting field for a sequence of polynomials.
class polySplitLim
Definition	df-cplmet 35948*	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 35949*	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 35950*	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 35951*	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 35952*	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 35953*	Longer proof of rexxfr3d 35952 using ax-11 2190 instead of ax-12 2211, 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 35954	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 35955*	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 19203 and elrspsn 21290. (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 35956*	Uniqueness of polynomial remainder: convert the subtraction in ply1divalg2 26179 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 35957*	Uniqueness of polynomial remainder in terms of a quotient structure in the sense of the right hand side of r1pid2 26202. (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 35958*	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 35959*	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 35960*	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 35961	Integral elements of a ring.
class ZRing
Syntax	cgf 35962	Galois finite field.
class GF
Syntax	cgfo 35963	Galois limit field.
class GF∞
Syntax	ceqp 35964	Equivalence relation for df-qp 35975.
class ~Qp
Syntax	crqp 35965	Equivalence relation representatives for df-qp 35975.
class /Qp
Syntax	cqp 35966	The set of 𝑝-adic rational numbers.
class Qp
Syntax	czp 35967	The set of 𝑝-adic integers. (Not to be confused with czn 21534.)
class Zp
Syntax	cqpa 35968	Algebraic completion of the 𝑝-adic rational numbers.
class _Qp
Syntax	ccp 35969	Metric completion of _Qp.
class Cp
Definition	df-zrng 35970	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 35971*	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 35972*	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 35973*	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 35974*	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 35975*	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 35976	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 35977*	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) ∣ ((𝑟deg1𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
Definition	df-cp 35978	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 35979	Practice problem 1. Clues: 5p4e9 12372 3p2e5 12365 eqtri 2784 oveq1i 7402. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ ((3 + 2) + 4) = 9
Theorem	problem2 35980	Practice problem 2. Clues: oveq12i 7404 adddiri 11192 add4i 11405 mulcli 11186 recni 11193 2re 12289 3eqtri 2788 10re 12708 5re 12302 1re 11178 4re 12299 eqcomi 2770 5p4e9 12372 oveq1i 7402 df-3 12278. (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 35981	Practice problem 3. Clues: eqcomi 2770 eqtri 2784 subaddrii 11517 recni 11193 4re 12299 3re 12295 1re 11178 df-4 12279 addcomi 11371. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 35982	Practice problem 4. Clues: pm3.2i 474 eqcomi 2770 eqtri 2784 subaddrii 11517 recni 11193 7re 12308 6re 12305 ax-1cn 11128 df-7 12282 ax-mp 5 oveq1i 7402 3cn 12296 2cn 12290 df-3 12278 mullidi 11184 subdiri 11634 mp3an 1481 mulcli 11186 subadd23 11439 oveq2i 7403 oveq12i 7404 3t2e6 12380 mulcomi 11187 subcli 11504 biimpri 230 subadd2i 11516. (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 35983	Practice problem 5. Clues: 3brtr3i 5128 mpbi 232 breqtri 5124 ltaddsubi 11745 remulcli 11195 2re 12289 3re 12295 9re 12314 eqcomi 2770 mvlladdi 11446 3cn 6cn 12306 eqtr3i 2786 6p3e9 12374 addcomi 11371 ltdiv1ii 12118 6re 12305 nngt0i 12249 2nn 12288 divcan3i 11934 recni 11193 2cn 12290 2ne0 12321 mpbir 233 eqtri 2784 mulcomi 11187 3t2e6 12380 divmuli 11942. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 35984	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 35985*	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 35986*	((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 35987*	((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 35988*	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 35989	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 35990*	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 35991	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 35992	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 35993	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 35994	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 35995	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 35996	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 35997	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 Hongxiu Chen
Theorem	2thALT 35998	Alternate proof of 2th 266. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	orbi2iALT 35999	Alternate proof of orbi2i 923. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
Theorem	pm3.48ALT 36000	Alternate proof of pm3.48 976. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))