P. 164 of Theorem List - Intuitionistic Logic Explorer

Type

Label

Description

Statement

Theorem

iseupth 16301

The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Theorem

iseupthf1o 16302

The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Theorem

eupthi 16303

Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Theorem

eupthf1o 16304

The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)

Theorem

eupthfi 16305

Any graph with an Eulerian path is of finite size, i.e. with a finite number of edges. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 18-Feb-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → dom 𝐼 ∈ Fin)

Theorem

eupthseg 16306

The 𝑁-th edge in an eulerian path is the edge having 𝑃(𝑁) and 𝑃(𝑁 + 1) as endpoints . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹(EulerPaths‘𝐺)𝑃 ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁)))

Theorem

eupthcl 16307

An Eulerian path has length ♯(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)

Theorem

eupthistrl 16308

An Eulerian path is a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.) (Revised by AV, 18-Feb-2021.)

⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)

Theorem

eupthiswlk 16309

An Eulerian path is a walk. (Contributed by AV, 6-Apr-2021.)

⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)

Theorem

eupthpf 16310

The 𝑃 function in an Eulerian path is a function from a finite sequence of nonnegative integers to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))

Theorem

eupthres 16311

The restriction ⟨𝐻, 𝑄⟩ of an Eulerian path ⟨𝐹, 𝑃⟩ to an initial segment of the path (of length 𝑁) forms an Eulerian path on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ 𝐻 = (𝐹 prefix 𝑁)    &   ⊢ 𝑄 = (𝑃 ↾ (0...𝑁))    &   ⊢ (Vtx‘𝑆) = 𝑉    ⇒   ⊢ (𝜑 → 𝐻(EulerPaths‘𝑆)𝑄)

Theorem

eupth2lem1 16312

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴 ≠ 𝐵 ∧ (𝑈 = 𝐴 ∨ 𝑈 = 𝐵))))

Theorem

eupth2lem2dc 16313

Lemma for eupth2 . (Contributed by Mario Carneiro, 8-Apr-2015.)

⊢ (𝜑 → 𝐵 ∈ 𝑋)    &   ⊢ (𝜑 → DECID 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝑈)    ⇒   ⊢ (𝜑 → (¬ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ 𝑈 ∈ if(𝐴 = 𝐶, ∅, {𝐴, 𝐶})))

Theorem

trlsegvdeglem1 16314

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → ((𝑃‘𝑁) ∈ 𝑉 ∧ (𝑃‘(𝑁 + 1)) ∈ 𝑉))

Theorem

trlsegvdeglem2 16315

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → Fun (iEdg‘𝑋))

Theorem

trlsegvdeglem3 16316

Lemma for trlsegvdeg . (Contributed by AV, 20-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → Fun (iEdg‘𝑌))

Theorem

trlsegvdeglem4 16317

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → dom (iEdg‘𝑋) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))

Theorem

trlsegvdeglem5 16318

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → dom (iEdg‘𝑌) = {(𝐹‘𝑁)})

Theorem

trlsegvdeglem6 16319

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → dom (iEdg‘𝑋) ∈ Fin)

Theorem

trlsegvdeglem7 16320

Lemma for trlsegvdeg . (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    ⇒   ⊢ (𝜑 → dom (iEdg‘𝑌) ∈ Fin)

Theorem

trlsegvdegfi 16321

The effect on vertex degree of adding one edge to a trail. In the following, a subgraph induced by a segment of a trail is called a "subtrail": For any subtrail 𝑍 of a trail ⟨𝐹, 𝑃⟩ in a pseudograph 𝐺 which is composed of subtrails 𝑋 and 𝑌, where 𝑌 consists of a single edge, the vertex degree of any vertex 𝑈 within 𝑍 is the sum of the vertex degree of 𝑈 within 𝑋 and the vertex degree of 𝑈 within 𝑌. Note that this theorem would not hold for arbitrary walks (if the last edge was identical with a previous edge, the degree of the vertices incident with this edge would not be increased because of this edge). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 20-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑍)‘𝑈) = (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)))

Theorem

eupth2lem3lem1fi 16322

Lemma for eupth2lem3fi 16330. (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑋)‘𝑈) ∈ ℕ₀)

Theorem

eupth2lem3lem2fi 16323

Lemma for eupth2lem3fi 16330. (Contributed by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ (𝜑 → ((VtxDeg‘𝑌)‘𝑈) ∈ ℕ₀)

Theorem

eupth2lem3lem3fi 16324*

Lemma for eupth2lem3fi 16330. If a loop {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} is added to a trail, the degree of the vertices with odd degree remains odd (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 21-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    &   ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))))    ⇒   ⊢ ((𝜑 ∧ (𝑃‘𝑁) = (𝑃‘(𝑁 + 1))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theorem

eupth2lem3lem6fi 16325*

If an edge (not a loop) is added to a trail, the degree of vertices not being end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). Remark: This seems to be not valid for hyperedges joining more vertices than (𝑃‘0) and (𝑃‘𝑁): if there is a third vertex in the edge, and this vertex is already contained in the trail, then the degree of this vertex could be affected by this edge! (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UPGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    &   ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})    ⇒   ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 ≠ (𝑃‘𝑁) ∧ 𝑈 ≠ (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theorem

eupth2lem3lem5 16326

Lemma for eupth2fi 16333. (Contributed by AV, 25-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    &   ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})    ⇒   ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉)

Theorem

eupth2lem3lem4fi 16327*

Lemma for eupth2lem3fi 16330. If an edge (not a loop) is added to a trail, the degree of the end vertices of this edge remains odd if it was odd before (regarding the subgraphs induced by the involved trails). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 25-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    &   ⊢ (𝜑 → if-((𝑃‘𝑁) = (𝑃‘(𝑁 + 1)), (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁)}, {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))} ⊆ (𝐼‘(𝐹‘𝑁))))    &   ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) ∈ 𝒫 𝑉)    ⇒   ⊢ ((𝜑 ∧ (𝑃‘𝑁) ≠ (𝑃‘(𝑁 + 1)) ∧ (𝑈 = (𝑃‘𝑁) ∨ 𝑈 = (𝑃‘(𝑁 + 1)))) → (¬ 2 ∥ (((VtxDeg‘𝑋)‘𝑈) + ((VtxDeg‘𝑌)‘𝑈)) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theorem

eupth2lem3lem7fi 16328*

Lemma for eupth2lem3fi 16330: Combining trlsegvdegfi 16321, eupth2lem3lem3fi 16324, eupth2lem3lem4fi 16327 and eupth2lem3lem6fi 16325. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 27-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹)))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    &   ⊢ (𝜑 → (Vtx‘𝑋) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑌) = 𝑉)    &   ⊢ (𝜑 → (Vtx‘𝑍) = 𝑉)    &   ⊢ (𝜑 → (iEdg‘𝑋) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   ⊢ (𝜑 → (iEdg‘𝑌) = {⟨(𝐹‘𝑁), (𝐼‘(𝐹‘𝑁))⟩})    &   ⊢ (𝜑 → (iEdg‘𝑍) = (𝐼 ↾ (𝐹 “ (0...𝑁))))    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    &   ⊢ (𝜑 → (𝐼‘(𝐹‘𝑁)) = {(𝑃‘𝑁), (𝑃‘(𝑁 + 1))})    ⇒   ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑍)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theorem

eupthvdres 16329

The vertex degree remains the same for all vertices if the edges are restricted to the edges of an Eulerian path. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(♯‘𝐹))))⟩    ⇒   ⊢ (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))

Theorem

eupth2lem3fi 16330*

Lemma for eupth2fi 16333. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    &   ⊢ 𝐻 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑁)))⟩    &   ⊢ 𝑋 = ⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑁 + 1))))⟩    &   ⊢ (𝜑 → 𝑁 ∈ ℕ₀)    &   ⊢ (𝜑 → (𝑁 + 1) ≤ (♯‘𝐹))    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐻)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑁), ∅, {(𝑃‘0), (𝑃‘𝑁)}))    ⇒   ⊢ (𝜑 → (¬ 2 ∥ ((VtxDeg‘𝑋)‘𝑈) ↔ 𝑈 ∈ if((𝑃‘0) = (𝑃‘(𝑁 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑁 + 1))})))

Theorem

eupth2lembfi 16331*

Lemma for eupth2fi 16333 (induction basis): There are no vertices of odd degree in an Eulerian path of length 0, having no edge and identical endpoints (the single vertex of the Eulerian path). (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^0)))⟩)‘𝑥)} = ∅)

Theorem

eupth2lemsfi 16332*

Lemma for eupth2fi 16333 (induction step): The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct, if the Eulerian path shortened by one edge has this property. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^𝑛)))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘𝑛), ∅, {(𝑃‘0), (𝑃‘𝑛)})) → ((𝑛 + 1) ≤ (♯‘𝐹) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘⟨𝑉, (𝐼 ↾ (𝐹 “ (0..^(𝑛 + 1))))⟩)‘𝑥)} = if((𝑃‘0) = (𝑃‘(𝑛 + 1)), ∅, {(𝑃‘0), (𝑃‘(𝑛 + 1))}))))

Theorem

eupth2fi 16333*

The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.) (Revised by AV, 26-Feb-2021.)

⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → Fun 𝐼)    &   ⊢ (𝜑 → 𝐹(EulerPaths‘𝐺)𝑃)    &   ⊢ (𝜑 → 𝑉 ∈ Fin)    ⇒   ⊢ (𝜑 → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))

PART 13  GUIDES AND MISCELLANEA

13.1  Guides (conventions, explanations, and examples)

13.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. The following sources lay out how mathematics is developed without the law of the excluded middle. Of course, there are a greater number of sources which assume excluded middle and most of what is in them applies here too (especially in a treatment such as ours which is built on first-order logic and set theory, rather than, say, type theory). Studying how a topic is treated in the Metamath Proof Explorer and the references therein is often a good place to start (and is easy to compare with the Intuitionistic Logic Explorer). The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

• Axioms of propositional calculus - Stanford Encyclopedia of Philosophy or [Heyting].
• Axioms of predicate calculus - our axioms are adapted from the ones in the Metamath Proof Explorer.
• Theorems of propositional calculus - [Heyting].
• Theorems of pure predicate calculus - Metamath Proof Explorer.
• Theorems of equality and substitution - Metamath Proof Explorer.
• Axioms of set theory - [Crosilla].
• Development of set theory - Chapter 10 of [HoTT].
• Construction of real and complex numbers - Chapter 11 of [HoTT]; [BauerTaylor].
• Theorems about real numbers - [Geuvers].

Theorem

conventions 16334

Unless there is a reason to diverge, we follow the conventions of the Metamath Proof Explorer (MPE, set.mm). This list of conventions is intended to be read in conjunction with the corresponding conventions in the Metamath Proof Explorer, and only the differences are described below.

• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, our choice of IZF (Intuitionistic Zermelo-Fraenkel) over CZF (Constructive Zermelo-Fraenkel, a weaker system) was just an expedient choice because IZF is easier to formalize in Metamath. You can find some development using CZF in BJ's mathbox starting at wbd 16428 (and the section header just above it). As for the axiom of choice, the full axiom of choice implies excluded middle as seen at acexmid 6017, although some authors will use countable choice or dependent choice. For example, countable choice or excluded middle is needed to show that the Cauchy reals coincide with the Dedekind reals - Corollary 11.4.3 of [HoTT], p. (varies).
• Junk/undefined results. Much of the discussion of this topic in the Metamath Proof Explorer applies except that certain techniques are not available to us. For example, the Metamath Proof Explorer will often say "if a function is evaluated within its domain, a certain result follows; if the function is evaluated outside its domain, the same result follows. Since the function must be evaluated within its domain or outside it, the result follows unconditionally" (the use of excluded middle in this argument is perhaps obvious when stated this way). Often, the easiest fix will be to prove we are evaluating functions within their domains, other times it will be possible to use a theorem like relelfvdm 5671 which says that if a function value produces an inhabited set, then the function is being evaluated within its domain.
• Bibliography references. The bibliography for the Intuitionistic Logic Explorer is separate from the one for the Metamath Proof Explorer but feel free to copy-paste a citation in either direction in order to cite it.

Label naming conventions

Here are a few of the label naming conventions:

• Suffixes. We follow the conventions of the Metamath Proof Explorer with a few additions. A biconditional in set.mm which is an implication in iset.mm should have a "r" (for the reverse direction), or "i"/"im" (for the forward direction) appended. A theorem in set.mm which has a decidability condition added should add "dc" to the theorem name. A theorem in set.mm where "nonempty class" is changed to "inhabited class" should add "m" (for member) to the theorem name.
• iset.mm versus set.mm names

  Theorems which are the same as in set.mm should be named the same (that is, where the statement of the theorem is the same; the proof can differ without a new name being called for). Theorems which are different should be named differently (we do have a small number of intentional exceptions to this rule but on the whole it serves us well).

  As for how to choose names so they are different between iset.mm and set.mm, when possible choose a name which reflect the difference in the theorems. For example, if a theorem in set.mm is an equality and the iset.mm analogue is a subset, add "ss" to the iset.mm name. If need be, add "i" to the iset.mm name (usually as a prefix to some portion of the name).

  As with set.mm, we welcome suggestions for better names (such as names which are more consistent with naming conventions).

  We do try to keep set.mm and iset.mm similar where we can. For example, if a theorem exists in both places but the name in set.mm isn't great, we tend to keep that name for iset.mm, or change it in both files together. This is mainly to make it easier to copy theorems, but also to generally help people browse proofs, find theorems, write proofs, etc.

The following table shows some commonly-used abbreviations in labels which are not found in the Metamath Proof Explorer, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME.

For the "g" abbreviation, this is related to the set.mm usage, in which "is a set" conditions are converted from hypotheses to antecedents, but is also used where "is a set" conditions are added relative to similar set.mm theorems.

Abbreviation	Mnenomic/Meaning	Source	Expression	Syntax?	Example(s)
ap	apart	df-pap 7467, df-ap 8762		Yes	apadd1 8788, apne 8803
g	with "is a set" condition			No	1stvalg 6305, brtposg 6420, setsmsbasg 15206
m	inhabited (from "member")		∃𝑥𝑥 ∈ 𝐴	No	r19.2m 3581, negm 9849, ctm 7308, basmex 13144
seq3, sum3	recursive sequence	df-seqfrec 10711		Yes	seq3-1 10725, fsum3 11950
tap	tight apartness	df-tap 7469		Yes	df-tap 7469

• Community. The Metamath mailing list also covers the Intuitionistic Logic Explorer and is at: https://groups.google.com/g/metamath 7469.

(Contributed by Jim Kingdon, 24-Feb-2020.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

13.1.2  Definitional examples

Theorem

ex-or 16335

Example for ax-io 716. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 3 ∨ 4 = 4)

Theorem

ex-an 16336

Example for ax-ia1 106. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

⊢ (2 = 2 ∧ 3 = 3)

Theorem

1kp2ke3k 16337

Example for df-dec 9612, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 9612 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000

Theorem

ex-fl 16338

Example for df-fl 10531. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

⊢ ((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theorem

ex-ceil 16339

Example for df-ceil 10532. (Contributed by AV, 4-Sep-2021.)

⊢ ((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theorem

ex-exp 16340

Example for df-exp 10802. (Contributed by AV, 4-Sep-2021.)

⊢ ((5↑2) = ;25 ∧ (-3↑-2) = (1 / 9))

Theorem

ex-fac 16341

Example for df-fac 10989. (Contributed by AV, 4-Sep-2021.)

⊢ (!‘5) = ;;120

Theorem

ex-bc 16342

Example for df-bc 11011. (Contributed by AV, 4-Sep-2021.)

⊢ (5C3) = ;10

Theorem

ex-dvds 16343

Example for df-dvds 12351: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)

⊢ 3 ∥ 6

Theorem

ex-gcd 16344

Example for df-gcd 12527. (Contributed by AV, 5-Sep-2021.)

⊢ (-6 gcd 9) = 3

PART 14  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)

14.1  Mathboxes for user contributions

14.1.1  Mathbox guidelines

Theorem

mathbox 16345

(This theorem is a dummy placeholder for these guidelines. The label of this theorem, "mathbox", is hard-coded into the Metamath program to identify the start of the mathbox section for web page generation.)

A "mathbox" is a user-contributed section that is maintained by its contributor independently from the main part of iset.mm.

For contributors:

By making a contribution, you agree to release it into the public domain, according to the statement at the beginning of iset.mm.

Guidelines:

Mathboxes in iset.mm follow the same practices as in set.mm, so refer to the mathbox guidelines there for more details.

(Contributed by NM, 20-Feb-2007.) (Revised by the Metamath team, 9-Sep-2023.) (New usage is discouraged.)

⊢ 𝜑    ⇒   ⊢ 𝜑

14.2  Mathbox for Matthew House

Theorem

depindlem1 16346*

Lemma for depind 16349. (Contributed by Matthew House, 14-Apr-2026.)

⊢ (𝜑 → 𝑃:ℕ₀⟶V)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   ⊢ (𝜑 → (𝐹:ℕ₀⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))

Theorem

depindlem2 16347*

Lemma for depind 16349. (Contributed by Matthew House, 14-Apr-2026.)

⊢ (𝜑 → 𝑃:ℕ₀⟶V)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   ⊢ (𝜑 → 𝐹 ∈ X𝑛 ∈ ℕ₀ (𝑃‘𝑛))

Theorem

depindlem3 16348*

Lemma for depind 16349. (Contributed by Matthew House, 14-Apr-2026.)

⊢ (𝜑 → 𝑃:ℕ₀⟶V)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    &   ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ₀ ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))    ⇒   ⊢ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ₀ (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

Theorem

depind 16349*

Theorem related to a dependently typed induction principle in type theory. (Contributed by Matthew House, 14-Apr-2026.)

⊢ (𝜑 → 𝑃:ℕ₀⟶V)    &   ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))    &   ⊢ (𝜑 → ∀𝑛 ∈ ℕ₀ (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))    ⇒   ⊢ (𝜑 → ∃!𝑓 ∈ X 𝑛 ∈ ℕ₀ (𝑃‘𝑛)((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ₀ (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))))

14.3  Mathbox for BJ

14.3.1  Propositional calculus

Theorem

bj-nnsn 16350

As far as implying a negated formula is concerned, a formula is equivalent to its double negation. (Contributed by BJ, 24-Nov-2023.)

⊢ ((𝜑 → ¬ 𝜓) ↔ (¬ ¬ 𝜑 → ¬ 𝜓))

Theorem

bj-nnor 16351

Double negation of a disjunction in terms of implication. (Contributed by BJ, 9-Oct-2019.)

⊢ (¬ ¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnim 16352

The double negation of an implication implies the implication with the consequent doubly negated. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))

Theorem

bj-nnan 16353

The double negation of a conjunction implies the conjunction of the double negations. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))

Theorem

bj-nnclavius 16354

Clavius law with doubly negated consequent. (Contributed by BJ, 4-Dec-2023.)

⊢ ((¬ 𝜑 → 𝜑) → ¬ ¬ 𝜑)

Theorem

bj-imnimnn 16355

If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 16354 as its last step. (Contributed by BJ, 27-Oct-2024.)

⊢ (𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → 𝜓)    ⇒   ⊢ ¬ ¬ 𝜓

14.3.1.1  Stable formulas

Some of the following theorems, like bj-sttru 16357 or bj-stfal 16359 could be deduced from their analogues for decidability, but stability is not provable from decidability in minimal calculus, so direct proofs have their interest.

Theorem

bj-trst 16356

A provable formula is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (𝜑 → STAB 𝜑)

Theorem

bj-sttru 16357

The true truth value is stable. (Contributed by BJ, 5-Aug-2024.)

⊢ STAB ⊤

Theorem

bj-fast 16358

A refutable formula is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ 𝜑 → STAB 𝜑)

Theorem

bj-stfal 16359

The false truth value is stable. (Contributed by BJ, 5-Aug-2024.)

⊢ STAB ⊥

Theorem

bj-nnst 16360

Double negation of stability of a formula. Intuitionistic logic refutes unstability (but does not prove stability) of any formula. This theorem can also be proved in classical refutability calculus (see https://us.metamath.org/mpeuni/bj-peircestab.html) but not in minimal calculus (see https://us.metamath.org/mpeuni/bj-stabpeirce.html). See nnnotnotr 16606 for the version not using the definition of stability. (Contributed by BJ, 9-Oct-2019.) Prove it in ( → , ¬ ) -intuitionistic calculus with definitions (uses of ax-ia1 106, ax-ia2 107, ax-ia3 108 are via sylibr 134, necessary for definition unpackaging), and in ( → , ↔ , ¬ )-intuitionistic calculus, following a discussion with Jim Kingdon. (Revised by BJ, 27-Oct-2024.)

⊢ ¬ ¬ STAB 𝜑

Theorem

bj-nnbist 16361

If a formula is not refutable, then it is stable if and only if it is provable. By double-negation translation, if 𝜑 is a classical tautology, then ¬ ¬ 𝜑 is an intuitionistic tautology. Therefore, if 𝜑 is a classical tautology, then 𝜑 is intuitionistically equivalent to its stability (and to its decidability, see bj-nnbidc 16374). (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (STAB 𝜑 ↔ 𝜑))

Theorem

bj-stst 16362

Stability of a proposition is stable if and only if that proposition is stable. STAB is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (STAB STAB 𝜑 ↔ STAB 𝜑)

Theorem

bj-stim 16363

A conjunction with a stable consequent is stable. See stabnot 840 for negation , bj-stan 16364 for conjunction , and bj-stal 16366 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓))

Theorem

bj-stan 16364

The conjunction of two stable formulas is stable. See bj-stim 16363 for implication, stabnot 840 for negation, and bj-stal 16366 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

⊢ ((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))

Theorem

bj-stand 16365

The conjunction of two stable formulas is stable. Deduction form of bj-stan 16364. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 16364 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

⊢ (𝜑 → STAB 𝜓)    &   ⊢ (𝜑 → STAB 𝜒)    ⇒   ⊢ (𝜑 → STAB (𝜓 ∧ 𝜒))

Theorem

bj-stal 16366

The universal quantification of a stable formula is stable. See bj-stim 16363 for implication, stabnot 840 for negation, and bj-stan 16364 for conjunction. (Contributed by BJ, 24-Nov-2023.)

⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)

Theorem

bj-pm2.18st 16367

Clavius law for stable formulas. See pm2.18dc 862. (Contributed by BJ, 4-Dec-2023.)

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜑) → 𝜑))

Theorem

bj-con1st 16368

Contraposition when the antecedent is a negated stable proposition. See con1dc 863. (Contributed by BJ, 11-Nov-2024.)

⊢ (STAB 𝜑 → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))

14.3.1.2  Decidable formulas

Theorem

bj-trdc 16369

A provable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

⊢ (𝜑 → DECID 𝜑)

Theorem

bj-dctru 16370

The true truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

⊢ DECID ⊤

Theorem

bj-fadc 16371

A refutable formula is decidable. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ 𝜑 → DECID 𝜑)

Theorem

bj-dcfal 16372

The false truth value is decidable. (Contributed by BJ, 5-Aug-2024.)

⊢ DECID ⊥

Theorem

bj-dcstab 16373

A decidable formula is stable. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

⊢ (DECID 𝜑 → STAB 𝜑)

Theorem

bj-nnbidc 16374

If a formula is not refutable, then it is decidable if and only if it is provable. See also comment of bj-nnbist 16361. (Contributed by BJ, 24-Nov-2023.)

⊢ (¬ ¬ 𝜑 → (DECID 𝜑 ↔ 𝜑))

Theorem

bj-nndcALT 16375

Alternate proof of nndc 858. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by BJ, 9-Oct-2019.)

⊢ ¬ ¬ DECID 𝜑

Theorem

bj-dcdc 16376

Decidability of a proposition is decidable if and only if that proposition is decidable. DECID is idempotent. (Contributed by BJ, 9-Oct-2019.)

⊢ (DECID DECID 𝜑 ↔ DECID 𝜑)

Theorem

bj-stdc 16377

Decidability of a proposition is stable if and only if that proposition is decidable. In particular, the assumption that every formula is stable implies that every formula is decidable, hence classical logic. (Contributed by BJ, 9-Oct-2019.)

⊢ (STAB DECID 𝜑 ↔ DECID 𝜑)

Theorem

bj-dcst 16378

Stability of a proposition is decidable if and only if that proposition is stable. (Contributed by BJ, 24-Nov-2023.)

⊢ (DECID STAB 𝜑 ↔ STAB 𝜑)

14.3.2  Predicate calculus

Theorem

bj-ex 16379*

Existential generalization. (Contributed by BJ, 8-Dec-2019.) Proof modification is discouraged because there are shorter proofs, but using less basic results (like exlimiv 1646 and 19.9ht 1689 or 19.23ht 1545). (Proof modification is discouraged.)

⊢ (∃𝑥𝜑 → 𝜑)

Theorem

bj-hbalt 16380

Closed form of hbal 1525 (copied from set.mm). (Contributed by BJ, 2-May-2019.)

⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑))

Theorem

bj-nfalt 16381

Closed form of nfal 1624 (copied from set.mm). (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)

⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥𝜑)

Theorem

spimd 16382

Deduction form of spim 1786. (Contributed by BJ, 17-Oct-2019.)

⊢ (𝜑 → Ⅎ𝑥𝜒)    &   ⊢ (𝜑 → ∀𝑥(𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Theorem

2spim 16383*

Double substitution, as in spim 1786. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜒    &   ⊢ Ⅎ𝑧𝜒    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑧∀𝑥𝜓 → 𝜒)

Theorem

ch2var 16384*

Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

ch2varv 16385*

Version of ch2var 16384 with nonfreeness hypotheses replaced with disjoint variable conditions. (Contributed by BJ, 17-Oct-2019.)

⊢ ((𝑥 = 𝑦 ∧ 𝑧 = 𝑡) → (𝜑 ↔ 𝜓))    &   ⊢ 𝜑    ⇒   ⊢ 𝜓

Theorem

bj-exlimmp 16386

Lemma for bj-vtoclgf 16393. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    ⇒   ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∃𝑥𝜒 → 𝜓))

Theorem

bj-exlimmpi 16387

Lemma for bj-vtoclgf 16393. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝜒 → 𝜑)    &   ⊢ (𝜒 → (𝜑 → 𝜓))    ⇒   ⊢ (∃𝑥𝜒 → 𝜓)

Theorem

bj-sbimedh 16388

A strengthening of sbiedh 1835 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))    &   ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 → 𝜒)))    ⇒   ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜒))

Theorem

bj-sbimeh 16389

A strengthening of sbieh 1838 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

Theorem

bj-sbime 16390

A strengthening of sbie 1839 (same proof). (Contributed by BJ, 16-Dec-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))    ⇒   ⊢ ([𝑦 / 𝑥]𝜑 → 𝜓)

14.3.3  Set theorey miscellaneous

Theorem

bj-el2oss1o 16391

Shorter proof of el2oss1o 6611 using more axioms. (Contributed by BJ, 21-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐴 ∈ 2_o → 𝐴 ⊆ 1_o)

14.3.4  Extensionality

Various utility theorems using FOL and extensionality.

Theorem

bj-vtoclgft 16392

Weakening two hypotheses of vtoclgf 2862. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ 𝑉 → 𝜓))

Theorem

bj-vtoclgf 16393

Weakening two hypotheses of vtoclgf 2862. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → 𝜑)    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝜓)

Theorem

elabgf0 16394

Lemma for elabgf 2948. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))

Theorem

elabgft1 16395

One implication of elabgf 2948, in closed form. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Theorem

elabgf1 16396

One implication of elabgf 2948. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabgf2 16397

One implication of elabgf 2948. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Theorem

elabf1 16398*

One implication of elabf 2949. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Theorem

elabf2 16399*

One implication of elabf 2949. (Contributed by BJ, 21-Nov-2019.)

⊢ Ⅎ𝑥𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))    ⇒   ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Theorem

elab1 16400*

One implication of elab 2950. (Contributed by BJ, 21-Nov-2019.)

⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))    ⇒   ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)