Type  Label  Description 
Statement 

Theorem  frgrwopreg1 28101* 
According to statement 5 in [Huneke] p. 2:
"If A ... is a singleton,
then that singleton is a universal friend". (Contributed by
Alexander
van der Vekens, 1Jan2018.) (Proof shortened by AV, 4Feb2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧
(♯‘𝐴) = 1)
→ ∃𝑣 ∈
𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) 

Theorem  frgrwopreg2 28102* 
According to statement 5 in [Huneke] p. 2:
"If ... B is a singleton,
then that singleton is a universal friend". (Contributed by
Alexander
van der Vekens, 1Jan2018.) (Proof shortened by AV, 4Feb2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧
(♯‘𝐵) = 1)
→ ∃𝑣 ∈
𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) 

Theorem  frgrwopreglem5lem 28103* 
Lemma for frgrwopreglem5 28104. (Contributed by AV, 5Feb2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦))) 

Theorem  frgrwopreglem5 28104* 
Lemma 5 for frgrwopreg 28106. If 𝐴 as well as 𝐵 contain at least
two vertices, there is a 4cycle in a friendship graph. This
corresponds to statement 6 in [Huneke]
p. 2: "... otherwise, there
are two different vertices in A, and they have two common neighbors in
B, ...". (Contributed by Alexander van der Vekens, 31Dec2017.)
(Proof shortened by AV, 5Feb2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 <
(♯‘𝐴) ∧ 1
< (♯‘𝐵))
→ ∃𝑎 ∈
𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))) 

Theorem  frgrwopreglem5ALT 28105* 
Alternate direct proof of frgrwopreglem5 28104, not using
frgrwopreglem5a 28094. This proof would be even a little bit
shorter
than the proof of frgrwopreglem5 28104 without using frgrwopreglem5lem 28103.
(Contributed by Alexander van der Vekens, 31Dec2017.) (Revised by
AV, 3Jan2022.) (Proof shortened by AV, 5Feb2022.)
(New usage is discouraged.) (Proof modification is discouraged.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 1 <
(♯‘𝐴) ∧ 1
< (♯‘𝐵))
→ ∃𝑎 ∈
𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸))) 

Theorem  frgrwopreg 28106* 
In a friendship graph there are either no vertices (𝐴 = ∅) or
exactly one vertex ((♯‘𝐴) = 1) having degree 𝐾, or all
(𝐵
= ∅) or all except one vertices ((♯‘𝐵) = 1) have
degree 𝐾. (Contributed by Alexander van der
Vekens, 31Dec2017.)
(Revised by AV, 10May2021.) (Proof shortened by AV, 3Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺)
& ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
& ⊢ 𝐵 = (𝑉 ∖ 𝐴) ⇒ ⊢ (𝐺 ∈ FriendGraph →
(((♯‘𝐴) = 1
∨ 𝐴 = ∅) ∨
((♯‘𝐵) = 1
∨ 𝐵 =
∅))) 

Theorem  frgrregorufr0 28107* 
In a friendship graph there are either no vertices having degree 𝐾,
or all vertices have degree 𝐾 for any (nonnegative integer) 𝐾,
unless there is a universal friend. This corresponds to claim 2 in
[Huneke] p. 2: "... all vertices
have degree k, unless there is a
universal friend." (Contributed by Alexander van der Vekens,
1Jan2018.) (Revised by AV, 11May2021.) (Proof shortened by AV,
3Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)) 

Theorem  frgrregorufr 28108* 
If there is a vertex having degree 𝐾 for each (nonnegative integer)
𝐾 in a friendship graph, then either
all vertices have degree 𝐾
or there is a universal friend. This corresponds to claim 2 in [Huneke]
p. 2: "Suppose there is a vertex of degree k > 1. ... all
vertices have
degree k, unless there is a universal friend. ... It follows that G is
kregular, i.e., the degree of every vertex is k". (Contributed by
Alexander van der Vekens, 1Jan2018.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺)
& ⊢ 𝐷 = (VtxDeg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) 

Theorem  frgrregorufrg 28109* 
If there is a vertex having degree 𝑘 for each nonnegative integer
𝑘 in a friendship graph, then there is
a universal friend. This
corresponds to claim 2 in [Huneke] p. 2:
"Suppose there is a vertex of
degree k > 1. ... all vertices have degree k, unless there is a
universal friend. ... It follows that G is kregular, i.e., the degree
of every vertex is k". Variant of frgrregorufr 28108 with generalization.
(Contributed by Alexander van der Vekens, 6Sep2018.) (Revised by AV,
26May2021.) (Proof shortened by AV, 12Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ_{0}
(∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))) 

Theorem  frgr2wwlkeu 28110* 
For two different vertices in a friendship graph, there is exactly one
third vertex being the middle vertex of a (simple) path/walk of length 2
between the two vertices. (Contributed by Alexander van der Vekens,
18Feb2018.) (Revised by AV, 12May2021.) (Proof shortened by AV,
4Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵)) 

Theorem  frgr2wwlkn0 28111 
In a friendship graph, there is always a path/walk of length 2 between
two different vertices. (Contributed by Alexander van der Vekens,
18Feb2018.) (Revised by AV, 12May2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅) 

Theorem  frgr2wwlk1 28112 
In a friendship graph, there is exactly one walk of length 2 between two
different vertices. (Contributed by Alexander van der Vekens,
19Feb2018.) (Revised by AV, 13May2021.) (Proof shortened by AV,
16Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1) 

Theorem  frgr2wsp1 28113 
In a friendship graph, there is exactly one simple path of length 2
between two different vertices. (Contributed by Alexander van der
Vekens, 3Mar2018.) (Revised by AV, 13May2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WSPathsNOn 𝐺)𝐵)) = 1) 

Theorem  frgr2wwlkeqm 28114 
If there is a (simple) path of length 2 from one vertex to another
vertex and a (simple) path of length 2 from the other vertex back to the
first vertex in a friendship graph, then the middle vertex is the same.
This is only an observation, which is not required to proof the
friendship theorem. (Contributed by Alexander van der Vekens,
20Feb2018.) (Revised by AV, 13May2021.) (Proof shortened by AV,
7Jan2022.)

⊢ ((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃)) 

Theorem  frgrhash2wsp 28115 
The number of simple paths of length 2 is n*(n1) in a friendship graph
with n vertices. This corresponds to the proof of claim 3 in [Huneke]
p. 2: "... the paths of length two in G: by assumption there are (
n
2 ) such paths.". However, Huneke counts undirected paths, so
obtains
the result ((𝑛C2) = ((𝑛 · (𝑛 − 1)) / 2)), whereas we
count directed paths, obtaining twice that number. (Contributed by
Alexander van der Vekens, 6Mar2018.) (Revised by AV, 10Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2
WSPathsN 𝐺)) =
((♯‘𝑉)
· ((♯‘𝑉) − 1))) 

Theorem  fusgreg2wsplem 28116* 
Lemma for fusgreg2wsp 28119 and related theorems. (Contributed by AV,
8Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))) 

Theorem  fusgr2wsp2nb 28117* 
The set of paths of length 2 with a given vertex in the middle for a
finite simple graph is the union of all paths of length 2 from one
neighbor to another neighbor of this vertex via this vertex.
(Contributed by Alexander van der Vekens, 9Mar2018.) (Revised by AV,
17May2021.) (Proof shortened by AV, 16Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪
𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩}) 

Theorem  fusgreghash2wspv 28118* 
According to statement 7 in [Huneke] p. 2:
"For each vertex v, there
are exactly ( k 2 ) paths with length two having v in the middle,
..."
in a finite kregular graph. For directed simple paths of length 2
represented by length 3 strings, we have again k*(k1) such paths, see
also comment of frgrhash2wsp 28115. (Contributed by Alexander van der
Vekens, 10Mar2018.) (Revised by AV, 17May2021.) (Proof shortened
by AV, 12Feb2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1)))) 

Theorem  fusgreg2wsp 28119* 
In a finite simple graph, the set of all paths of length 2 is the union
of all the paths of length 2 over the vertices which are in the middle
of such a path. (Contributed by Alexander van der Vekens, 10Mar2018.)
(Revised by AV, 18May2021.) (Proof shortened by AV, 10Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥)) 

Theorem  2wspmdisj 28120* 
The sets of paths of length 2 with a given vertex in the middle are
distinct for different vertices in the middle. (Contributed by
Alexander van der Vekens, 11Mar2018.) (Revised by AV, 18May2021.)
(Proof shortened by AV, 10Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ⊢ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥) 

Theorem  fusgreghash2wsp 28121* 
In a finite kregular graph with N vertices there are N times "k choose
2" paths with length 2, according to statement 8 in [Huneke] p. 2: "...
giving n * ( k 2 ) total paths of length two.", if the direction of
traversing the path is not respected. For simple paths of length 2
represented by length 3 strings, however, we have again n*k*(k1) such
paths. (Contributed by Alexander van der Vekens, 11Mar2018.)
(Revised by AV, 19May2021.) (Proof shortened by AV, 12Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1))))) 

Theorem  frrusgrord0lem 28122* 
Lemma for frrusgrord0 28123. (Contributed by AV, 12Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧
(♯‘𝑉) ≠
0)) 

Theorem  frrusgrord0 28123* 
If a nonempty finite friendship graph is kregular, its order is
k(k1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim
that the number n of vertices in G is exactly k(k1)+1.".
(Contributed
by Alexander van der Vekens, 11Mar2018.) (Revised by AV,
26May2021.) (Proof shortened by AV, 12Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) 

Theorem  frrusgrord 28124 
If a nonempty finite friendship graph is kregular, its order is
k(k1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim
that the number n of vertices in G is exactly k(k1)+1.". Variant
of
frrusgrord0 28123, using the definition RegUSGraph (dfrusgr 27346).
(Contributed by Alexander van der Vekens, 25Aug2018.) (Revised by AV,
26May2021.) (Proof shortened by AV, 12Jan2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1))) 

Theorem  numclwwlk2lem1lem 28125 
Lemma for numclwwlk2lem1 28159. (Contributed by Alexander van der Vekens,
3Oct2018.) (Revised by AV, 27May2021.) (Revised by AV,
15Mar2022.)

⊢ ((𝑋 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0))) 

Theorem  2clwwlklem 28126 
Lemma for clwwnonrepclwwnon 28128 and extwwlkfab 28135. (Contributed by
Alexander van der Vekens, 18Sep2018.) (Revised by AV, 10May2022.)
(Revised by AV, 30Oct2022.)

⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0)) 

Theorem  clwwnrepclwwn 28127 
If the initial vertex of a closed walk occurs another time in the walk,
the walk starts with a closed walk. Notice that 3 ≤
𝑁 is required,
because for 𝑁 = 2,
(𝑤
prefix (𝑁 − 2)) =
(𝑤 prefix 0) =
∅, but ∅ (and
anything else) is not a representation of an empty closed walk as word,
see clwwlkn0 27811. (Contributed by Alexander van der Vekens,
15Sep2018.)
(Revised by AV, 28May2021.) (Revised by AV, 30Oct2022.)

⊢ ((𝑁 ∈ (ℤ_{≥}‘3)
∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊 prefix (𝑁 − 2)) ∈ ((𝑁 − 2) ClWWalksN 𝐺)) 

Theorem  clwwnonrepclwwnon 28128 
If the initial vertex of a closed walk occurs another time in the walk,
the walk starts with a closed walk on this vertex. See also the remarks
in clwwnrepclwwn 28127. (Contributed by AV, 24Apr2022.)
(Revised by AV,
10May2022.) (Revised by AV, 30Oct2022.)

⊢ ((𝑁 ∈ (ℤ_{≥}‘3)
∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊 prefix (𝑁 − 2)) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) 

Theorem  2clwwlk2clwwlklem 28129 
Lemma for 2clwwlk2clwwlk 28133. (Contributed by AV, 27Apr2022.)

⊢ ((𝑁 ∈ (ℤ_{≥}‘3)
∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)2)) 

Theorem  2clwwlk 28130* 
Value of operation 𝐶, mapping a vertex v and an integer n
greater
than 1 to the "closed nwalks v(0) ... v(n2) v(n1) v(n) from v =
v(0)
= v(n) with v(n2) = v" according to definition 6 in [Huneke] p. 2.
Such closed walks are "double loops" consisting of a closed
(n2)walk v
= v(0) ... v(n2) = v and a closed 2walk v = v(n2) v(n1) v(n) = v,
see 2clwwlk2clwwlk 28133. (𝑋𝐶𝑁) is called the "set of double
loops
of length 𝑁 on vertex 𝑋 " in the following.
(Contributed by
Alexander van der Vekens, 14Sep2018.) (Revised by AV, 29May2021.)
(Revised by AV, 20Apr2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}) 

Theorem  2clwwlk2 28131* 
The set (𝑋𝐶2) of double loops of length 2 on a
vertex 𝑋 is
equal to the set of closed walks with length 2 on 𝑋. Considered as
"double loops", the first of the two closed walks/loops is
degenerated,
i.e., has length 0. (Contributed by AV, 18Feb2022.) (Revised by AV,
20Apr2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶2) = (𝑋(ClWWalksNOn‘𝐺)2)) 

Theorem  2clwwlkel 28132* 
Characterization of an element of the value of operation 𝐶, i.e.,
of a word being a double loop of length 𝑁 on vertex 𝑋.
(Contributed by Alexander van der Vekens, 24Sep2018.) (Revised by AV,
29May2021.) (Revised by AV, 20Apr2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))) 

Theorem  2clwwlk2clwwlk 28133* 
An element of the value of operation 𝐶, i.e., a word being a double
loop of length 𝑁 on vertex 𝑋, is composed of two
closed walks.
(Contributed by AV, 28Apr2022.) (Proof shortened by AV,
3Nov2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑊 ∈ (𝑋𝐶𝑁) ↔ ∃𝑎 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))∃𝑏 ∈ (𝑋(ClWWalksNOn‘𝐺)2)𝑊 = (𝑎 ++ 𝑏))) 

Theorem  numclwwlk1lem2foalem 28134 
Lemma for numclwwlk1lem2foa 28137. (Contributed by AV, 29May2021.)
(Revised by AV, 1Nov2022.)

⊢ (((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ((((𝑊 ++
⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 2)) = 𝑋)) 

Theorem  extwwlkfab 28135* 
The set (𝑋𝐶𝑁) of double loops of length 𝑁 on
vertex 𝑋
can be constructed from the set 𝐹 of closed walks on 𝑋 with
length smaller by 2 than the fixed length by appending a neighbor of the
last vertex and afterwards the last vertex (which is the first vertex)
itself ("walking forth and back" from the last vertex). 3 ≤ 𝑁
is
required since for 𝑁 = 2: 𝐹 = (𝑋(ClWWalksNOn‘𝐺)0)
= ∅ (see clwwlk0on0 27875 stating that a closed walk of length 0 is
not represented as word), which would result in an empty set on the
right hand side, but (𝑋𝐶𝑁) needs not be empty, see 2clwwlk2 28131.
(Contributed by Alexander van der Vekens, 18Sep2018.) (Revised by AV,
29May2021.) (Revised by AV, 31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑤‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)}) 

Theorem  extwwlkfabel 28136* 
Characterization of an element of the set (𝑋𝐶𝑁), i.e., a double
loop of length 𝑁 on vertex 𝑋 with a construction from
the set
𝐹 of closed walks on 𝑋 with
length smaller by 2 than the fixed
length by appending a neighbor of the last vertex and afterwards the
last vertex (which is the first vertex) itself ("walking forth and
back"
from the last vertex). (Contributed by AV, 22Feb2022.) (Revised by
AV, 31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ ((𝑊 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑊‘(𝑁 − 2)) = 𝑋)))) 

Theorem  numclwwlk1lem2foa 28137* 
Going forth and back from the end of a (closed) walk: 𝑊 represents
the closed walk p_{0}, ..., p_{(n}2), p_{0} =
p_{(n}2). With 𝑋 = p_{(n}2)
= p_{0} and 𝑌 = p_{(n}1), ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)
represents the closed walk p_{0}, ...,
p_{(n}2), p_{(n}1), p_{n} =
p_{0} which
is a double loop of length 𝑁 on vertex 𝑋. (Contributed by
Alexander van der Vekens, 22Sep2018.) (Revised by AV, 29May2021.)
(Revised by AV, 5Mar2022.) (Proof shortened by AV, 2Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ ((𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋𝐶𝑁))) 

Theorem  numclwwlk1lem2f 28138* 
𝑇
is a function, mapping a double loop of length 𝑁 on vertex
𝑋 to the ordered pair of the first
loop and the successor of 𝑋
in the second loop, which must be a neighbor of 𝑋. (Contributed
by Alexander van der Vekens, 19Sep2018.) (Revised by AV,
29May2021.) (Proof shortened by AV, 23Feb2022.) (Revised by AV,
31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)⟶(𝐹 × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1lem2fv 28139* 
Value of the function 𝑇. (Contributed by Alexander van der
Vekens, 20Sep2018.) (Revised by AV, 29May2021.) (Revised by AV,
31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 −
1))⟩) ⇒ ⊢ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩) 

Theorem  numclwwlk1lem2f1 28140* 
𝑇
is a 11 function. (Contributed by AV, 26Sep2018.) (Revised
by AV, 29May2021.) (Proof shortened by AV, 23Feb2022.) (Revised
by AV, 31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–11→(𝐹 × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1lem2fo 28141* 
𝑇
is an onto function. (Contributed by Alexander van der Vekens,
20Sep2018.) (Revised by AV, 29May2021.) (Proof shortened by AV,
13Feb2022.) (Revised by AV, 31Oct2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–onto→(𝐹 × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1lem2f1o 28142* 
𝑇
is a 11 onto function. (Contributed by Alexander van der
Vekens, 26Sep2018.) (Revised by AV, 29May2021.) (Revised by AV,
6Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & ⊢ 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 −
1))⟩) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ 𝑇:(𝑋𝐶𝑁)–11onto→(𝐹 × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1lem2 28143* 
The set of double loops of length 𝑁 on vertex 𝑋 and the set of
closed walks of length less by 2 on 𝑋 combined with the neighbors of
𝑋 are equinumerous. (Contributed by
Alexander van der Vekens,
6Jul2018.) (Revised by AV, 29May2021.) (Revised by AV,
31Jul2022.) (Proof shortened by AV, 3Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3))
→ (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋))) 

Theorem  numclwwlk1 28144* 
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) with v(n2) = v
is kf(n2)". Since 𝐺 is kregular, the vertex v(n2) = v
has k
neighbors v(n1), so there are k walks from v(n2) = v to v(n) = v (via
each of v's neighbors) completing each of the f(n2) walks from v=v(0)
to v(n2)=v. This theorem holds even for k=0, but not for n=2, since
𝐹 =
∅, but (𝑋𝐶2), the set of closed walks with
length 2
on 𝑋, see 2clwwlk2 28131, needs not be ∅ in this case. This is
because of the special definition of 𝐹 and the usage of words to
represent (closed) walks, and does not contradict Huneke's statement,
which would read "the number of closed 2walks v(0) v(1) v(2) from
v =
v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty
closed
walks on v, see numclwlk1lem1 28152. If the general representation of
(closed) walk is used, Huneke's statement can be proven even for n = 2,
see numclwlk1 28154. This case, however, is not required to
prove the
friendship theorem. (Contributed by Alexander van der Vekens,
26Sep2018.) (Revised by AV, 29May2021.) (Revised by AV,
6Mar2022.) (Proof shortened by AV, 31Jul2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹))) 

Theorem  clwwlknonclwlknonf1o 28145* 
𝐹
is a bijection between the two representations of closed walks of
a fixed positive length on a fixed vertex. (Contributed by AV,
26May2022.) (Proof shortened by AV, 7Aug2022.) (Revised by AV,
1Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋)} & ⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^{nd} ‘𝑐) prefix
(♯‘(1^{st} ‘𝑐)))) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–11onto→(𝑋(ClWWalksNOn‘𝐺)𝑁)) 

Theorem  clwwlknonclwlknonen 28146* 
The sets of the two representations of closed walks of a fixed positive
length on a fixed vertex are equinumerous. (Contributed by AV,
27May2022.) (Proof shortened by AV, 3Nov2022.)

⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) 

Theorem  dlwwlknondlwlknonf1olem1 28147 
Lemma 1 for dlwwlknondlwlknonf1o 28148. (Contributed by AV, 29May2022.)
(Revised by AV, 1Nov2022.)

⊢ (((♯‘(1^{st}
‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (((2^{nd} ‘𝑐) prefix (♯‘(1^{st}
‘𝑐)))‘(𝑁 − 2)) = ((2^{nd}
‘𝑐)‘(𝑁 − 2))) 

Theorem  dlwwlknondlwlknonf1o 28148* 
𝐹
is a bijection between the two representations of double loops
of a fixed positive length on a fixed vertex. (Contributed by AV,
30May2022.) (Revised by AV, 1Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋 ∧ ((2^{nd}
‘𝑤)‘(𝑁 − 2)) = 𝑋)} & ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}
& ⊢ 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^{nd} ‘𝑐) prefix
(♯‘(1^{st} ‘𝑐)))) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ 𝐹:𝑊–11onto→𝐷) 

Theorem  dlwwlknondlwlknonen 28149* 
The sets of the two representations of double loops of a fixed length on
a fixed vertex are equinumerous. (Contributed by AV, 30May2022.)
(Proof shortened by AV, 3Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋 ∧ ((2^{nd}
‘𝑤)‘(𝑁 − 2)) = 𝑋)} & ⊢ 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ 𝑊 ≈ 𝐷) 

Theorem  wlkl0 28150* 
There is exactly one walk of length 0 on each vertex 𝑋.
(Contributed by AV, 4Jun2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 0 ∧
((2^{nd} ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩}) 

Theorem  clwlknon2num 28151* 
There are k walks of length 2 on each vertex 𝑋 in a kregular simple
graph. Variant of clwwlknon2num 27888, using the general definition of
walks instead of walks as words. (Contributed by AV, 4Jun2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 2 ∧
((2^{nd} ‘𝑤)‘0) = 𝑋)}) = 𝐾) 

Theorem  numclwlk1lem1 28152* 
Lemma 1 for numclwlk1 28154 (Statement 9 in [Huneke] p. 2 for n=2): "the
number of closed 2walks v(0) v(1) v(2) from v = v(0) = v(2) ... is
kf(0)". (Contributed by AV, 23May2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋 ∧ ((2^{nd}
‘𝑤)‘(𝑁 − 2)) = 𝑋)} & ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = (𝑁 − 2) ∧
((2^{nd} ‘𝑤)‘0) = 𝑋)} ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹))) 

Theorem  numclwlk1lem2 28153* 
Lemma 2 for numclwlk1 28154 (Statement 9 in [Huneke] p. 2 for n>2). This
theorem corresponds to numclwwlk1 28144, using the general definition of
walks instead of walks as words. (Contributed by AV, 4Jun2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋 ∧ ((2^{nd}
‘𝑤)‘(𝑁 − 2)) = 𝑋)} & ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = (𝑁 − 2) ∧
((2^{nd} ‘𝑤)‘0) = 𝑋)} ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (♯‘𝐶) =
(𝐾 ·
(♯‘𝐹))) 

Theorem  numclwlk1 28154* 
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) with v(n2) = v
is kf(n2)". Since 𝐺 is kregular, the vertex v(n2) = v
has k
neighbors v(n1), so there are k walks from v(n2) = v to v(n) = v (via
each of v's neighbors) completing each of the f(n2) walks from v=v(0)
to v(n2)=v. This theorem holds even for k=0. (Contributed by AV,
23May2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = 𝑁 ∧ ((2^{nd}
‘𝑤)‘0) = 𝑋 ∧ ((2^{nd}
‘𝑤)‘(𝑁 − 2)) = 𝑋)} & ⊢ 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^{st}
‘𝑤)) = (𝑁 − 2) ∧
((2^{nd} ‘𝑤)‘0) = 𝑋)} ⇒ ⊢ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2)))
→ (♯‘𝐶) =
(𝐾 ·
(♯‘𝐹))) 

Theorem  numclwwlkovh0 28155* 
Value of operation 𝐻, mapping a vertex 𝑣 and an
integer 𝑛
greater than 1 to the "closed nwalks v(0) ... v(n2) v(n1) v(n)
from v
= v(0) = v(n) ... with v(n2) =/= v" according to definition 7 in
[Huneke] p. 2. (Contributed by AV,
1May2022.)

⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋}) 

Theorem  numclwwlkovh 28156* 
Value of operation 𝐻, mapping a vertex 𝑣 and an
integer 𝑛
greater than 1 to the "closed nwalks v(0) ... v(n2) v(n1) v(n)
from v
= v(0) = v(n) ... with v(n2) =/= v" according to definition 7 in
[Huneke] p. 2. Definition of ClWWalksNOn resolved. (Contributed by
Alexander van der Vekens, 26Aug2018.) (Revised by AV, 30May2021.)
(Revised by AV, 1May2022.)

⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))}) 

Theorem  numclwwlkovq 28157* 
Value of operation 𝑄, mapping a vertex 𝑣 and a
positive integer
𝑛 to the not closed walks v(0) ... v(n)
of length 𝑛 from a fixed
vertex 𝑣 = v(0). "Not closed" means
v(n) =/= v(0). Remark:
𝑛
∈ ℕ_{0} would not be useful: numclwwlkqhash 28158 would not hold,
because (𝐾↑0) = 1! (Contributed by
Alexander van der Vekens,
27Sep2018.) (Revised by AV, 30May2021.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}) 

Theorem  numclwwlkqhash 28158* 
In a 𝐾regular graph, the size of the set
of walks of length 𝑁
starting with a fixed vertex 𝑋 and ending not at this vertex is the
difference between 𝐾 to the power of 𝑁 and the
size of the set
of closed walks of length 𝑁 on vertex 𝑋. (Contributed by
Alexander van der Vekens, 30Sep2018.) (Revised by AV, 30May2021.)
(Revised by AV, 5Mar2022.) (Proof shortened by AV, 7Jul2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
(♯‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)))) 

Theorem  numclwwlk2lem1 28159* 
In a friendship graph, for each walk of length 𝑛 starting at a fixed
vertex 𝑣 and ending not at this vertex, there
is a unique vertex so
that the walk extended by an edge to this vertex and an edge from this
vertex to the first vertex of the walk is a value of operation 𝐻.
If the walk is represented as a word, it is sufficient to add one vertex
to the word to obtain the closed walk contained in the value of
operation 𝐻, since in a word representing a
closed walk the
starting vertex is not repeated at the end. This theorem generally
holds only for friendship graphs, because these guarantee that for the
first and last vertex there is a (unique) third vertex "in
between".
(Contributed by Alexander van der Vekens, 3Oct2018.) (Revised by AV,
30May2021.) (Revised by AV, 1May2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwlk2lem2f 28160* 
𝑅
is a function mapping the "closed (n+2)walks v(0) ... v(n2)
v(n1) v(n) v(n+1) v(n+2) starting at 𝑋 = v(0) = v(n+2) with
v(n)
=/= X" to the words representing the prefix v(0) ... v(n2)
v(n1)
v(n) of the walk. (Contributed by Alexander van der Vekens,
5Oct2018.) (Revised by AV, 31May2021.) (Proof shortened by AV,
23Mar2022.) (Revised by AV, 1Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
& ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1))) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁)) 

Theorem  numclwlk2lem2fv 28161* 
Value of the function 𝑅. (Contributed by Alexander van der
Vekens, 6Oct2018.) (Revised by AV, 31May2021.) (Revised by AV,
1Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
& ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1))) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1)))) 

Theorem  numclwlk2lem2f1o 28162* 
𝑅
is a 11 onto function. (Contributed by Alexander van der
Vekens, 6Oct2018.) (Revised by AV, 21Jan2022.) (Proof shortened
by AV, 17Mar2022.) (Revised by AV, 1Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})
& ⊢ 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1))) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–11onto→(𝑋𝑄𝑁)) 

Theorem  numclwwlk2lem3 28163* 
In a friendship graph, the size of the set of walks of length 𝑁
starting with a fixed vertex 𝑋 and ending not at this vertex equals
the size of the set of all closed walks of length (𝑁 + 2)
starting
at this vertex 𝑋 and not having this vertex as last
but 2 vertex.
(Contributed by Alexander van der Vekens, 6Oct2018.) (Revised by AV,
31May2021.) (Proof shortened by AV, 3Nov2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) →
(♯‘(𝑋𝑄𝑁)) = (♯‘(𝑋𝐻(𝑁 + 2)))) 

Theorem  numclwwlk2 28164* 
Statement 10 in [Huneke] p. 2: "If n >
1, then the number of closed
nwalks v(0) ... v(n2) v(n1) v(n) from v = v(0) = v(n) ... with v(n2)
=/= v is k^(n2)  f(n2)." According to rusgrnumwlkg 27761, we have
k^(n2) different walks of length (n2): v(0) ... v(n2). From this
number, the number of closed walks of length (n2), which is f(n2) per
definition, must be subtracted, because for these walks v(n2) =/= v(0)
= v would hold. Because of the friendship condition, there is exactly
one vertex v(n1) which is a neighbor of v(n2) as well as of
v(n)=v=v(0), because v(n2) and v(n)=v are different, so the number of
walks v(0) ... v(n2) is identical with the number of walks v(0) ...
v(n), that means each (not closed) walk v(0) ... v(n2) can be extended
by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way.
(Contributed by Alexander van der Vekens, 6Oct2018.) (Revised by AV,
31May2021.) (Revised by AV, 1May2022.)

⊢ 𝑉 = (Vtx‘𝐺)
& ⊢ 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))) 

Theorem  numclwwlk3lem1 28165 
Lemma 2 for numclwwlk3 28168. (Contributed by Alexander van der Vekens,
26Aug2018.) (Proof shortened by AV, 23Jan2022.)

⊢ ((𝐾 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (((𝐾↑(𝑁 − 2)) − 𝑌) + (𝐾 · 𝑌)) = (((𝐾 − 1) · 𝑌) + (𝐾↑(𝑁 − 2)))) 

Theorem  numclwwlk3lem2lem 28166* 
Lemma for numclwwlk3lem2 28167: The set of closed vertices of a fixed
length 𝑁 on a fixed vertex 𝑉 is the
union of the set of closed
walks of length 𝑁 at 𝑉 with the last but one
vertex being 𝑉
and the set of closed walks of length 𝑁 at 𝑉 with the last but
one vertex not being 𝑉. (Contributed by AV, 1May2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁))) 

Theorem  numclwwlk3lem2 28167* 
Lemma 1 for numclwwlk3 28168: The number of closed vertices of a fixed
length 𝑁 on a fixed vertex 𝑉 is the
sum of the number of closed
walks of length 𝑁 at 𝑉 with the last but one
vertex being 𝑉
and the set of closed walks of length 𝑁 at 𝑉 with the last but
one vertex not being 𝑉. (Contributed by Alexander van der
Vekens,
6Oct2018.) (Revised by AV, 1Jun2021.) (Revised by AV,
1May2022.)

⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})
& ⊢ 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_{≥}‘2)
↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ⊢ (((𝐺 ∈ FinUSGraph ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_{≥}‘2))
→ (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = ((♯‘(𝑋𝐻𝑁)) + (♯‘(𝑋𝐶𝑁)))) 

Theorem  numclwwlk3 28168 
Statement 12 in [Huneke] p. 2: "Thus f(n)
= (k  1)f(n  2) + k^(n2)."
 the number of the closed walks v(0) ... v(n2) v(n1) v(n) is the sum
of the number of the closed walks v(0) ... v(n2) v(n1) v(n) with
v(n2) = v(n) (see numclwwlk1 28144) and with v(n2) =/= v(n) (see
numclwwlk2 28164): f(n) = kf(n2) + k^(n2)  f(n2) =
(k1)f(n2) +
k^(n2). (Contributed by Alexander van der Vekens, 26Aug2018.)
(Revised by AV, 6Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2)))) 

Theorem  numclwwlk4 28169* 
The total number of closed walks in a finite simple graph is the sum of
the numbers of closed walks starting at each of its vertices.
(Contributed by Alexander van der Vekens, 7Oct2018.) (Revised by AV,
2Jun2021.) (Revised by AV, 7Mar2022.) (Proof shortened by AV,
28Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ) →
(♯‘(𝑁
ClWWalksN 𝐺)) =
Σ𝑥 ∈ 𝑉 (♯‘(𝑥(ClWWalksNOn‘𝐺)𝑁))) 

Theorem  numclwwlk5lem 28170 
Lemma for numclwwlk5 28171. (Contributed by Alexander van der Vekens,
7Oct2018.) (Revised by AV, 2Jun2021.) (Revised by AV,
7Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ_{0}) → (2
∥ (𝐾 − 1)
→ ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1)) 

Theorem  numclwwlk5 28171 
Statement 13 in [Huneke] p. 2: "Let p be
a prime divisor of k1; then
f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander
van
der Vekens, 7Oct2018.) (Revised by AV, 2Jun2021.) (Revised by AV,
7Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)𝑃)) mod 𝑃) = 1) 

Theorem  numclwwlk7lem 28172 
Lemma for numclwwlk7 28174, frgrreggt1 28176 and frgrreg 28177: If a finite,
nonempty friendship graph is 𝐾regular, the 𝐾 is a nonnegative
integer. (Contributed by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈
ℕ_{0}) 

Theorem  numclwwlk6 28173 
For a prime divisor 𝑃 of 𝐾 − 1, the total
number of closed
walks of length 𝑃 in a 𝐾regular friendship graph
is equal
modulo 𝑃 to the number of vertices.
(Contributed by Alexander van
der Vekens, 7Oct2018.) (Revised by AV, 3Jun2021.) (Proof shortened
by AV, 7Mar2022.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((♯‘𝑉) mod 𝑃)) 

Theorem  numclwwlk7 28174 
Statement 14 in [Huneke] p. 2: "The total
number of closed walks of
length p [in a friendship graph] is (k(k1)+1)f(p)=1 (mod p)",
since the
number of vertices in a friendship graph is (k(k1)+1), see
frrusgrord0 28123 or frrusgrord 28124, and p divides (k1), i.e. (k1) mod p =
0 => k(k1) mod p = 0 => k(k1)+1 mod p = 1. Since the null graph
is a
friendship graph, see frgr0 28048, as well as kregular (for any k), see
0vtxrgr 27364, but has no closed walk, see 0clwlk0 27915, this theorem would
be false for a null graph: ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0
≠ 1, so this case must be excluded (by
assuming 𝑉
≠ ∅).
(Contributed by Alexander van der Vekens, 1Sep2018.) (Revised by AV,
3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1) 

Theorem  numclwwlk8 28175 
The size of the set of closed walks of length 𝑃, 𝑃 prime, is
divisible by 𝑃. This corresponds to statement 9 in
[Huneke] p. 2:
"It follows that, if p is a prime number, then the number of closed
walks
of length p is divisible by p", see also clwlksndivn 27869. (Contributed by
Alexander van der Vekens, 7Oct2018.) (Revised by AV, 3Jun2021.)
(Proof shortened by AV, 2Mar2022.)

⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) →
((♯‘(𝑃
ClWWalksN 𝐺)) mod 𝑃) = 0) 

Theorem  frgrreggt1 28176 
If a finite nonempty friendship graph is 𝐾regular with 𝐾 > 1,
then 𝐾 must be 2.
(Contributed by Alexander van der Vekens,
7Oct2018.) (Revised by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2)) 

Theorem  frgrreg 28177 
If a finite nonempty friendship graph is 𝐾regular, then 𝐾 must
be 2 (or 0).
(Contributed by Alexander van der Vekens,
9Oct2018.) (Revised by AV, 3Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))) 

Theorem  frgrregord013 28178 
If a finite friendship graph is 𝐾regular, then it must have order
0, 1 or 3. (Contributed by Alexander van der Vekens, 9Oct2018.)
(Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) 

Theorem  frgrregord13 28179 
If a nonempty finite friendship graph is 𝐾regular, then it must
have order 1 or 3. Special case of frgrregord013 28178. (Contributed by
Alexander van der Vekens, 9Oct2018.) (Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) 

Theorem  frgrogt3nreg 28180* 
If a finite friendship graph has an order greater than 3, it cannot be
𝑘regular for any 𝑘.
(Contributed by Alexander van der Vekens,
9Oct2018.) (Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 <
(♯‘𝑉)) →
∀𝑘 ∈
ℕ_{0} ¬ 𝐺 RegUSGraph 𝑘) 

Theorem  friendshipgt3 28181* 
The friendship theorem for big graphs: In every finite friendship graph
with order greater than 3 there is a vertex which is adjacent to all
other vertices. (Contributed by Alexander van der Vekens, 9Oct2018.)
(Revised by AV, 4Jun2021.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 <
(♯‘𝑉)) →
∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) 

Theorem  friendship 28182* 
The friendship theorem: In every finite (nonempty) friendship graph
there is a vertex which is adjacent to all other vertices. This is
Metamath 100 proof #83. (Contributed by Alexander van der Vekens,
9Oct2018.)

⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) 

PART 17 GUIDES AND
MISCELLANEA


17.1 Guides (conventions, explanations, and
examples)


17.1.1 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.
They are organized as follows:
Logic and set theory provide a foundation for all of mathematics. To learn
about them, you should study one or more of the references listed below. We
indicate references using square brackets. 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  [Margaris].
 Axioms of predicate calculus  [Megill] (System S3' in the article
referenced).
 Theorems of propositional calculus  [WhiteheadRussell].
 Theorems of pure predicate calculus  [Margaris].
 Theorems of equality and substitution  [Monk2], [Tarski], [Megill].
 Axioms of set theory  [BellMachover].
 Development of set theory  [TakeutiZaring]. (The first part of [Quine]
has a good explanation of the powerful device of "virtual" or
class abstractions, which is essential to our development.)
 Construction of real and complex numbers  [Gleason].
 Theorems about real numbers  [Apostol].


Theorem  conventions 28183 
Here are some of the conventions we use in the Metamath Proof
Explorer (MPE, set.mm), and how they correspond to typical textbook
language (skipping the many cases where they are identical).
For more specific conventions, see:
Notation.
Where possible, the notation attempts to conform to modern
conventions, with variations due to our choice of the axiom system
or to make proofs shorter. However, our notation is strictly
sequential (lefttoright). For example, summation is written in the
form Σ𝑘 ∈ 𝐴𝐵 (dfsum 15034) which denotes that index
variable 𝑘 ranges over 𝐴 when evaluating 𝐵. Thus,
Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 15229).
The notation is usually explained in more detail when first introduced.
Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " ⊢ " have labels starting
with "ax" (axioms) or "df" (definitions). A statement with a
label starting with "ax" corresponds to what is traditionally
called an axiom. A statement with a label starting with "df"
introduces new symbols or a new relationship among symbols
that can be eliminated; they always extend the definition of
a wff or class. Metamath blindly treats $a statements as new
given facts but does not try to justify them. The mmj2 program
will justify the definitions as sound as discussed below,
except for 4 definitions (dfbi 210, dfcleq 2815, dfclel 2894, dfclab 2801)
that require a more complex metalogical justification by hand.
Proven axioms.
In some cases we wish to treat an expression as an axiom in
later theorems, even though it can be proved. For example,
we derive the postulates or axioms of complex arithmetic as
theorems of ZFC set theory. For convenience, after deriving
the postulates, we reintroduce them as new axioms on
top of set theory. This lets us easily identify which axioms
are needed for a particular complex number proof, without the
obfuscation of the set theory used to derive them. For more, see
mmcomplex.html 2801. When we wish
to use a previouslyproven assertion as an axiom, our convention
is that we use the
regular "axNAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is the complex arithmetic axiom ax1cn 10584,
proven by the preceding theorem ax1cn 10560.
The Metamath program will warn if an axiom does not match the preceding
theorem that justifies it if the names match in this way.
Definitions (df...).
We encourage definitions to include hypertext links to proven examples.
Statements with hypotheses.
Many theorems and some axioms, such as axmp 5, have hypotheses that
must be satisfied in order for the conclusion to hold, in this case min
and maj. When displayed in summarized form such as in the "Theorem
List" page (to get to it, click on "Nearby theorems" on the axmp 5
page), the hypotheses are connected with an ampersand and separated from
the conclusion with a double right arrow, such as in
" ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 𝜓". These symbols are not part of
the Metamath language but are just informal notation meaning "and" and
"implies".
Discouraged use and modification.
If something should only be used in limited ways, it is marked with
"(New usage is discouraged.)". This is used, for example, when something
can be constructed in more than one way, and we do not want later
theorems to depend on that specific construction.
This marking is also used if we want later proofs to use proven axioms.
For example, we want later proofs to
use ax1cn 10584 (not ax1cn 10560) and ax1ne0 10595 (not ax1ne0 10571), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 10560 and ax1ne0 10571 are marked as "(New usage is discouraged.)".
In some cases a proof should not normally be changed, e.g., when it
demonstrates some specific technique.
These are marked with "(Proof modification is discouraged.)".
New definitions infrequent.
Typically, we are minimalist when introducing new definitions; they are
introduced only when a clear advantage becomes apparent for reducing
the number of symbols, shortening proofs, etc. We generally avoid
the introduction of gratuitous definitions because each one requires
associated theorems and additional elimination steps in proofs.
For example, we use < and ≤ for inequality expressions, and
use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴)
for the hyperbolic sine.
Minimizing axiom dependencies.
We prefer proofs that depend on fewer and/or weaker axioms, even if
the proofs are longer. In particular, because of the nonconstructive
nature of the axiom of choice dfac 9531, we prefer proofs that do not use
it, or use weaker versions like countable choice axcc 9846 or dependent
choice axdc 9857. An example is our proof of the SchroederBernstein
Theorem sbth 8625, which does not use the axiom of choice. Similarly,
any theorem in firstorder logic (FOL) that contains only setvar
variables that are all mutually distinct, and has no wff variables, can
be proved without using ax10 2145 through ax13 2391, by using ax10w 2133
through ax13w 2140 instead.
We do not try to similarly reduce dependencies on definitions, since
definitions are conservative (they do not increase the proving power of
a deductive system), and are introduced in order to be used to increase
readability). An exception is made for the definitions dfclab 2801,
dfcleq 2815, dfclel 2894, since they can be considered as axioms under
some definitions of what a definition is exactly (see their comments).
Alternate proofs (ALT).
If a different proof is shorter or clearer but uses more or stronger
axioms, we make that proof an "alternate" proof (marked with an ALT
label suffix), even if this alternate proof was formalized first.
We then make the proof that requires fewer axioms the main proof.
Alternate proofs can also occur in other cases when an alternate proof
gives some particular insight. Their comment should begin with
"Alternate proof of ~ xxx " followed by a description of the
specificity of that alternate proof. There can be multiple alternates.
Alternate (*ALT) theorems should have "(Proof modification is
discouraged.) (New usage is discouraged.)" in their comment and should
follow the main statement, so that people reading the text in order will
see the main statement first. The alternate and main statement comments
should use hyperlinks to refer to each other.
Alternate versions (ALTV).
The suffix ALTV is reserved for theorems (or definitions) which are
alternate versions, or variants, of an existing theorem. This is
reserved to statements in mathboxes and is typically used temporarily,
when it is not clear yet which variant to use. If it is decided that
both variants should be kept and moved to the main part of set.mm, then
a label for the variant should be found with a more explicit suffix
indicating how it is a variant (e.g., commutation of some subformula,
antecedent replaced with hypothesis, (un)curried variant, biconditional
instead of implication, etc.). There is no requirement to add
discouragement tags, but their comment should have a link to the main
version of the statement and describe how it is a variant of it.
Old (OLD) versions or proofs.
If a proof, definition, axiom, or theorem is going to be removed, we
often stage that change by first renaming its label with an OLD suffix
(to make it clear that it is going to be removed). Old (*OLD)
statements should have
"(Proof modification is discouraged.) (New usage is discouraged.)" and
"Obsolete version of ~ xxx as of ddMmmyyyy." (not enclosed in
parentheses) in the comment. An old statement should follow the main
statement, so that people reading the text in order will see the main
statement first. This typically happens when a shorter proof to an
existing theorem is found: the existing theorem is kept as an *OLD
statement for one year. When a proof is shortened automatically (using
the Metamath program "MMPA> MINIMIZE__WITH *" command), then it is not
necessary to keep the old proof, nor to add credit for the shortening.
Variables.
Propositional variables (variables for wellformed formulas or wffs) are
represented with lowercase Greek letters and are generally used
in this order:
𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta,
𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma.
Individual setvar variables are represented with lowercase Latin letters
and are generally used in this order:
𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡.
Variables that represent classes are often represented by
uppercase Latin letters:
𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on.
There are other symbols that also represent class variables and suggest
specific purposes, e.g., 0 for a zero element (e.g., fsuppcor 8855)
and connective symbols such as + for some group addition operation
(e.g., grprinvd ).
Class variables are selected in alphabetical order starting
from 𝐴 if there is no reason to do otherwise, but many
assertions select different class variables or a different order
to make their intended meaning clearer.
Turnstile.
"⊢ ", meaning "It is provable that", is the first token
of all assertions
and hypotheses that aren't syntax constructions. This is a standard
convention in logic. For us, it also prevents any ambiguity with
statements that are syntax constructions, such as "wff ¬ 𝜑".
Biconditional (↔).
There are basically two ways to maximize the effectiveness of
biconditionals (↔):
you can either have onedirectional simplifications of all theorems
that produce biconditionals, or you can have onedirectional
simplifications of theorems that consume biconditionals.
Some tools (like Lean) follow the first approach, but set.mm follows
the second approach. Practically, this means that in set.mm, for
every theorem that uses an implication in the hypothesis, like
axmp 5, there is a corresponding version with a biconditional or a
reversed biconditional, like mpbi 233 or mpbir 234. We prefer this
second approach because the number of duplications in the second
approach is bounded by the size of the propositional calculus section,
which is much smaller than the number of possible theorems in all later
sections that produce biconditionals. So although theorems like
biimpi 219 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 710, sylbir 238,
or 3imtr4i 295.
Quantifiers.
The quantifiers are named as follows:
 ∀: universal quantifier (wal 1536);
 ∃: existential quantifier (dfex 1782);
 ∃*: atmostone quantifier (dfmo 2622);
 ∃!: unique existential quantifier (dfeu 2653).
The phrase "uniqueness quantifier" is avoided since it is ambiguous:
it can be understood as claiming either uniqueness (∃*) or unique
existence (∃!).
Substitution.
The expression "[𝑦 / 𝑥]𝜑" should be read "the formula that
results from the proper substitution of 𝑦 for 𝑥 in the formula
𝜑". See dfsb 2070 and the related dfsbc 3748 and dfcsb 3856.
Isaset.
"𝐴 ∈ V" should be read "Class 𝐴 is a set (i.e. exists)."
This is a convention based on Definition 2.9 of [Quine] p. 19.
See dfv 3471 and isset 3481.
However, instead of using 𝐼 ∈ V in the antecedent of a theorem for
some variable 𝐼, we now prefer to use 𝐼 ∈ 𝑉 (or another
variable if 𝑉 is not available) to make it more general. That way we
can often avoid needing extra uses of elex 3487 and syl 17 in the common
case where 𝐼 is already a member of something.
For hypotheses ($e statement) of theorems (mostly in inference form),
however, ⊢ 𝐴 ∈ V is used rather than ⊢ 𝐴 ∈ 𝑉 (e.g.,
difexi 5208). This is because 𝐴 ∈ V is almost always satisfied using
an existence theorem stating "... ∈ V", and a hardcoded V in
the $e statement saves a couple of syntax building steps that substitute
V into 𝑉. Notice that this does not hold for hypotheses of
theorems in deduction form: Here still ⊢ (𝜑 → 𝐴 ∈ 𝑉) should be
used rather than ⊢ (𝜑 → 𝐴 ∈ V).
Converse.
"^{◡}𝑅" should be read "converse of (relation) 𝑅"
and is the same as the more standard notation R^{1}
(the standard notation is ambiguous). See dfcnv 5540.
This can be used to define a subset, e.g., dftan 15416 notates
"the set of values whose cosine is a nonzero complex number" as
(^{◡}cos “ (ℂ ∖ {0})).
Function application.
"(𝐹‘𝑥)" should be read "the value
of function 𝐹 at 𝑥" and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(cos‘0) = 1 (see cos0 15494). The left apostrophe notation
originated with Peano and was adopted in Definition *30.01 of
[WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and
Definition 6.11 of [TakeutiZaring] p. 26. See dffv 6342.
In the ASCII (input) representation there are spaces around the grave
accent; there is a single accent when it is used directly,
and it is doubled within comments.
Infix and parentheses.
When a function that takes two classes and produces a class
is applied as part of an infix expression, the expression is always
surrounded by parentheses (see dfov 7143).
For example, the + in (2 + 2); see 2p2e4 11760.
Function application is itself an example of this.
Similarly, predicate expressions
in infix form that take two or three wffs and produce a wff
are also always surrounded by parentheses, such as
(𝜑 → 𝜓), (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓), and
(𝜑 ↔ 𝜓)
(see wi 4, dfor 845, dfan 400, and dfbi 210 respectively).
In contrast, a binary relation (which compares two _classes_ and
produces a _wff_) applied in an infix expression is _not_
surrounded by parentheses.
This includes set membership 𝐴 ∈ 𝐵 (see wel 2115),
equality 𝐴 = 𝐵 (see dfcleq 2815),
subset 𝐴 ⊆ 𝐵 (see dfss 3925), and
lessthan 𝐴 < 𝐵 (see dflt 10539). For the general definition
of a binary relation in the form 𝐴𝑅𝐵, see dfbr 5043.
For example, 0 < 1 (see 0lt1 11151) does not use parentheses.
Unary minus.
The symbol  is used to indicate a unary minus, e.g., 1.
It is specially defined because it is so commonly used.
See cneg 10860.
Function definition.
Functions are typically defined by first defining the constant symbol
(using $c) and declaring that its symbol is a class with the
label cNAME (e.g., ccos 15409).
The function is then defined labeled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 15415).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 15471), its
function application form labeled NAMEval (e.g., cosval 15467),
and at least one simple value (e.g., cos0 15494).
Another way to define functions is to use recursion (for more details
about recursion see below). For an example of how to define functions
that aren't primitive recursive using recursion, see the Ackermann
function definition dfack 45013 (which is based on the sequence builder
seq, see dfseq 13365).
Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
(!‘4) = ;24 (dffac 13630 and fac4 13637).
Unambiguous symbols.
A given symbol has a single unambiguous meaning in general.
Thus, where the literature might use the same symbol with different
meanings, here we use different (variant) symbols for different
meanings. These variant symbols often have suffixes, subscripts,
or underlines to distinguish them. For example, here
"0" always means the value zero (df0 10533), while
"0_{g}" is the group identity element (df0g 16706),
"0." is the poset zero (dfp0 17640),
"0_{𝑝}" is the zero polynomial (df0p 24272),
"0_{vec}" is the zero vector in a normed subcomplex vector space
(df0v 28379), and
"0" is a class variable for use as a connective symbol
(this is used, for example, in p0val 17642).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"1", "+", "∗", and "∥".
These symbols are very similar to traditional notation, but because
they are different symbols they eliminate ambiguity.
ASCII representation of symbols.
We must have an ASCII representation for each symbol.
We generally choose short sequences, ideally digraphs, and generally
choose sequences that vaguely resemble the mathematical symbol.
Here are some of the conventions we use when selecting an
ASCII representation.
We generally do not include parentheses inside a symbol because
that confuses text editors (such as emacs).
Greek letters for wff variables always use the first two letters
of their English names, making them easy to type and easy to remember.
Symbols that almost look like letters, such as ∀,
are often represented by that letter followed by a period.
For example, "A." is used to represent ∀,
"e." is used to represent ∈, and
"E." is used to represent ∃.
Single letters are now always variable names, so constants that are
often shown as single letters are now typically preceded with "_"
in their ASCII representation, for example,
"_i" is the ASCII representation for the imaginary unit i.
A script font constant is often the letter
preceded by "~" meaning "curly", such as "~P" to represent
the power class 𝒫.
Originally, all setvar and class variables used only single letters
az and AZ, respectively. A big change in recent years was to
allow the use of certain symbols as variable names to make formulas
more readable, such as a variable representing an additive group
operation. The convention is to take the original constant token
(in this case "+" which means complex number addition) and put
a period in front of it to result in the ASCII representation of the
variable ".+", shown as +, that can
be used instead of say the letter "P" that had to be used before.
Choosing tokens for more advanced concepts that have no standard
symbols but are represented by words in books, is hard. A few are
reasonably obvious, like "Grp" for group and "Top" for topology,
but often they seem to end up being either too long or too
cryptic. It would be nice if the math community came up with
standardized short abbreviations for English math terminology,
like they have more or less done with symbols, but that probably
won't happen any time soon.
Another informal convention that we've somewhat followed, that is
also not uncommon in the literature, is to start tokens with a
capital letter for collectionlike objects and lower case for
functionlike objects. For example, we have the collections On
(ordinal numbers), Fin, Prime, Grp, and we have the functions sin,
tan, log, sup. Predicates like Ord and Lim also tend to start
with upper case, but in a sense they are really collectionlike,
e.g. Lim indirectly represents the collection of limit ordinals,
but it can't be an actual class since not all limit ordinals
are sets.
This initial upper versus lower case letter convention is sometimes
ambiguous. In the past there's been a debate about whether
domain and range are collectionlike or functionlike, thus whether
we should use Dom, Ran or dom, ran. Both are used in the literature.
In the end dom, ran won out for aesthetic reasons
(Norm Megill simply just felt they looked nicer).
Typography conventions.
Class symbols for functions (e.g., abs, sin)
should usually not have leading or trailing blanks in their
HTML representation.
This is in contrast to class symbols for operations
(e.g., gcd, sadd, eval), which usually do
include leading and trailing blanks in their representation.
If a class symbol is used for a function as well as an operation
(according to the definition dfov 7143, each operation value can be
written as function value of an ordered pair), the convention for its
primary usage should be used, e.g. (iEdg‘𝐺) versus
(𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸⟩.
LaTeX definitions.
Each token has a "LaTeX definition" which is used by the Metamath
program to output tex files. When writing LaTeX definitions,
contributors should favor simplicity over perfection of the display, and
should only use core LaTeX symbols or symbols from standard packages; if
packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic,
graphicx are needed, this should be discussed. A useful resource is
The Comprehensive LaTeX
Symbol List.
Number construction independence.
There are many ways to model complex numbers.
After deriving the complex number postulates we
reintroduce them as new axioms on top of set theory.
This lets us easily identify which axioms are needed
for a particular complex number proof, without the obfuscation
of the set theory used to derive them.
This also lets us be independent of the specific construction,
which we believe is valuable.
See mmcomplex.html 7143 for details.
Thus, for example, we don't allow the use of ∅ ∉ ℂ,
as handy as that would be, because that would be
constructionspecific. We want proofs about ℂ to be independent
of whether or not ∅ ∈ ℂ.
Minimize hypotheses.
In most cases we try to minimize hypotheses, so that the statement be
more general and easier to use. There are exceptions. For example, we
intentionally add hypotheses if they help make proofs independent of a
particular construction (e.g., the contruction of the complex numbers
ℂ). We also intentionally add hypotheses for many real and
complex number theorems to expressly state their domains even when they
are not needed. For example, we could show that
⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) without any hypotheses, but we require that
theorems using this result prove that 𝐴 and 𝐵 are real numbers,
so that the statement we use is ltnei 10753. Here are the reasons as
discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 10753:
 Having the hypotheses immediately shows the intended domain of
applicability (is it ℝ, ℝ^{*}, ω, or something else?),
without having to trace back to definitions.
 Having the hypotheses forces the intended use of the statement,
which generally is desirable.
 Many outofdomain values are dependent on contingent details of
definitions, so hypothesisfree theorems would be nonportable and
"brittle".
 Only a few theorems can have their hypotheses removed in this
fashion, due to coincidences for our particular settheoretical
definitions. The poor user (especially a novice learning, e.g., real
number arithmetic) is going to be confused not knowing when hypotheses
are needed and when they are not. For someone who has not traced back
the settheoretical foundations of the definitions, it is seemingly
random and is not intuitive at all.
 Ultimately, this is a matter of consensus, and the consensus in
the group was in favor of keeping sometimes redundant hypotheses.
Natural numbers.
There are different definitions of "natural" numbers in the literature.
We use ℕ (dfnn 11626) for the set of positive integers starting
from 1, and ℕ_{0} (dfn0 11886) for the set of nonnegative integers
starting at zero.
Decimal numbers.
Numbers larger than nine are often expressed in base 10 using the
decimal constructor dfdec 12087, e.g., ;;;4001 (see 4001prm 16469
for a proof that 4001 is prime).
Theorem forms.
We will use the following descriptive terms to categorize theorems:
 A theorem is in "closed form" if it has no $e hypotheses
(e.g., unss 4135). The term "tautology" is also used, especially in
propositional calculus. This form was formerly called "theorem form"
or "closed theorem form".
 A theorem is in "deduction form" (or is a "deduction") if it
has zero or more $e hypotheses, and the hypotheses and the conclusion
are implications that share the same antecedent. More precisely, the
conclusion is an implication with a wff variable as the antecedent
(usually 𝜑), and every hypothesis ($e statement) is either:
 an implication with the same antecedent as the conclusion, or
 a definition. A definition can be for a class variable (this is a
class variable followed by =, e.g. the definition of 𝐷 in
lhop 24617) or a wff variable (this is a wff variable followed by
↔); class variable definitions are more common.
In practice, a proof of a theorem in deduction form will also contain
many steps that are implications where the antecedent is either that
wff variable (usually 𝜑) or is a conjunction (𝜑 ∩ ...)
including that wff variable (𝜑). E.g. a1d 25, unssd 4137.
Although they are no real deductions, theorems without $e hypotheses,
but in the form (𝜑 → ...), are also said to be in "deduction
form". Such theorems usually have a two step proof, applying a1i 11 to a
given theorem, and are used as convenience theorems to shorten many
proofs. E.g. eqidd 2823, which is used more than 1500 times.
 A theorem is in "inference form" (or is an "inference") if
it has one or more $e hypotheses, but is not in deduction form,
i.e. there is no common antecedent (e.g., unssi 4136).
Any theorem whose conclusion is an implication has an associated
inference, whose hypotheses are the hypotheses of that theorem
together with the antecedent of its conclusion, and whose conclusion is
the consequent of that conclusion. When both theorems are in set.mm,
then the associated inference is often labeled by adding the suffix "i"
to the label of the original theorem (for instance, con3i 157 is the
inference associated with con3 156). The inference associated with a
theorem is easily derivable from that theorem by a simple use of
axmp 5. The other direction is the subject of the Deduction Theorem
discussed below. We may also use the term "associated inference" when
the above process is iterated. For instance, syl 17 is an
inference associated with imim1 83 because it is the inference
associated with imim1i 63 which is itself the inference
associated with imim1 83.
"Deduction form" is the preferred form for theorems because this form
allows us to easily use the theorem in places where (in traditional
textbook formalizations) the standard Deduction Theorem (see below)
would be used. We call this approach "deduction style".
In contrast, we usually avoid theorems in "inference form" when that
would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other
forms of the same theorem (e.g., pm2.43d 53). The labels for inferences
usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems
in "closed form" would have no special suffix (e.g., pm2.43 56) or, if
the nonsuffixed label is already used, then we add the suffix "t" (for
"theorem" or "tautology", e.g., ancomst 468 or nfimt 1896). When an
inference with an "is a set" hypothesis (e.g., 𝐴 ∈ V) is converted
to a theorem (in closed form) by replacing the hypothesis with an
antecedent of the form (𝐴 ∈ 𝑉 →, we sometimes suffix the closed
form with "g" (for "more general") as in uniex 7452 versus uniexg 7451. In
this case, the inference often has no suffix "i".
When submitting a new theorem, a revision of a theorem, or an upgrade
of a theorem from a Mathbox to the Main database, please use the
general form to be the default form of the theorem, without the suffix
"g" . For example, "brresg" lost its suffix "g" when it was revised for
some other reason, and now it is brres 5838. Its inference form which was
the original "brres", now is brresi 5840. The same holds for the suffix
"t".
Deduction theorem.
The Deduction Theorem is a metalogical theorem that provides an
algorithm for constructing a proof of a theorem from the proof of its
corresponding deduction (its associated inference). See for instance
Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually
carries out the algorithm, because (in its most basic form) it involves
an exponential explosion of the number of proof steps as more hypotheses
are eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see mmdeduction.html 5840.
The Deduction Theorem is a metalogical theorem that cannot be applied
directly in Metamath, and the explosion of steps would be a problem
anyway, so alternatives are used. One alternative we use sometimes is
the "weak deduction theorem" dedth 4495, which works in certain cases in
set theory. We also sometimes use dedhb 3670. However, the primary
mechanism we use today for emulating the deduction theorem is to write
proofs in deduction form (aka "deduction style") as described earlier;
the prefixed 𝜑 → mimics the context in a deduction proof system.
In practice this mechanism works very well. This approach is described
in the deduction form and natural deduction page mmnatded.html 3670; a
list of translations for common natural deduction rules is given in
natded 28186.
Recursion.
We define recursive functions using various "recursion constructors".
These allow us to define, with compact direct definitions, functions
that are usually defined in textbooks with indirect selfreferencing
recursive definitions. This produces compact definition and much
simpler proofs, and greatly reduces the risk of creating unsound
definitions. Examples of recursion constructors include
recs(𝐹) in dfrecs 7995, rec(𝐹, 𝐼) in dfrdg 8033,
seq_{ω}(𝐹, 𝐼) in dfseqom 8071, and seq𝑀( + , 𝐹) in
dfseq 13365. These have characteristic function 𝐹 and initial value
𝐼. (Σ_{g} in dfgsum 16707 isn't really designed for arbitrary
recursion, but you could do it with the right magma.) The logically
primary one is dfrecs 7995, but for the "average user" the most useful
one is probably dfseq 13365 provided that a countable sequence is
sufficient for the recursion.
Extensible structures.
Mathematics includes many structures such as ring, group, poset, etc.
We define an "extensible structure" which is then used to define group,
ring, poset, etc. This allows theorems from more general structures
(groups) to be reused for more specialized structures (rings) without
having to reprove them. See dfstruct 16476.
Undefined results and "junk theorems".
Some expressions are only expected to be meaningful in certain contexts.
For example, consider Russell's definition description binder iota,
where (℩𝑥𝜑) is meant to be "the 𝑥 such that 𝜑"
(where 𝜑 typically depends on x).
What should that expression produce when there is no such 𝑥?
In set.mm we primarily use one of two approaches.
One approach is to make the expression evaluate to the empty set
whenever the expression is being used outside of its expected context.
While not perfect, it makes it a bit more clear when something
is undefined, and it has the advantage that it makes more
things equal outside their domain which can remove hypotheses when
you feel like exploiting these socalled junk theorems.
Note that Quine does this with iota (his definition of iota
evaluates to the empty set when there is no unique value of 𝑥).
Quine has no problem with that and we don't see why we should,
so we define iota exactly the same way that Quine does.
The main place where you see this being systematically exploited is in
"reverse closure" theorems like 𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹,
which is useful when 𝐹 is a family of sets. (by this we
mean it's a set set even in a type theoretic interpretation.)
The second approach uses "(New usage is discouraged.)" to prevent
unintentional uses of certain properties.
For example, you could define some construct dfNAME whose
usage is discouraged, and prove only the specific properties
you wish to use (and add those proofs to the list of permitted uses
of "discouraged" information). From then on, you can only use
those specific properties without a warning.
Other approaches often have hidden problems.
For example, you could try to "not define undefined terms"
by creating definitions like ${ $d 𝑦𝑥 $. $d 𝑦𝜑 $.
dfiota $a ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) = ∪ {𝑥 ∣ 𝜑}) $. $}.
This will be rejected by the definition checker, but the bigger
theoretical reason to reject this axiom is that it breaks equality 
the metatheorem (𝑥 = 𝑦 → P(x) = P(y) ) fails
to hold if definitions don't unfold without some assumptions.
(That is, iotabidv 6318 is no longer provable and must be added
as an axiom.) It is important for every syntax constructor to
satisfy equality theorems *unconditionally*, e.g., expressions
like (1 / 0) = (1 / 0) should not be rejected.
This is forced on us by the context free term
language, and anything else requires a lot more infrastructure
(e.g., a type checker) to support without making everything else
more painful to use.
Another approach would be to try to make nonsensical
statements syntactically invalid, but that can create its own
complexities; in some cases that would make parsing itself undecidable.
In practice this does not seem to be a serious issue.
No one does these things deliberately in "real" situations,
and some knowledgeable people (such as Mario Carneiro)
have never seen this happen accidentally.
Norman Megill doesn't agree that these "junk" consequences are
necessarily bad anyway, and they can significantly shorten proofs
in some cases. This database would be much larger if, for example,
we had to condition fvex 6665 on the argument being in the domain
of the function. It is impossible to derive a contradiction
from sound definitions (i.e. that pass the definition check),
assuming ZFC is consistent, and he doesn't see the point of all the
extra busy work and huge increase in set.mm size that would result
from restricting *all* definitions.
So instead of implementing a complex system to counter a
problem that does not appear to occur in practice, we use
a significantly simpler set of approaches.
Organizing proofs.
Humans have trouble understanding long proofs. It is often preferable
to break longer proofs into smaller parts (just as with traditional
proofs). In Metamath this is done by creating separate proofs of the
separate parts.
A proof with the sole purpose of supporting a final proof is a lemma;
the naming convention for a lemma is the final proof label followed by
"lem", and a number if there is more than one. E.g., sbthlem1 8615 is the
first lemma for sbth 8625. The comment should begin with "Lemma for",
followed by the final proof label, so that it can be suppressed in
theorem lists (see the Metamath program "MM> WRITE THEOREM_LIST"
command).
Also, consider proving reusable results separately, so that others will
be able to easily reuse that part of your work.
Limit proof size.
It is often preferable to break longer proofs into
smaller parts, just as you would do with traditional proofs.
One reason is that humans have trouble understanding long proofs.
Another reason is that it's generally best to prove
reusable results separately,
so that others will be able to easily reuse them.
Finally, the Metamath program "MMPA> MINIMIZE__WITH *" command can take
much longer with very long proofs.
We encourage proofs to be no more than 200 essential steps, and
generally no more than 500 essential steps,
though these are simply guidelines and not hardandfast rules.
Much smaller proofs are fine!
We also acknowledge that some proofs, especially autogenerated ones,
should sometimes not be broken up (e.g., because
breaking them up might be useless and inefficient due to many
interconnections and reused terms within the proof).
In Metamath, breaking up longer proofs is done by creating multiple
separate proofs of separate parts.
A proof with the sole purpose of supporting a final proof is a
lemma; the naming convention for a lemma is the final proof's name
followed by "lem", and a number if there is more than one. E.g.,
sbthlem1 8615 is the first lemma for sbth 8625.
Hypertext links.
We strongly encourage comments to have many links to related material,
with accompanying text that explains the relationship. These can help
readers understand the context. Links to other statements, or to
HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a
spaceseparated tilde (e.g., " ~ dfprm " results in " dfprm 16005").
When the Metamath program is used to generate HTML, it automatically
inserts hypertext links for syntax used (e.g., every symbol used), every
axiom and definition depended on, the justification for each step in a
proof, and to both the next and previous assertions.
Hypertext links to section headers.
Some section headers have text under them that describes or explains the
section. However, they are not part of the description of axioms or
theorems, and there is no way to link to them directly. To provide for
this, section headers with accompanying text (indicated with "*"
prefixed to mmtheorems.html#mmdtoc 16005 entries) have an anchor in
mmtheorems.html 16005 whose name is the first $a or $p statement that
follows the header. For example there is a glossary under the section
heading called GRAPH THEORY. The first $a or $p statement that follows
is cedgf 26780. To reference it we link to the anchor using a
spaceseparated tilde followed by the spaceseparated link
mmtheorems.html#cedgf, which will become the hyperlink
mmtheorems.html#cedgf 26780. Note that no theorem in set.mm is allowed to
begin with "mm" (this is enforced by the Metamath program "MM> VERIFY
MARKUP" command). Whenever the program sees a tilde reference beginning
with "http:", "https:", or "mm", the reference is assumed to be a link
to something other than a statement label, and the tilde reference is
used as is. This can also be useful for relative links to other pages
such as mmcomplex.html 26780.
Bibliography references.
Please include a bibliographic reference to any external material used.
A name in square brackets in a comment indicates a
bibliographic reference. The full reference must be of the form
KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER 
note that this is a very specific form that requires a page number.
There should be no comma between the author reference and the
"p." (a constant indicator).
Whitespace, comma, period, or semicolon should follow NUMBER.
An example is Theorem 3.1 of [Monk1] p. 22,
The KEYWORD, which is not casesensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g., mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we _require_ page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
If this is a new reference, please also add it to the
"Bibliography" section of mmset.html.
(The file mmbiblio.html is automatically rebuilt, e.g.,
using the Metamath program "MM> WRITE BIBLIOGRAPHY" command.)
Acceptable shorter proofs.
Shorter proofs are welcome, and any shorter proof we accept
will be acknowledged in the theorem description. However,
in some cases a proof may be "shorter" or not depending on
how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1)
shorten the set.mm file (with compressed proofs), (2) reduce
the size of the HTML file generated with SHOW STATEMENT xx
/ HTML, (3) use only existing, unmodified theorems in the
database (the order of theorems may be changed, though), and
(4) use no additional axioms.
Usually we will also automatically accept a _new_ theorem
that is used to shorten multiple proofs, if the total size
of set.mm (including the comment of the new theorem, not
including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on
the number of compressed proof steps and less on the length
of the label section (since the names are in principle
arbitrary). If two proofs have the same number of compressed
proof steps, we will typically give preference to the one
with the smaller number of different labels, or if these
numbers are the same, the proof with the fewest number of
characters that the proofs happen to have by chance when
label lengths are included.
A few theorems have a longer proof than necessary in order
to avoid the use of certain axioms, for pedagogical purposes,
and for other reasons. These theorems will (or should) have
a "(Proof modification is discouraged.)" tag in their
description. For example, idALT 23 shows a proof directly from
axioms. Shorter proofs for such cases won't be accepted,
of course, unless the criteria described continues to be
satisfied.
Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
mmdefinitions.html 23.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
Reserved symbols: 'LETTER.
Some symbols are reserved for potential future use.
Symbols with the pattern 'LETTER are reserved for possibly
representing characters (this is somewhat similar to Lisp).
We would expect '\n to represent newline, 'sp for space, and perhaps
'\x24 for the dollar character.
The challenge of varying mathematical conventions
We try to follow mathematical conventions, but in many cases
different texts use different conventions.
In those cases we pick some reasonably common convention and stick to
it.
We have already mentioned that the term "natural number" has
varying definitions (some start from 0, others start from 1), but
that is not the only such case.
A useful example is the set of metavariables used to represent
arbitrary wellformed formulas (wffs).
We use an open phi, φ, to represent the first arbitrary wff in an
assertion with one or more wffs; this is a common convention and
this symbol is easily distinguished from the empty set symbol.
That said, it is impossible to please everyone or simply "follow
the literature" because there are many different conventions for
a variable that represents any arbitrary wff.
To demonstrate the point,
here are some conventions for variables that represent an arbitrary
wff and some texts that use each convention:
 open phi φ (and so on): Tarski's papers,
Rasiowa & Sikorski's
The Mathematics of Metamathematics (1963),
Monk's Introduction to Set Theory (1969),
Enderton's Elements of Set Theory (1977),
Bell & Machover's A Course in Mathematical Logic (1977),
Jech's Set Theory (1978),
Takeuti & Zaring's
Introduction to Axiomatic Set Theory (1982).
 closed phi ϕ (and so on):
Levy's Basic Set Theory (1979),
Kunen's Set Theory (1980),
Paulson's Isabelle: A Generic Theorem Prover (1994),
Huth and Ryan's Logic in Computer Science (2004/2006).
 Greek α, β, γ:
Duffy's Principles of Automated Theorem Proving (1991).
 Roman A, B, C:
Kleene's Introduction to Metamathematics (1974),
Smullyan's FirstOrder Logic (1968/1995).
 script A, B, C:
Hamilton's Logic for Mathematicians (1988).
 italic A, B, C:
Mendelson's Introduction to Mathematical Logic (1997).
 italic P, Q, R:
Suppes's Axiomatic Set Theory (1972),
Gries and Schneider's A Logical Approach to Discrete Math
(1993/1994),
Rosser's Logic for Mathematicians (2008).
 italic p, q, r:
Quine's Set Theory and Its Logic (1969),
Kuratowski & Mostowski's Set Theory (1976).
 italic X, Y, Z:
Dijkstra and Scholten's
Predicate Calculus and Program Semantics (1990).
 Fraktur letters:
Fraenkel et. al's Foundations of Set Theory (1973).
Distinctness or freeness
Here are some conventions that address distinctness or freeness of a
variable:
 Ⅎ𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑";
see dfnf 1786 (whose description has some important technical
details). Similarly, Ⅎ𝑥𝐴 is read 𝑥 is not free in (class)
𝐴, see dfnfc 2962.
 "$d 𝑥𝑦 $." should be read "Assume 𝑥 and 𝑦 are distinct
variables."
 "$d 𝜑𝑥 $." should be read "Assume 𝑥 does not occur in
ϕ." Sometimes a theorem is proved using Ⅎ𝑥𝜑 (dfnf 1786)
in place of "$d 𝜑𝑥 $." when a more general result is desired;
ax5 1911 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2660 from eu6 2658.
 "$d 𝐴𝑥 $." should be read "Assume 𝑥 is not a variable
occurring in class 𝐴."
 "$d 𝐴𝑥 $. $d 𝜓𝑥 $.
$e  (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) $." is an idiom often used instead
of explicit substitution, meaning "Assume ψ results from the
proper substitution of 𝐴 for 𝑥 in ϕ." Therefore, we often
use the term "implicit substitution" for such a hypothesis.
 Class and wff variables should appear at the beginning of distinct
variable conditions, and setvars should be in alphabetical order.
E.g., "$d 𝑍𝑥𝑦 $.", "$d 𝜓𝑎𝑥 $.". This convention should
be applied for new theorems (formerly, the class and wff variables
mostly appear at the end) and will be assured by a formatter in the
future.
 " ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ..." occurs early in some cases, and
should be read "If x and y are distinct
variables, then..." This antecedent provides us with a technical
device (called a "distinctor" in Section 7 of [Megill] p. 444)
to avoid the need for the
$d statement early in our development of predicate calculus, permitting
unrestricted substitutions as conceptually simple as those in
propositional calculus. However, the $d eventually becomes a
requirement, and after that this device is rarely used.
There is a general technique to replace a $d x A or
$d x ph condition in a theorem with the corresponding
Ⅎ𝑥𝐴 or Ⅎ𝑥𝜑; here it is.
⊢ T[x, A] where $d 𝑥𝐴,
and you wish to prove ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ T[x, A].
You apply the theorem substituting 𝑦 for 𝑥 and 𝐴 for 𝐴,
where 𝑦 is a new dummy variable, so that
$d y A is satisfied.
You obtain ⊢ T[y, A], and apply chvar to obtain ⊢
T[x, A] (or just use mpbir 234 if T[x, A] binds 𝑥).
The side goal is ⊢ (𝑥 = 𝑦 → ( T[y, A] ↔ T[x, A] )),
where you can use equality theorems, except
that when you get to a bound variable you use a nondv bound variable
renamer theorem like cbval 2417. The section
mmtheorems32.html#mm3146s 2417 also describes the
metatheorem that underlies this.
Additional rules for definitions
Standard Metamath verifiers do not distinguish between axioms and
definitions (both are $a statements).
In practice, we require that definitions (1) be conservative
(a definition should not allow an expression
that previously qualified as a wff but was not provable
to become provable) and be eliminable
(there should exist an algorithmic method for converting any
expression using the definition into
a logically equivalent expression that previously qualified as a wff).
To ensure this, we have additional rules on almost all definitions
($a statements with a label that does not begin with ax).
These additional rules are not applied in a few cases where they
are too strict (dfbi 210, dfclab 2801, dfcleq 2815, and dfclel 2894);
see those definitions for more information.
These additional rules for definitions are checked by at least
mmj2's definition check (see
mmj2 master file mmj2jar/macros/definitionCheck.js).
This definition check relies on the database being very much like
set.mm, down to the names of certain constants and types, so it
cannot apply to all Metamath databases... but it is useful in set.mm.
In this definition check, a $astatement with a given label and
typecode ⊢ passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
The expression must be a biconditional or an equality (i.e. its
rootsymbol must be ↔ or =).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the rootsymbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "rootsymbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 1083 or wsb 2069).
The defined expression must not appear in any statement
between its syntax axiom ($a wff ) and its definition,
and the defined expression must not be used in its definiens.
See df3an 1086 for an example where the same symbol is used in
different ways (this is allowed).
No two variables occurring in the LHS may share a
disjoint variable (DV) condition.
All dummy variables are required to be disjoint from any
other (dummy or not) variable occurring in this labeled expression.
Either
(a) there must be no nonsetvar dummy variables, or
(b) there must be a justification theorem.
The justification theorem must be of form
⊢ ( definiens rootsymbol definiens' )
where definiens' is definiens but the dummy variables are all
replaced with other unused dummy variables of the same type.
Note that rootsymbol is ↔ or =, and that setvar
variables are simply variables with the setvar typecode.
One of the following must be true:
(a) there must be no setvar dummy variables,
(b) there must be a justification theorem as described in rule 5, or
(c) if there are setvar dummy variables, every one must not be free.
That is, it must be true that
(𝜑 → ∀𝑥𝜑) for each setvar dummy variable 𝑥
where 𝜑 is the definiens.
We use two different tests for nonfreeness; one must succeed
for each setvar dummy variable 𝑥.
The first test requires that the setvar dummy variable 𝑥
be syntactically bound
(this is sometimes called the "fast" test, and this implies
that we must track binding operators).
The second test requires a successful
search for the directlystated proof of (𝜑 → ∀𝑥𝜑)
Part c of this rule is how most setvar dummy variables
are handled.
Rule 3 may seem unnecessary, but it is needed.
Without this rule, you can define something like
cbar $a wff Foo x y $.
${ $d x y $. dffoo $a  ( Foo x y <> x = y ) $. $}
and now "Foo x x" is not eliminable;
there is no way to prove that it means anything in particular,
because the definitional theorem that is supposed to be
responsible for connecting it to the original language wants
nothing to do with this expression, even though it is well formed.
A justification theorem for a definition (if used this way)
must be proven before the definition that depends on it.
One example of a justification theorem is vjust 3470.
The definition dfv 3471 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} is justified
by the justification theorem vjust 3470
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}.
Another example of a justification theorem is trujust 1540;
the definition dftru 1541 ⊢ (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥))
is justified by trujust 1540 ⊢ ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).
Here is more information about our processes for checking and
contributing to this work:
Multiple verifiers.
This entire file is verified by multiple independentlyimplemented
verifiers when it is checked in, giving us extremely high
confidence that all proofs follow from the assumptions.
The checkers also check for various other problems such as
overly long lines.
Discouraged information.
A separate file named "discouraged" lists all
discouraged statements and uses of them, and this file is checked.
If you change the use of discouraged things, you will need to change
this file.
This makes it obvious when there is a change to anything discouraged
(triggering further review).
LRParser check.
Metamath verifiers ensure that $p statements follow from previous
$a and $p statements.
However, by itself the Metamath language permits certain kinds of
syntactic ambiguity that we choose to avoid in this database.
Thus, we require that this database unambiguously parse
using the "LRParser" check (implemented by at least mmj2).
(For details, see mmj2 master file src/mmj/verify/LRParser.java).
This check
counters, for example, a devious ambiguous construct
developed by saueran at oregonstate dot edu
posted on Mon, 11 Feb 2019 17:32:32 0800 (PST)
based on creating definitions with mismatched parentheses.
Proposing specific changes.
Please propose specific changes as pull requests (PRs) against the
"develop" branch of set.mm, at:
https://github.com/metamath/set.mm/tree/develop 1540.
Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/g/metamath 1540.
(Contributed by the Metamath team, 27Dec2016.) Date of last revision.
(Revised by the Metamath team, 22Sep2022.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

Theorem  conventionslabels 28184 
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 28183 and links therein.
Every statement has a unique identifying label, which serves the
same purpose as an equation number in a book.
We use various label naming conventions to provide
easytoremember hints about their contents.
Labels are not a 1to1 mapping, because that would create
long names that would be difficult to remember and tedious to type.
Instead, label names are relatively short while
suggesting their purpose.
Names are occasionally changed to make them more consistent or
as we find better ways to name them.
Here are a few of the label naming conventions:
 Axioms, definitions, and wff syntax.
As noted earlier, axioms are named "axNAME",
proofs of proven axioms are named "axNAME", and
definitions are named "dfNAME".
Wff syntax declarations have labels beginning with "w"
followed by short fragment suggesting its purpose.
 Hypotheses.
Hypotheses have the name of the final axiom or theorem, followed by
".", followed by a unique id (these ids are usually consecutive integers
starting with 1, e.g. for rgen 3140"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 21209: "mdet0.d $e  D = ( N maDet R ) $.").
 Common names.
If a theorem has a wellknown name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2749 and stirling 42670.
 Principia Mathematica.
Proofs of theorems from Principia Mathematica often use a special
naming convention: "pm" followed by its identifier.
For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named
pm2.27 42.
 19.x series of theorems.
Similar to the conventions for the theorems from Principia Mathematica,
theorems from Section 19 of [Margaris] p. 90 often use a special naming
convention: "19." resp. "r19." (for corresponding restricted quantifier
versions) followed by its identifier.
For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled
19.38 1840, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 3204.
 Characters to be used for labels.
Although the specification of Metamath allows for dots/periods "." in
any label, it is usually used only in labels for hypotheses (see above).
Exceptions are the labels of theorems from Principia Mathematica and the
19.x series of theorems from Section 19 of [Margaris] p. 90 (see above)
and 0.999... 15228. Furthermore, the underscore "_" should not be used.
Finally, only lower case characters should be used (except the special
suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at
least in main set.mm (exceptions are tolerated in mathboxes).
 Syntax label fragments.
Most theorems are named using a concatenation of syntax label fragments
(omitting variables) that represent the important part of the theorem's
main conclusion. Almost every syntactic construct has a definition
labeled "dfNAME", and normally NAME is the syntax label fragment. For
example, the class difference construct (𝐴 ∖ 𝐵) is defined in
dfdif 3911, and thus its syntax label fragment is "dif". Similarly, the
subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss"
because it is defined in dfss 3925. Most theorem names follow from
these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 4083. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in dfsn 4540), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from dfpr 4542). Digits are used to represent
themselves. Suffixes (e.g., with numbers) are sometimes used to
distinguish multiple theorems that would otherwise produce the same
label.
 Phantom definitions.
In some cases there are common label fragments for something that could
be in a definition, but for technical reasons is not. The iselementof
(is member of) construct 𝐴 ∈ 𝐵 does not have a dfNAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses iselementof
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4693. An "n" is often used for negation (¬), e.g.,
nan 828.
 Exceptions.
Sometimes there is a definition dfNAME but the label fragment is not
the NAME part. The definition should note this exception as part of its
definition. In addition, the table below attempts to list all such
cases and marks them in bold. For example, the label fragment "cn"
represents complex numbers ℂ (even though its definition is in
dfc 10532) and "re" represents real numbers ℝ ( definition dfr 10536).
The empty set ∅ often uses fragment 0, even though it is defined
in dfnul 4266. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with dfadd 10537), but "p" is used as
the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 11760.
 Other markings.
In labels we sometimes use "com" for "commutative", "ass" for
"associative", "rot" for "rotation", and "di" for "distributive".
 Focus on the important part of the conclusion.
Typically the conclusion is the part the user is most interested in.
So, a rough guideline is that a label typically provides a hint
about only the conclusion; a label rarely says anything about the
hypotheses or antecedents.
If there are multiple theorems with the same conclusion
but different hypotheses/antecedents, then the labels will need
to differ; those label differences should emphasize what is different.
There is no need to always fully describe the conclusion; just
identify the important part. For example,
cos0 15494 is the theorem that provides the value for the cosine of 0;
we would need to look at the theorem itself to see what that value is.
The label "cos0" is concise and we use it instead of "cos0eq1".
There is no need to add the "eq1", because there will never be a case
where we have to disambiguate between different values produced by
the cosine of zero, and we generally prefer shorter labels if
they are unambiguous.
 Closures and values.
As noted above, if a function dfNAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl".
E.g., for cosine (dfcos 15415) we have value cosval 15467 and closure
coscl 15471.
 Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implicationonly theorems. They are grouped in a more adhoc way that
attempts to make their distinctions clearer. These often use
abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and
"id" for "identity". It is especially hard to give good names in the
propositional calculus section because there are so few primitives.
However, in most cases this is not a serious problem. There are a few
very common theorems like axmp 5 and syl 17 that you will have no
trouble remembering, a few theorem series like syl*anc and simp* that
you can use parametrically, and a few other useful glue things for
destructuring 'and's and 'or's (see natded 28186 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
 Suffixes.
Suffixes are used to indicate the form of a theorem (inference,
deduction, or closed form, see above).
Additionally, we sometimes suffix with "v" the label of a theorem adding
a disjoint variable condition, as in 19.21v 1940 versus 19.21 2208. This
often permits to prove the result using fewer axioms, and/or to
eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2208).
If no constraint is put on axiom use, then the vversion can be proved
from the original theorem using nfv 1915. If two (resp. three) such
disjoint variable conditions are added, then the suffix "vv" (resp.
"vvv") is used, e.g., exlimivv 1933.
Conversely, we sometimes suffix with "f" the label of a theorem
introducing such a hypothesis to eliminate the need for the disjoint
variable condition; e.g. euf 2660 derived from eu6 2658. The "f" stands
for "not free in" which is less restrictive than "does not occur in."
The suffix "b" often means "biconditional" (↔, "iff" , "if and
only if"), e.g., sspwb 5319.
We sometimes suffix with "s" the label of an inference that manipulates
an antecedent, leaving the consequent unchanged. The "s" means that the
inference eliminates the need for a syllogism (syl 17) type inference
in a proof. A theorem label is suffixed with "ALT" if it provides an
alternate lesspreferred proof of a theorem (e.g., the proof is
clearer but uses more axioms than the preferred version).
The "ALT" may be further suffixed with a number if there is more
than one alternate theorem.
Furthermore, a theorem label is suffixed with "OLD" if there is a new
version of it and the OLD version is obsolete (and will be removed
within one year).
Finally, it should be mentioned that suffixes can be combined, for
example in cbvaldva 2431 (cbval 2417 in deduction form "d" with a not free
variable replaced by a disjoint variable condition "v" with a
conjunction as antecedent "a"). As a general rule, the suffixes for
the theorem forms ("i", "d" or "g") should be the first of multiple
suffixes, as for example in vtocldf 3530.
Here is a nonexhaustive list of common suffixes:
 a : theorem having a conjunction as antecedent
 b : theorem expressing a logical equivalence
 c : contraction (e.g., sylc 65, syl2anc 587), commutes
(e.g., biimpac 482)
 d : theorem in deduction form
 f : theorem with a hypothesis such as Ⅎ𝑥𝜑
 g : theorem in closed form having an "is a set" antecedent
 i : theorem in inference form
 l : theorem concerning something at the left
 r : theorem concerning something at the right
 r : theorem with something reversed (e.g., a biconditional)
 s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
 t : theorem in closed form (not having an "is a set" antecedent)
 v : theorem with one (main) disjoint variable condition
 vv : theorem with two (main) disjoint variable conditions
 w : weak(er) form of a theorem
 ALT : alternate proof of a theorem
 ALTV : alternate version of a theorem or definition (mathbox
only)
 OLD : old/obsolete version of a theorem (or proof) or definition
 Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the 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 dfNAME definition but the label fragment is not NAME. This is
not a complete list of abbreviations, though we do want this to
eventually be a complete list of exceptions.
Abbreviation  Mnenomic  Source 
Expression  Syntax?  Example(s) 
a  and (suffix)  
 No  biimpa 480, rexlimiva 3267 
abl  Abelian group  dfabl 18900 
Abel  Yes  ablgrp 18902, zringabl 20165 
abs  absorption    No 
ressabs 16554 
abs  absolute value (of a complex number) 
dfabs 14586  (abs‘𝐴)  Yes 
absval 14588, absneg 14628, abs1 14648 
ad  adding  
 No  adantr 484, ad2antlr 726 
add  add (see "p")  dfadd 10537 
(𝐴 + 𝐵)  Yes 
addcl 10608, addcom 10815, addass 10613 
al  "for all"  
∀𝑥𝜑  No  alim 1812, alex 1827 
ALT  alternative/less preferred (suffix)  
 No  idALT 23 
an  and  dfan 400 
(𝜑 ∧ 𝜓)  Yes 
anor 980, iman 405, imnan 403 
ant  antecedent  
 No  adantr 484 
ass  associative  
 No  biass 389, orass 919, mulass 10614 
asym  asymmetric, antisymmetric  
 No  intasym 5953, asymref 5954, posasymb 17553 
ax  axiom  
 No  ax6dgen 2132, ax1cn 10560 
bas, base 
base (set of an extensible structure)  dfbase 16480 
(Base‘𝑆)  Yes 
baseval 16533, ressbas 16545, cnfldbas 20093 
b, bi  biconditional ("iff", "if and only if")
 dfbi 210  (𝜑 ↔ 𝜓)  Yes 
impbid 215, sspwb 5319 
br  binary relation  dfbr 5043 
𝐴𝑅𝐵  Yes  brab1 5090, brun 5093 
cbv  change bound variable   
No  cbvalivw 2014, cbvrex 3421 
cl  closure    No 
ifclda 4473, ovrcl 7181, zaddcl 12010 
cn  complex numbers  dfc 10532 
ℂ  Yes  nnsscn 11630, nncn 11633 
cnfld  field of complex numbers  dfcnfld 20090 
ℂ_{fld}  Yes  cnfldbas 20093, cnfldinv 20120 
cntz  centralizer  dfcntz 18438 
(Cntz‘𝑀)  Yes 
cntzfval 18441, dprdfcntz 19128 
cnv  converse  dfcnv 5540 
^{◡}𝐴  Yes  opelcnvg 5728, f1ocnv 6609 
co  composition  dfco 5541 
(𝐴 ∘ 𝐵)  Yes  cnvco 5733, fmptco 6873 
com  commutative  
 No  orcom 867, bicomi 227, eqcomi 2831 
con  contradiction, contraposition  
 No  condan 817, con2d 136 
csb  class substitution  dfcsb 3856 
⦋𝐴 / 𝑥⦌𝐵  Yes 
csbid 3868, csbie2g 3895 
cyg  cyclic group  dfcyg 18988 
CycGrp  Yes 
iscyg 18989, zringcyg 20182 
d  deduction form (suffix)  
 No  idd 24, impbid 215 
df  (alternate) definition (prefix)  
 No  dfrel2 6024, dffn2 6496 
di, distr  distributive  
 No 
andi 1005, imdi 394, ordi 1003, difindi 4232, ndmovdistr 7322 
dif  class difference  dfdif 3911 
(𝐴 ∖ 𝐵)  Yes 
difss 4083, difindi 4232 
div  division  dfdiv 11287 
(𝐴 / 𝐵)  Yes 
divcl 11293, divval 11289, divmul 11290 
dm  domain  dfdm 5542 
dom 𝐴  Yes  dmmpt 6072, iswrddm0 13881 
e, eq, equ  equals (equ for setvars, eq for
classes)  dfcleq 2815 
𝐴 = 𝐵  Yes 
2p2e4 11760, uneqri 4102, equtr 2028 
edg  edge  dfedg 26839 
(Edg‘𝐺)  Yes 
edgopval 26842, usgredgppr 26984 
el  element of  
𝐴 ∈ 𝐵  Yes 
eldif 3918, eldifsn 4693, elssuni 4843 
en  equinumerous  dfen 
𝐴 ≈ 𝐵  Yes  domen 8509, enfi 8722 
eu  "there exists exactly one"  eu6 2658 
∃!𝑥𝜑  Yes  euex 2661, euabsn 4636 
ex  exists (i.e. is a set)  
∈ V  No  brrelex1 5582, 0ex 5187 
ex, e  "there exists (at least one)" 
dfex 1782 
∃𝑥𝜑  Yes  exim 1835, alex 1827 
exp  export  
 No  expt 180, expcom 417 
f  "not free in" (suffix)  
 No  equs45f 2483, sbf 2272 
f  function  dff 6338 
𝐹:𝐴⟶𝐵  Yes  fssxp 6515, opelf 6520 
fal  false  dffal 1551 
⊥  Yes  bifal 1554, falantru 1573 
fi  finite intersection  dffi 8863 
(fi‘𝐵)  Yes  fival 8864, inelfi 8870 
fi, fin  finite  dffin 8500 
Fin  Yes 
isfi 8520, snfi 8581, onfin 8698 
fld  field (Note: there is an alternative
definition Fld of a field, see dffld 35388)  dffield 19496 
Field  Yes  isfld 19502, fldidom 20069 
fn  function with domain  dffn 6337 
𝐴 Fn 𝐵  Yes  ffn 6494, fndm 6434 
frgp  free group  dffrgp 18827 
(freeGrp‘𝐼)  Yes 
frgpval 18875, frgpadd 18880 
fsupp  finitely supported function 
dffsupp 8822  𝑅 finSupp 𝑍  Yes 
isfsupp 8825, fdmfisuppfi 8830, fsuppco 8853 
fun  function  dffun 6336 
Fun 𝐹  Yes  funrel 6351, ffun 6497 
fv  function value  dffv 6342 
(𝐹‘𝐴)  Yes  fvres 6671, swrdfv 14001 
fz  finite set of sequential integers 
dffz 12886 
(𝑀...𝑁)  Yes  fzval 12887, eluzfz 12897 
fz0  finite set of sequential nonnegative integers 

(0...𝑁)  Yes  nn0fz0 13000, fz0tp 13003 
fzo  halfopen integer range  dffzo 13029 
(𝑀..^𝑁)  Yes 
elfzo 13035, elfzofz 13048 
g  more general (suffix); eliminates "is a set"
hypotheses  
 No  uniexg 7451 
gr  graph  
 No  uhgrf 26853, isumgr 26886, usgrres1 27103 
grp  group  dfgrp 18097 
Grp  Yes  isgrp 18100, tgpgrp 22681 
gsum  group sum  dfgsum 16707 
(𝐺 Σ_{g} 𝐹)  Yes 
gsumval 17878, gsumwrev 18485 
hash  size (of a set)  dfhash 13687 
(♯‘𝐴)  Yes 
hashgval 13689, hashfz1 13702, hashcl 13713 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1826, hbald 2175, hbequid 36163 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 17949, isghm 18349, isrhm 19467 
i  inference (suffix)  
 No  eleq1i 2904, tcsni 9173 
i  implication (suffix)  
 No  brwdomi 9020, infeq5i 9087 
id  identity  
 No  biid 264 
iedg  indexed edge  dfiedg 26790 
(iEdg‘𝐺)  Yes 
iedgval0 26831, edgiedgb 26845 
idm  idempotent  
 No  anidm 568, tpidm13 4666 
im, imp  implication (label often omitted) 
dfim 14451  (𝐴 → 𝐵)  Yes 
iman 405, imnan 403, impbidd 213 
ima  image  dfima 5545 
(𝐴 “ 𝐵)  Yes  resima 5865, imaundi 5986 
imp  import  
 No  biimpa 480, impcom 411 
in  intersection  dfin 3915 
(𝐴 ∩ 𝐵)  Yes  elin 3924, incom 4152 
inf  infimum  dfinf 8895 
inf(ℝ^{+}, ℝ^{*}, < )  Yes 
fiinfcl 8953, infiso 8960 
is...  is (something a) ...?  
 No  isring 19292 
j  joining, disjoining  
 No  jc 164, jaoi 854 
l  left  
 No  olcd 871, simpl 486 
map  mapping operation or set exponentiation 
dfmap 8395  (𝐴 ↑_{m} 𝐵)  Yes 
mapvalg 8403, elmapex 8414 
mat  matrix  dfmat 21011 
(𝑁 Mat 𝑅)  Yes 
matval 21014, matring 21046 
mdet  determinant (of a square matrix) 
dfmdet 21188  (𝑁 maDet 𝑅)  Yes 
mdetleib 21190, mdetrlin 21205 
mgm  magma  dfmgm 17843 
Magma  Yes 
mgmidmo 17861, mgmlrid 17868, ismgm 17844 
mgp  multiplicative group  dfmgp 19231 
(mulGrp‘𝑅)  Yes 
mgpress 19241, ringmgp 19294 
mnd  monoid  dfmnd 17903 
Mnd  Yes  mndass 17911, mndodcong 18661 
mo  "there exists at most one"  dfmo 2622 
∃*𝑥𝜑  Yes  eumo 2662, moim 2626 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 20 
mpo  mapsto notation for an operation 
dfmpo 7145  (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)  Yes 
mpompt 7250, resmpo 7256 
mpt  modus ponendo tollens  
 No  mptnan 1770, mptxor 1771 
mpt  mapsto notation for a function 
dfmpt 5123  (𝑥 ∈ 𝐴 ↦ 𝐵)  Yes 
fconstmpt 5591, resmpt 5883 
mpt2  mapsto notation for an operation (deprecated).
We are in the process of replacing mpt2 with mpo in labels. 
dfmpo 7145  (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)  Yes 
mpompt 7250, resmpo 7256 
mul  multiplication (see "t")  dfmul 10538 
(𝐴 · 𝐵)  Yes 
mulcl 10610, divmul 11290, mulcom 10612, mulass 10614 
n, not  not  
¬ 𝜑  Yes 
nan 828, notnotr 132 
ne  not equal  dfne  𝐴 ≠ 𝐵 
Yes  exmidne 3021, neeqtrd 3080 
nel  not element of  dfnel  𝐴 ∉ 𝐵

Yes  neli 3117, nnel 3124 
ne0  not equal to zero (see n0)  
≠ 0  No 
negne0d 10984, ine0 11064, gt0ne0 11094 
nf  "not free in" (prefix)  
 No  nfnd 1859 
ngp  normed group  dfngp 23188 
NrmGrp  Yes  isngp 23200, ngptps 23206 
nm  norm (on a group or ring)  dfnm 23187 
(norm‘𝑊)  Yes 
nmval 23194, subgnm 23237 
nn  positive integers  dfnn 11626 
ℕ  Yes  nnsscn 11630, nncn 11633 
nn0  nonnegative integers  dfn0 11886 
ℕ_{0}  Yes  nnnn0 11892, nn0cn 11895 
n0  not the empty set (see ne0)  
≠ ∅  No  n0i 4271, vn0 4276, ssn0 4326 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1884 
on  ordinal number  dfon 6173 
𝐴 ∈ On  Yes 
elon 6178, 1on 8096 onelon 6194 
op  ordered pair  dfop 4546 
⟨𝐴, 𝐵⟩  Yes  dfopif 4773, opth 5345 
or  or  dfor 845 
(𝜑 ∨ 𝜓)  Yes 
orcom 867, anor 980 
ot  ordered triple  dfot 4548 
⟨𝐴, 𝐵, 𝐶⟩  Yes 
euotd 5380, fnotovb 7190 
ov  operation value  dfov 7143 
(𝐴𝐹𝐵)  Yes
 fnotovb 7190, fnovrn 7308 
p  plus (see "add"), for allconstant
theorems  dfadd 10537 
(3 + 2) = 5  Yes 
3p2e5 11776 
pfx  prefix  dfpfx 14024 
(𝑊 prefix 𝐿)  Yes 
pfxlen 14036, ccatpfx 14054 
pm  Principia Mathematica  
 No  pm2.27 42 
pm  partial mapping (operation)  dfpm 8396 
(𝐴 ↑_{pm} 𝐵)  Yes  elpmi 8412, pmsspw 8428 
pr  pair  dfpr 4542 
{𝐴, 𝐵}  Yes 
elpr 4562, prcom 4642, prid1g 4670, prnz 4686 
prm, prime  prime (number)  dfprm 16005 
ℙ  Yes  1nprm 16012, dvdsprime 16020 
pss  proper subset  dfpss 3927 
𝐴 ⊊ 𝐵  Yes  pssss 4047, sspsstri 4054 
q  rational numbers ("quotients")  dfq 12337 
ℚ  Yes  elq 12338 
r  right  
 No  orcd 870, simprl 770 
rab  restricted class abstraction 
dfrab 3139  {𝑥 ∈ 𝐴 ∣ 𝜑}  Yes 
rabswap 3464, dfoprab 7144 
ral  restricted universal quantification 
dfral 3135  ∀𝑥 ∈ 𝐴𝜑  Yes 
ralnex 3224, ralrnmpo 7273 
rcl  reverse closure  
 No  ndmfvrcl 6683, nnarcl 8229 
re  real numbers  dfr 10536 
ℝ  Yes  recn 10616, 0re 10632 
rel  relation  dfrel 5539  Rel 𝐴 
Yes  brrelex1 5582, relmpoopab 7776 
res  restriction  dfres 5544 
(𝐴 ↾ 𝐵)  Yes 
opelres 5837, f1ores 6611 
reu  restricted existential uniqueness 
dfreu 3137  ∃!𝑥 ∈ 𝐴𝜑  Yes 
nfreud 3353, reurex 3404 
rex  restricted existential quantification 
dfrex 3136  ∃𝑥 ∈ 𝐴𝜑  Yes 
rexnal 3226, rexrnmpo 7274 
rmo  restricted "at most one" 
dfrmo 3138  ∃*𝑥 ∈ 𝐴𝜑  Yes 
nfrmod 3354, nrexrmo 3408 
rn  range  dfrn 5543  ran 𝐴 
Yes  elrng 5739, rncnvcnv 5781 
rng  (unital) ring  dfring 19290 
Ring  Yes 
ringidval 19244, isring 19292, ringgrp 19293 
rot  rotation  
 No  3anrot 1097, 3orrot 1089 
s  eliminates need for syllogism (suffix) 
  No  ancoms 462 
sb  (proper) substitution (of a set) 
dfsb 2070  [𝑦 / 𝑥]𝜑  Yes 
spsbe 2088, sbimi 2079 
sbc  (proper) substitution of a class 
dfsbc 3748  [𝐴 / 𝑥]𝜑  Yes 
sbc2or 3756, sbcth 3762 
sca  scalar  dfsca 16572 
(Scalar‘𝐻)  Yes 
resssca 16641, mgpsca 19237 
simp  simple, simplification  
 No  simpl 486, simp3r3 1280 
sn  singleton  dfsn 4540 
{𝐴}  Yes  eldifsn 4693 
sp  specialization  
 No  spsbe 2088, spei 2413 
ss  subset  dfss 3925 
𝐴 ⊆ 𝐵  Yes  difss 4083 
struct  structure  dfstruct 16476 
Struct  Yes  brstruct 16483, structfn 16491 
sub  subtract  dfsub 10861 
(𝐴 − 𝐵)  Yes 
subval 10866, subaddi 10962 
sup  supremum  dfsup 8894 
sup(𝐴, 𝐵, < )  Yes 
fisupcl 8921, supmo 8904 
supp  support (of a function)  dfsupp 7818 
(𝐹 supp 𝑍)  Yes 
ressuppfi 8847, mptsuppd 7840 
swap  swap (two parts within a theorem) 
  No  rabswap 3464, 2reuswap 3712 
syl  syllogism  syl 17 
 No  3syl 18 
sym  symmetric  
 No  dfsymdif 4193, cnvsym 5952 
symg  symmetric group  dfsymg 18487 
(SymGrp‘𝐴)  Yes 
symghash 18497, pgrpsubgsymg 18528 
t 
times (see "mul"), for allconstant theorems 
dfmul 10538 
(3 · 2) = 6  Yes 
3t2e6 11791 
th, t 
theorem 


No 
nfth 1803, sbcth 3762, weth 9906, ancomst 468 
tp  triple  dftp 4544 
{𝐴, 𝐵, 𝐶}  Yes 
eltpi 4599, tpeq1 4652 
tr  transitive  
 No  bitrd 282, biantr 805 
tru, t 
true, truth 
dftru 1541 
⊤ 
Yes 
bitru 1547, truanfal 1572, biimt 364 
un  union  dfun 3913 
(𝐴 ∪ 𝐵)  Yes 
uneqri 4102, uncom 4104 
unit  unit (in a ring) 
dfunit 19386  (Unit‘𝑅)  Yes 
isunit 19401, nzrunit 20031 
v 
setvar (especially for specializations of
theorems when a class is replaced by a setvar variable) 

x 
Yes 
cv 1537, vex 3472, velpw 4516, vtoclf 3533 
v 
disjoint variable condition used in place of nonfreeness
hypothesis (suffix) 


No 
spimv 2409 
vtx 
vertex 
dfvtx 26789 
(Vtx‘𝐺) 
Yes 
vtxval0 26830, opvtxov 26796 
vv 
two disjoint variable conditions used in place of nonfreeness
hypotheses (suffix) 


No 
19.23vv 1944 
w  weak (version of a theorem) (suffix)  
 No  ax11w 2134, spnfw 1984 
wrd  word 
dfword 13858  Word 𝑆  Yes 
iswrdb 13863, wrdfn 13871, ffz0iswrd 13884 
xp  cross product (Cartesian product) 
dfxp 5538  (𝐴 × 𝐵)  Yes 
elxp 5555, opelxpi 5569, xpundi 5597 
xr  eXtended reals  dfxr 10668 
ℝ^{*}  Yes  ressxr 10674, rexr 10676, 0xr 10677 
z  integers (from German "Zahlen") 
dfz 11970  ℤ  Yes 
elz 11971, zcn 11974 
zn  ring of integers mod 𝑁  dfzn 20198 
(ℤ/nℤ‘𝑁)  Yes 
znval 20225, zncrng 20234, znhash 20248 
zring  ring of integers  dfzring 20162 
ℤ_{ring}  Yes  zringbas 20167, zringcrng 20163

0, z 
slashed zero (empty set)  dfnul 4266 
∅  Yes 
n0i 4271, vn0 4276; snnz 4685, prnz 4686 
(Contributed by the Metamath team, 27Dec2016.) Date of last revision.
(Revised by the Metamath team, 22Sep2022.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

Theorem  conventionscomments 28185 
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding comments, and more generally nonmathematical
conventions.
For other conventions, see conventions 28183 and links therein.
 Input format.
The input format is ASCII. Tab characters are not allowed. If
nonASCII characters have to be displayed in comments, use embedded
mathematical symbols when they have been defined (e.g., "` > `" for
" →") or HTML entities (e.g., "é" for "é").
Default indentation is by two spaces. Lines are hardwrapped to be at
most 79character long, excluding the newline character (this can be
achieved, except currently for section comments, by the Metamath program
"MM> WRITE SOURCE set.mm / REWRAP" command or by running the script
scripts/rewrap). The file ends with an empty line. There are no
trailing spaces. As for line wrapping in statements, we try to break
lines before the most important token.
 Language and spelling.
The MPE uses American English, e.g., we write "neighborhood" instead of
the British English "neighbourhood". An exception is the word "analog",
which can be either a noun or an adjective (furthermore, "analog" has
the confounding meaning "not digital"); therefore, "analogue" is used
for the noun and "analogous" for the adjective. We favor regular
plurals, e.g., "formulas" instead of "formulae", "lemmas" instead of
"lemmata".
Since comments may contain many spaceseparated symbols, we use the
older convention of two spaces after a period ending a sentence, to
better separate sentences (this is also achieved by the Metamath program
"MM> WRITE SOURCE set.mm / REWRAP" command).
When compound words have several variants, we prefer the concatenated
variant (e.g., nonempty, nontrivial, nonpositive, nonzero,
nonincreasing, nondegenerate...).
 Quotation style.
We use the "logical quotation style", which means that when a quoted
text is followed by punctuation not pertaining to the quote, then the
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single quote also serves as apostrophe), and the single quote in the
case of a nested quotation.
 Sectioning and section headers.
The database set.mm has a sectioning system with four levels of titles,
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Sections of any level are separated by two blank lines (if there is a
"@( Begin $[ ... $] @)" comment (where "@" is actually "$") before a
section header, then the double blank line should go before that
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of section headers is best seen in the source file (set.mm); it is as
follows:
 a line with "@(" (with the "@" replaced by "$");
 a decoration line;
 section title indented with two spaces;
 a (matching) decoration line;
 [blank line; header comment indented with two spaces;
blank line;]
 a line with "@)" (with the "@" replaced by "$");
 one blank line.
As everywhere else, lines are hardwrapped to be 79character long. It
is expected that in a future version, the Metamath program "MM> WRITE
SOURCE set.mm / REWRAP" command will reformat section headers to
automatically conform with this format.
 Comments.
As for formatting of the file set.mm, and in particular formatting and
layout of the comments, the foremost rule is consistency. The first
sections of set.mm, in particular Part 1 "Classical firstorder logic
with equality" can serve as a model for contributors. Some formatting
rules are enforced when using the Metamath program "MM> WRITE SOURCE
set.mm / REWRAP" command. Here are a few other rules, which are
not enforced, but that we try to follow:

A math string in a comment should be surrounded by spaceseparated
backquotes on the same line, and if it is too long it should be broken
into multiple adjacent math strings on multiple lines.

The file set.mm should have a double blank line between sections, and at
no other places. In particular, there are no triple blank lines.

The header comments should be spaced as those of Part 1, namely, with
a blank line before and after the comment, and an indentation of two
spaces.

As of 20Sep2022, section comments are not rewrapped by the Metamath
program "MM> WRITE SOURCE set.mm / REWRAP" command, though this is
expected in a future version. Similar spacing and wrapping should be
used as for other comments: double spaces after a period ending a
sentence, line wrapping with line width of 79, and no trailing spaces
at the end of lines.
 Contributors.
Each assertion (theorem, definition or axiom) has a contribution tag of
the form "(Contributed by xxx, ddMmmyyyy.)" (see Metamath Book,
p. 142). The date cannot serve as a proof of anteriority since there is
currently no formal guarantee that the date is correct (a claim of
anterioty can be backed, for instance, by the uploading of a result to a
public repository with verifiable date). The contributor is the first
person who proved (or stated, in the case of a definition or axiom) the
statement. The list of contributors appears at the beginning of set.mm.
An exception should be made if a theorem is essentially an extract or a
variant of an already existing theorem, in which case the contributor
should be that of the statement from which it is derived, with the
modification signaled by a "(Revised by xxx, ddMmmyyyy.)" tag.
 Usage of parentheticals.
Usually, the comment of a theorem should contain at most one of the
"Revised by" and "Proof shortened by" parentheticals, see Metamath Book,
pp. 142143 (there must always be a "Contributed by" parenthetical for
every theorem). Exceptions for "Proof shortened by" parentheticals
are essential additional shortenings by a different person. If a proof
is shortened by the same person, the date within the "Proof shortened
by" parenthetical should be updated only. This also holds for "Revised
by" parentheticals, except that also more than one of such
parentheticals for the same person are acceptable (if there are good
reasons for this). A revision tag is optionally preceded by a short
description of the revision. Since this is somewhat subjective,
judgment and intellectual honesty should be applied, with collegial
settlement in case of dispute.
 Explaining new labels.
A comment should explain the first use of an abbreviation within a
label. This is often in a definition (e.g., the definition dfan 400
introduces the abbreviation "an" for conjunction ("and")), but not
always (e.g., the theorem alim 1812 introduces the abbreviation "al" for
the universal quantifier ("for all")). See conventionslabels 28184 for a
table of abbreviations.
(Contributed by the Metamath team, 27Dec2016.) Date of last revision.
(Revised by the Metamath team, 22Sep2022.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

17.1.2 Natural deduction


Theorem  natded 28186 
Here are typical natural deduction (ND) rules in the style of Gentzen
and Jaśkowski, along with MPE translations of them. This also
shows the recommended theorems when you find yourself needing these
rules (the recommendations encourage a slightly different proof style
that works more naturally with set.mm). A decent list of the standard
rules of natural deduction can be found beginning with definition /\I in
[Pfenning] p. 18. For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer. Many more citations could be added.
Name  Natural Deduction Rule  Translation 
Recommendation  Comments 
IT 
Γ⊢ 𝜓 => Γ⊢ 𝜓 
idi 1 
nothing  Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. 
∧I 
Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒 
jca 515 
jca 515, pm3.2i 474 
Definition ∧I in [Pfenning] p. 18,
definition I∧m,n in [Clemente] p. 10, and
definition ∧I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{L} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓 
simpld 498 
simpld 498, adantr 484 
Definition ∧E_{L} in [Pfenning] p. 18,
definition E∧(1) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
∧E_{R} 
Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒 
simprd 499 
simpr 488, adantl 485 
Definition ∧E_{R} in [Pfenning] p. 18,
definition E∧(2) in [Clemente] p. 11, and
definition ∧E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and Jaśkowski) 
→I 
Γ, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒 
ex 416  ex 416 
Definition →I in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition →I in [Indrzejczak] p. 33. 
→E 
Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒 
mpd 15  axmp 5, mpd 15, mpdan 686, imp 410 
Definition →E in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition →E in [Indrzejczak] p. 33. 
∨I_{L}  Γ⊢ 𝜓 =>
Γ⊢ 𝜓 ∨ 𝜒 
olcd 871 
olc 865, olci 863, olcd 871 
Definition ∨I in [Pfenning] p. 18,
definition I∨n(1) in [Clemente] p. 12 
∨I_{R}  Γ⊢ 𝜒 =>
Γ⊢ 𝜓 ∨ 𝜒 
orcd 870 
orc 864, orci 862, orcd 870 
Definition ∨I_{R} in [Pfenning] p. 18,
definition I∨n(2) in [Clemente] p. 12. 
∨E  Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 &
Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃 
mpjaodan 956 
mpjaodan 956, jaodan 955, jaod 856 
Definition ∨E in [Pfenning] p. 18,
definition E∨m,n,p in [Clemente] p. 12. 
¬I  Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓 
inegd 1558  pm2.01d 193 

¬I  Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 =>
Γ⊢ ¬ 𝜓 
mtand 815  mtand 815 
definition I¬m,n,p in [Clemente] p. 13. 
¬I  Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 =>
Γ⊢ ¬ 𝜓 
pm2.65da 816  pm2.65da 816 
Contradiction. 
¬I 
Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓 
pm2.01da 798  pm2.01d 193, pm2.65da 816, pm2.65d 199 
For an alternative falsumfree natural deduction ruleset 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥ 
pm2.21fal 1560 
pm2.21dd 198  
¬E 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 

pm2.21dd 198 
definition →E in [Indrzejczak] p. 33. 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃 
pm2.21dd 198  pm2.21dd 198, pm2.21d 121, pm2.21 123 
For an alternative falsumfree natural deduction ruleset.
Definition ¬E in [Pfenning] p. 18. 
⊤I  Γ⊢ ⊤ 
trud 1548  tru 1542, trud 1548, mptru 1545 
Definition ⊤I in [Pfenning] p. 18. 
⊥E  Γ, ⊥⊢ 𝜃 
falimd 1556  falim 1555 
Definition ⊥E in [Pfenning] p. 18. 
∀I 
Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓 
alrimiv 1928  alrimiv 1928, ralrimiva 3174 
Definition ∀I^{a} in [Pfenning] p. 18,
definition I∀n in [Clemente] p. 32. 
∀E 
Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓 
spsbcd 3761  spcv 3581, rspcv 3593 
Definition ∀E in [Pfenning] p. 18,
definition E∀n,t in [Clemente] p. 32. 
∃I 
Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓 
spesbcd 3839  spcev 3582, rspcev 3598 
Definition ∃I in [Pfenning] p. 18,
definition I∃n,t in [Clemente] p. 32. 
∃E 
Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 =>
Γ⊢ 𝜃 
exlimddv 1936  exlimddv 1936, exlimdd 2221,
exlimdv 1934, rexlimdva 3270 
Definition ∃E^{a,u} in [Pfenning] p. 18,
definition E∃m,n,p,a in [Clemente] p. 32. 
⊥C 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 
efald 1559  efald 1559 
Proof by contradiction (classical logic),
definition ⊥C in [Pfenning] p. 17. 
⊥C 
Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓 
pm2.18da 799  pm2.18da 799, pm2.18d 127, pm2.18 128 
For an alternative falsumfree natural deduction ruleset 
¬ ¬C 
Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓 
notnotrd 135  notnotrd 135, notnotr 132 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. 
EM  Γ⊢ 𝜓 ∨ ¬ 𝜓 
exmidd 893  exmid 892 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
=I  Γ⊢ 𝐴 = 𝐴 
eqidd 2823  eqid 2822, eqidd 2823 
Introduce equality,
definition =I in [Pfenning] p. 127. 
=E  Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 =>
Γ⊢ [𝐵 / 𝑥]𝜓 
sbceq1dd 3753  sbceq1d 3752, equality theorems 
Eliminate equality,
definition =E in [Pfenning] p. 127. (Both E1 and E2.) 
Note that MPE uses classical logic, not intuitionist logic. As is
conventional, the "I" rules are introduction rules, "E" rules are
elimination rules, the "C" rules are conversion rules, and Γ
represents the set of (current) hypotheses. We use wff variable names
beginning with 𝜓 to provide a closer representation
of the Metamath
equivalents (which typically use the antedent 𝜑 to represent the
context Γ).
Most of this information was developed by Mario Carneiro and posted on
3Feb2017. For more information, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
For annotated examples where some traditional ND rules
are directly applied in MPE, see exnatded5.2 28187, exnatded5.3 28190,
exnatded5.5 28193, exnatded5.7 28194, exnatded5.8 28196, exnatded5.13 28198,
exnatded9.20 28200, and exnatded9.26 28202.
(Contributed by DAW, 4Feb2017.) (New usage is discouraged.)

⊢ 𝜑 ⇒ ⊢ 𝜑 

17.1.3 Natural deduction examples
These are examples of how natural deduction rules can be applied in Metamath
(both as lineforline translations of ND rules, and as a way to apply
deduction forms without being limited to applying ND rules). For more
information, see natded 28186 and mmnatded.html 28186. Since these examples should
not be used within proofs of other theorems, especially in mathboxes, they
are marked with "(New usage is discouraged.)".


Theorem  exnatded5.2 28187 
Theorem 5.2 of [Clemente] p. 15, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  5  ((𝜓 ∧ 𝜒) → 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) 
Given 
$e. 
2  2  (𝜒 → 𝜓) 
(𝜑 → (𝜒 → 𝜓)) 
Given 
$e. 
3  1  𝜒 
(𝜑 → 𝜒) 
Given 
$e. 
4  3  𝜓 
(𝜑 → 𝜓) 
→E 2,3 
mpd 15, the MPE equivalent of →E, 1,2 
5  4  (𝜓 ∧ 𝜒) 
(𝜑 → (𝜓 ∧ 𝜒)) 
∧I 4,3 
jca 515, the MPE equivalent of ∧I, 3,1 
6  6  𝜃 
(𝜑 → 𝜃) 
→E 1,5 
mpd 15, the MPE equivalent of →E, 4,5 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
Below is the final Metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.22 28188.
A proof without context is shown in exnatded5.2i 28189.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded5.22 28188 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 28187 and exnatded5.2i 28189. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

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

Theorem  exnatded5.2i 28189 
The same as exnatded5.2 28187 and exnatded5.22 28188 but with no context.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ ((𝜓 ∧ 𝜒) → 𝜃)
& ⊢ (𝜒 → 𝜓)
& ⊢ 𝜒 ⇒ ⊢ 𝜃 

Theorem  exnatded5.3 28190 
Theorem 5.3 of [Clemente] p. 16, translated line by line using an
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.32 28191.
A proof without context is shown in exnatded5.3i 28192.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3  (𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 484 to move it into the ND hypothesis 
2  5;6  (𝜒 → 𝜃) 
(𝜑 → (𝜒 → 𝜃)) 
Given 
$e; adantr 484 to move it into the ND hypothesis 
3  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 488, to access the new assumption 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15, the MPE equivalent of →E, 1.3.
adantr 484 was used to transform its dependency
(we could also use imp 410 to get this directly from 1)

5  7  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15, the MPE equivalent of →E, 4,6.
adantr 484 was used to transform its dependency 
6  8  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) 
∧I 4,5 
jca 515, the MPE equivalent of ∧I, 4,7 
7  9  (𝜓 → (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 
→I 3,6 
ex 416, the MPE equivalent of →I, 8 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded5.32 28191 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 28190 and exnatded5.3i 28192. (Contributed by Mario Carneiro,
9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

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

Theorem  exnatded5.3i 28192 
The same as exnatded5.3 28190 and exnatded5.32 28191 but with no context.
Identical to jccir 525, which should be used instead. (Contributed
by
Mario Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜓 → 𝜒)
& ⊢ (𝜒 → 𝜃) ⇒ ⊢ (𝜓 → (𝜒 ∧ 𝜃)) 

Theorem  exnatded5.5 28193 
Theorem 5.5 of [Clemente] p. 18, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  2;3 
(𝜓 → 𝜒) 
(𝜑 → (𝜓 → 𝜒)) 
Given 
$e; adantr 484 to move it into the ND hypothesis 
2  5  ¬ 𝜒 
(𝜑 → ¬ 𝜒)  Given 
$e; we'll use adantr 484 to move it into the ND hypothesis 
3  1 
... 𝜓  ((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 488 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15 1,3 
5  6  ... ¬ 𝜒 
((𝜑 ∧ 𝜓) → ¬ 𝜒) 
IT 2 
adantr 484 5 
6  7  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
∧I 3,4,5 
pm2.65da 816 4,6 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 484; simpr 488 is useful when you want to
depend directly on the new assumption).
Below is the final Metamath proof (which reorders some steps).
A much more efficient proof is mtod 201;
a proof without context is shown in mto 200.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded5.7 28194 
Theorem 5.7 of [Clemente] p. 19, translated line by line using the
interpretation of natural deduction in Metamath.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.72 28195.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer
.
The original proof, which uses Fitch style, was written as follows:
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  6 
(𝜓 ∨ (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) 
Given 
$e. No need for adantr 484 because we do not move this
into an ND hypothesis 
2  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption (new scope) 
simpr 488 
3  2  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) 
∨I_{L} 2 
orcd 870, the MPE equivalent of ∨I_{L}, 1 
4  3  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜒 ∧ 𝜃)) 
ND hypothesis assumption (new scope) 
simpr 488 
5  4  ... 𝜒 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜒) 
∧E_{L} 4 
simpld 498, the MPE equivalent of ∧E_{L}, 3 
6  6  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒)) 
∨I_{R} 5 
olcd 871, the MPE equivalent of ∨I_{R}, 4 
7  7  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
∨E 1,3,6 
mpjaodan 956, the MPE equivalent of ∨E, 2,5,6 
The original used Latin letters for predicates;
we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.72 28195 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 28194. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ ⊢ (𝜑 → (𝜓 ∨ 𝜒)) 

Theorem  exnatded5.8 28196 
Theorem 5.8 of [Clemente] p. 20, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  10;11 
((𝜓 ∧ 𝜒) → ¬ 𝜃) 
(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) 
Given 
$e; adantr 484 to move it into the ND hypothesis 
2  3;4  (𝜏 → 𝜃) 
(𝜑 → (𝜏 → 𝜃))  Given 
$e; adantr 484 to move it into the ND hypothesis 
3  7;8 
𝜒  (𝜑 → 𝜒) 
Given 
$e; adantr 484 to move it into the ND hypothesis 
4  1;2  𝜏  (𝜑 → 𝜏) 
Given 
$e. adantr 484 to move it into the ND hypothesis 
5  6  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND Hypothesis/Assumption 
simpr 488. New ND hypothesis scope, each reference outside
the scope must change antecedent 𝜑 to (𝜑 ∧ 𝜓). 
6  9  ... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜒)) 
∧I 5,3 
jca 515 (∧I), 6,8 (adantr 484 to bring in scope) 
7  5  ... ¬ 𝜃 
((𝜑 ∧ 𝜓) → ¬ 𝜃) 
→E 1,6 
mpd 15 (→E), 2,4 
8  12  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 (→E), 9,11;
note the contradiction with ND line 7 (MPE line 5) 
9  13  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
¬I 5,7,8 
pm2.65da 816 (¬I), 5,12; proof by contradiction.
MPE step 6 (ND#5) does not need a reference here, because
the assumption is embedded in the antecedents 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 484; simpr 488 is useful when you want to
depend directly on the new assumption).
Below is the final Metamath proof (which reorders some steps).
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.82 28197.
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.82 28197 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 28196. (Contributed by Mario
Carneiro, 9Feb2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & ⊢ (𝜑 → (𝜏 → 𝜃)) & ⊢ (𝜑 → 𝜒)
& ⊢ (𝜑 → 𝜏) ⇒ ⊢ (𝜑 → ¬ 𝜓) 

Theorem  exnatded5.13 28198 
Theorem 5.13 of [Clemente] p. 20, translated line by line using the
interpretation of natural deduction in Metamath.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded5.132 28199.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  15  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
Given 
$e. 
2;3  2  (𝜓 → 𝜃) 
(𝜑 → (𝜓 → 𝜃))  Given 
$e. adantr 484 to move it into the ND hypothesis 
3  9  (¬ 𝜏 → ¬ 𝜒) 
(𝜑 → (¬ 𝜏 → ¬ 𝜒)) 
Given 
$e. ad2antrr 725 to move it into the ND subhypothesis 
4  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 488 
5  4  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 1,3 
6  5  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏)) 
∨I 5 
orcd 870 4 
7  6  ... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 488 
8  8  ... ... ¬ 𝜏 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) 
(sub) ND hypothesis assumption 
simpr 488 
9  11  ... ... ¬ 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) 
→E 3,8 
mpd 15 8,10 
10  7  ... ... 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒) 
IT 7 
adantr 484 6 
11  12  ... ¬ ¬ 𝜏 
((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏) 
¬I 8,9,10 
pm2.65da 816 7,11 
12  13  ... 𝜏 
((𝜑 ∧ 𝜒) → 𝜏) 
¬E 11 
notnotrd 135 12 
13  14  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏)) 
∨I 12 
olcd 871 13 
14  16  (𝜃 ∨ 𝜏) 
(𝜑 → (𝜃 ∨ 𝜏)) 
∨E 1,6,13 
mpjaodan 956 5,14,15 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 484; simpr 488 is useful when you want to
depend directly on the new assumption).
(Contributed by Mario Carneiro, 9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded5.132 28199 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 28198. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∨ 𝜒)) & ⊢ (𝜑 → (𝜓 → 𝜃)) & ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒)) ⇒ ⊢ (𝜑 → (𝜃 ∨ 𝜏)) 

Theorem  exnatded9.20 28200 
Theorem 9.20 of [Clemente] p. 43, translated line by line using the
usual translation of natural deduction (ND) in the
Metamath Proof Explorer (MPE) notation.
For information about ND and Metamath, see the
page on Deduction Form and Natural Deduction
in Metamath Proof Explorer.
The original proof, which uses Fitch style, was written as follows
(the leading "..." shows an embedded ND hypothesis, beginning with
the initial assumption of the ND hypothesis):
#  MPE#  ND Expression 
MPE Translation  ND Rationale 
MPE Rationale 
1  1 
(𝜓 ∧ (𝜒 ∨ 𝜃)) 
(𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) 
Given 
$e 
2  2  𝜓 
(𝜑 → 𝜓) 
∧E_{L} 1 
simpld 498 1 
3  11 
(𝜒 ∨ 𝜃) 
(𝜑 → (𝜒 ∨ 𝜃)) 
∧E_{R} 1 
simprd 499 1 
4  4 
... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 488 
5  5 
... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) 
∧I 2,4 
jca 515 3,4 
6  6 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{R} 5 
orcd 870 5 
7  8 
... 𝜃 
((𝜑 ∧ 𝜃) → 𝜃) 
ND hypothesis assumption 
simpr 488 
8  9 
... (𝜓 ∧ 𝜃) 
((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃)) 
∧I 2,7 
jca 515 7,8 
9  10 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{L} 8 
olcd 871 9 
10  12 
((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
(𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨E 3,6,9 
mpjaodan 956 6,10,11 
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath lineforline translation of this
natural deduction approach precedes every line with an antecedent
including 𝜑 and uses the Metamath equivalents
of the natural deduction rules.
To add an assumption, the antecedent is modified to include it
(typically by using adantr 484; simpr 488 is useful when you want to
depend directly on the new assumption).
Below is the final Metamath proof (which reorders some steps).
A much more efficient proof is exnatded9.202 28201.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → (𝜓 ∧ (𝜒 ∨ 𝜃))) ⇒ ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 