Type  Label  Description 
Statement 

Theorem  numclwwlkovh 27801* 
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 27802* 
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 27803 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 27803* 
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)}) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) →
(♯‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)))) 

Theorem  numclwwlk2lem1 27804* 
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 27805* 
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 27806* 
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 27807* 
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  numclwlk2lem2fOLD 27808* 
Obsolete version of numclwlk2lem2f 27805 as of 12Oct2022. (Contributed
by Alexander van der Vekens, 5Oct2018.) (Revised by AV,
31May2021.) (Proof shortened by AV, 23Mar2022.) (Revised by AV,
1May2022.) (New usage is discouraged.)
(Proof modification is discouraged.)

⊢ 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 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁)) 

Theorem  numclwlk2lem2fvOLD 27809* 
Obsolete version of numclwlk2lem2fv 27806 as of 12Oct2022. (Contributed
by Alexander van der Vekens, 6Oct2018.) (Revised by AV,
31May2021.) (Revised by AV, 1May2022.)
(New usage is discouraged.) (Proof modification is discouraged.)

⊢ 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 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩))) 

Theorem  numclwlk2lem2f1oOLD 27810* 
Obsolete version of numclwlk2lem2f1o 27807 as of 12Oct2022.
(Contributed by Alexander van der Vekens, 6Oct2018.) (Revised by
AV, 21Jan2022.) (Proof shortened by AV, 17Mar2022.) (Revised by
AV, 1May2022.) (New usage is discouraged.)
(Proof modification is discouraged.)

⊢ 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 substr ⟨0, (𝑁 + 1)⟩))
⇒ ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–11onto→(𝑋𝑄𝑁)) 

Theorem  numclwwlk2lem3 27811* 
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  numclwwlk2lem3OLD 27812* 
Obsolete proof of numclwwlk2lem3 27811 as of 12Oct2022. (Contributed by
Alexander van der Vekens, 6Oct2018.) (Revised by AV, 31May2021.)
(Proof shortened by AV, 21Jan2022.) (Revised by AV, 1May2022.)
(New usage is discouraged.) (Proof modification is discouraged.)

⊢ 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 27813* 
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 27358, 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}) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_{≥}‘3)))
→ (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))))) 

Theorem  numclwwlk3lem1 27814 
Lemma 2 for numclwwlk3 27817. (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 27815* 
Lemma for numclwwlk3lem2 27816: 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 27816* 
Lemma 1 for numclwwlk3 27817: 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 27817 
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 27784) and with v(n2) =/= v(n) (see
numclwwlk2 27813): 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) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 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 27818* 
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 27819 
Lemma for numclwwlk5 27820. (Contributed by Alexander van der Vekens,
7Oct2018.) (Revised by AV, 2Jun2021.) (Revised by AV,
7Mar2022.)

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

Theorem  numclwwlk5 27820 
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) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)𝑃)) mod 𝑃) = 1) 

Theorem  numclwwlk7lem 27821 
Lemma for numclwwlk7 27823, frgrreggt1 27825 and frgrreg 27826: If a finite,
nonempty friendship graph is 𝐾regular, the 𝐾 is a nonnegative
integer. (Contributed by AV, 3Jun2021.)

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

Theorem  numclwwlk6 27822 
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) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((♯‘𝑉) mod 𝑃)) 

Theorem  numclwwlk7 27823 
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 27748 or frrusgrord 27749, 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 27672, as well as kregular (for any k), see
0vtxrgr 26924, but has no closed walk, see 0clwlk0 27535, 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) ⇒ ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1) 

Theorem  numclwwlk8 27824 
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 27488. (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 27825 
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 ≠ ∅) → ((𝐺RegUSGraph𝐾 ∧ 1 < 𝐾) → 𝐾 = 2)) 

Theorem  frgrreg 27826 
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 ∧ 𝐺RegUSGraph𝐾) → (𝐾 = 0 ∨ 𝐾 = 2))) 

Theorem  frgrregord013 27827 
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 ∧ 𝐺RegUSGraph𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3)) 

Theorem  frgrregord13 27828 
If a nonempty finite friendship graph is 𝐾regular, then it must
have order 1 or 3. Special case of frgrregord013 27827. (Contributed by
Alexander van der Vekens, 9Oct2018.) (Revised by AV, 4Jun2021.)

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

Theorem  frgrogt3nreg 27829* 
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} ¬ 𝐺RegUSGraph𝑘) 

Theorem  friendshipgt3 27830* 
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 27831* 
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 27832 
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 14825) 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 15017).
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 199, dfcleq 2770, dfclel 2774, dfclab 2764)
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. 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 complex arithmetic axiom ax1cn 10330,
proven by the preceding theorem ax1cn 10306.
The metamath.exe 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 presented in
summarized form such as in the Theorem List (click on "Nearby theorems"
on the axmp 5 page), the hypotheses are connected with an ampersand and
separated from the conclusion with a big 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 10330 (not ax1cn 10306) and ax1ne0 10341 (not ax1ne0 10317), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 10306 and ax1ne0 10317 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 axioms and the axiom of choice.
We prefer proofs that depend on fewer and/or weaker axioms,
even if the proofs are longer. In particular, we prefer proofs that do
not use the axiom of choice (dfac 9272) where such proofs can be found.
The axiom of choice is widely accepted, and ZFC is the most
commonlyaccepted fundamental set of axioms for mathematics.
However, there have been and still are some lingering controversies
about the Axiom of Choice. Therefore, where a proof
does not require the axiom of choice, we prefer that proof instead.
E.g., our proof of the SchroederBernstein Theorem (sbth 8368)
does not use the axiom of choice.
In some cases, the weaker axiom of countable choice (axcc 9592)
or axiom of dependent choice (axdc 9603) can be used instead.
Similarly, any theorem in first order logic (FOL) that
contains only set variables that are all mutually distinct,
and has no wff variables, can be proved *without* using
ax10 2135 through ax13 2334, by invoking ax10w 2123 through ax13w 2130.
We encourage proving theorems *without* ax10 2135 through ax13 2334
and moving them up to the ax4 1853 through ax9 2116 section.
 Alternate (ALT) proofs.
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.
 Alternative (ALTV) versions.
If a theorem or definition is an alternate/variant of an
existing theorem or definition, its label should have the same name
with suffix ALTV. Such alternates are often temporary only, until it
is decided which alternative should be used in the future. Alternate
(*ALTV) theorems or definitions are usually contained in mathboxes.
Their comments need not to contain "(Proof modification is discouraged.)
(New usage is discouraged.)". Their comment should begin with
"Alternate version of ~ xxx " followed by a description of the
specificity of the difference from the main variant. Alternate
statements should generally follow the main statement.
 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 8597)
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 222 or mpbir 223. 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 208 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 701, sylbir 227,
or 3imtr4i 284.
 Quantifiers.
The quantifiers are named as follows:
 ∀: universal quantifier (wal 1599);
 ∃: existential quantifier (dfex 1824);
 ∃*: atmostone quantifier (dfmo 2551);
 ∃!: unique existential quantifier (dfeu 2587);
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 2012 and the related dfsbc 3653 and dfcsb 3752.
 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 3400 and isset 3409.
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 3414 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 5046). 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 5363.
This can be used to define a subset, e.g., dftan 15204 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 15282). 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 6143.
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 6925).
For example, the + in (2 + 2); see 2p2e4 11517.
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 837, dfan 387, and dfbi 199 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 2108),
equality 𝐴 = 𝐵 (see dfcleq 2770),
subset 𝐴 ⊆ 𝐵 (see dfss 3806), and
lessthan 𝐴 < 𝐵 (see dflt 10285). For the general definition
of a binary relation in the form 𝐴𝑅𝐵, see dfbr 4887.
For example, 0 < 1 (see 0lt1 10897) 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 10607.
 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 15197).
The function is then defined labeled dfNAME; definitions
are typically given using the mapsto notation (e.g., dfcos 15203).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 15259), its
function application form labeled NAMEval (e.g., cosval 15255),
and at least one simple value (e.g., cos0 15282).
 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 13379 and fac4 13386).
 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 10279), while
"0_{g}" is the group identity element (df0g 16488),
"0." is the poset zero (dfp0 17425),
"0_{𝑝}" is the zero polynomial (df0p 23874),
"0_{vec}" is the zero vector in a normed subcomplex vector space
(df0v 28025), and
"0" is a class variable for use as a connective symbol
(this is used, for example, in p0val 17427).
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 6925, 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 metamath.exe 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 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 10500. Here are the reasons as
discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ:
 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 11375) for the set of positive integers starting
from 1, and ℕ_{0} (dfn0 11643) 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 11846, e.g., ;;;4001 (see 4001prm 16250
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 4010). 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 24216) 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 4012.
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 2779, 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 4011).
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 152 is the
inference associated with con3 151). 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). When
an inference is converted to a theorem by eliminating an "is a set"
hypothesis, we sometimes suffix the closed form with "g" (for "more
general") as in uniex 7230 versus uniexg 7232. 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 5649. Its inference form which was the
original "brres", now is brresi 5651.
 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.
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 4363, which works in certain cases in
set theory. We also sometimes use dedhb 3586. 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; a
list of translations for common natural deduction rules is given in
natded 27835.
 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 7751, rec(𝐹, 𝐼) in dfrdg 7789,
seq_{𝜔}(𝐹, 𝐼) in dfseqom 7826, and seq𝑀( + , 𝐹) in
dfseq 13120. These have characteristic function 𝐹 and initial value
𝐼. (Σ_{g} in dfgsum 16489 isn't really designed for arbitrary
recursion, but you could do it with the right magma.) The logically
primary one is dfrecs 7751, but for the "average user" the most useful
one is probably dfseq 13120 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 16257.
 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 6120 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 6459 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 8358 is the
first lemma for sbth 8368. 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 8358 is the first lemma for sbth 8368.
 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 15791").
When metamath.exe 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 entries) have an anchor in
mmtheorems.html 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 26337. 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. 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.
 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.exe "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.
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 1828 (whose description has some important technical
details). Similarly, Ⅎ𝑥𝐴 is read 𝑥 is not free in (class)
𝐴, see dfnfc 2921.
 "$d x y $." should be read "Assume x and y are distinct
variables."
 "$d x 𝜑 $." should be read "Assume x does not occur in phi $."
Sometimes a theorem is proved using
Ⅎ𝑥𝜑 (dfnf 1828) in place of
"$d 𝑥𝜑 $." when a more general result is desired;
ax5 1953 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2595 from eu6 2592.
 "$d x A $." should be read "Assume x is not a variable occurring in
class A."
 "$d x A $. $d x ps $. $e  (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) $."
is an idiom
often used instead of explicit substitution, meaning "Assume psi results
from the proper substitution of A for x in phi."
 " ⊢ (¬ ∀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 223 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 2368. The section
mmtheorems32.html#mm3146s 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 199, dfclab 2764, dfcleq 2770, and dfclel 2774);
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 $astatement with a given label and
typecode ⊢ passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
 The expression must be a biconditional or an equality (i.e. its
rootsymbol must be ↔ or =).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the rootsymbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "rootsymbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 1070 or wsb 2011).
 The defined expression must not appear in any statement
between its syntax axiom ($a wff ) and its definition,
and the defined expression must not be used in its definiens.
See df3an 1073 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 3399.
The definition dfv 3400 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} is justified
by the justification theorem vjust 3399
⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}.
Another example of a justification theorem is trujust 1603;
the definition dftru 1605 ⊢ (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥))
is justified by trujust 1603 ⊢ ((∀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.
 Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/forum/#!forum/metamath.
(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 27833 
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 27832 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 3104"rgen.1 $e  ( x e. A > ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g. for mdet0 20817: "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 2695 and stirling 41233.
 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 1882, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 3138.
 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... 15016. 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 3795, 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 3806. 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 3960. There are many other syntax label fragments, e.g.,
singleton construct {𝐴} has syntax label fragment "sn" (because it
is defined in dfsn 4399), and the pair construct {𝐴, 𝐵} has
fragment "pr" ( from dfpr 4401). 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 4550. An "n" is often used for negation (¬), e.g.,
nan 820.
 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 10278) and "re" represents real numbers ℝ ( definition dfr 10282).
The empty set ∅ often uses fragment 0, even though it is defined
in dfnul 4142. The syntax construct (𝐴 + 𝐵) usually uses the
fragment "add" (which is consistent with dfadd 10283), 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 11517.
 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 15282 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 15203) we have value cosval 15255 and closure
coscl 15259.
 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 27835 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 (see above).
Additionally, we sometimes suffix with "v" the label of a theorem
eliminating a hypothesis such as Ⅎ𝑥𝜑 in 19.21 2192 via the use of
disjoint variable conditions combined with nfv 1957. If two (or three)
such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used,
e.g. exlimivv 1975.
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 2595 derived from eu6 2592. 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 5149.
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 2376 (cbval 2368 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 3457 or rabeqif 3388.
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 579), commutes
(e.g., biimpac 472)
 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)
 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
 OLD : old/obsolete version of a theorem/definition/proof
 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 470, rexlimiva 3210 
abl  Abelian group  dfabl 18582 
Abel  Yes  ablgrp 18584, zringabl 20218 
abs  absorption    No 
ressabs 16336 
abs  absolute value (of a complex number) 
dfabs 14383  (abs‘𝐴)  Yes 
absval 14385, absneg 14424, abs1 14444 
ad  adding  
 No  adantr 474, ad2antlr 717 
add  add (see "p")  dfadd 10283 
(𝐴 + 𝐵)  Yes 
addcl 10354, addcom 10562, addass 10359 
al  "for all"  
∀𝑥𝜑  No  alim 1854, alex 1869 
ALT  alternative/less preferred (suffix)  
 No  idALT 23 
an  and  dfan 387 
(𝜑 ∧ 𝜓)  Yes 
anor 968, iman 392, imnan 390 
ant  antecedent  
 No  adantr 474 
ass  associative  
 No  biass 376, orass 908, mulass 10360 
asym  asymmetric, antisymmetric  
 No  intasym 5766, asymref 5767, posasymb 17338 
ax  axiom  
 No  ax6dgen 2122, ax1cn 10306 
bas, base 
base (set of an extensible structure)  dfbase 16261 
(Base‘𝑆)  Yes 
baseval 16314, ressbas 16326, cnfldbas 20146 
b, bi  biconditional ("iff", "if and only if")
 dfbi 199  (𝜑 ↔ 𝜓)  Yes 
impbid 204, sspwb 5149 
br  binary relation  dfbr 4887 
𝐴𝑅𝐵  Yes  brab1 4934, brun 4937 
cbv  change bound variable   
No  cbvalivw 2054, cbvrex 3364 
cl  closure    No 
ifclda 4341, ovrcl 6962, zaddcl 11769 
cn  complex numbers  dfc 10278 
ℂ  Yes  nnsscn 11379, nncn 11383 
cnfld  field of complex numbers  dfcnfld 20143 
ℂ_{fld}  Yes  cnfldbas 20146, cnfldinv 20173 
cntz  centralizer  dfcntz 18133 
(Cntz‘𝑀)  Yes 
cntzfval 18136, dprdfcntz 18801 
cnv  converse  dfcnv 5363 
^{◡}𝐴  Yes  opelcnvg 5548, f1ocnv 6403 
co  composition  dfco 5364 
(𝐴 ∘ 𝐵)  Yes  cnvco 5553, fmptco 6661 
com  commutative  
 No  orcom 859, bicomi 216, eqcomi 2787 
con  contradiction, contraposition  
 No  condan 808, con2d 132 
csb  class substitution  dfcsb 3752 
⦋𝐴 / 𝑥⦌𝐵  Yes 
csbid 3759, csbie2g 3782 
cyg  cyclic group  dfcyg 18666 
CycGrp  Yes 
iscyg 18667, zringcyg 20235 
d  deduction form (suffix)  
 No  idd 24, impbid 204 
df  (alternate) definition (prefix)  
 No  dfrel2 5837, dffn2 6293 
di, distr  distributive  
 No 
andi 993, imdi 381, ordi 991, difindi 4108, ndmovdistr 7100 
dif  class difference  dfdif 3795 
(𝐴 ∖ 𝐵)  Yes 
difss 3960, difindi 4108 
div  division  dfdiv 11033 
(𝐴 / 𝐵)  Yes 
divcl 11039, divval 11035, divmul 11036 
dm  domain  dfdm 5365 
dom 𝐴  Yes  dmmpt 5884, iswrddm0 13626 
e, eq, equ  equals  dfcleq 2770 
𝐴 = 𝐵  Yes 
2p2e4 11517, uneqri 3978, equtr 2068 
edg  edge  dfedg 26396 
(Edg‘𝐺)  Yes 
edgopval 26399, usgredgppr 26542 
el  element of  
𝐴 ∈ 𝐵  Yes 
eldif 3802, eldifsn 4550, elssuni 4702 
en  equinumerous  dfen 
𝐴 ≈ 𝐵  Yes  domen 8254, enfi 8464 
eu  "there exists exactly one"  eu6 2592 
∃!𝑥𝜑  Yes  euex 2597, euabsn 4493 
ex  exists (i.e. is a set)  
∈ V  No  brrelex1 5403, 0ex 5026 
ex  "there exists (at least one)"  dfex 1824 
∃𝑥𝜑  Yes  exim 1877, alex 1869 
exp  export  
 No  expt 170, expcom 404 
f  "not free in" (suffix)  
 No  equs45f 2425, sbf 2456 
f  function  dff 6139 
𝐹:𝐴⟶𝐵  Yes  fssxp 6310, opelf 6315 
fal  false  dffal 1615 
⊥  Yes  bifal 1618, falantru 1637 
fi  finite intersection  dffi 8605 
(fi‘𝐵)  Yes  fival 8606, inelfi 8612 
fi, fin  finite  dffin 8245 
Fin  Yes 
isfi 8265, snfi 8326, onfin 8439 
fld  field (Note: there is an alternative
definition Fld of a field, see dffld 34415)  dffield 19142 
Field  Yes  isfld 19148, fldidom 19702 
fn  function with domain  dffn 6138 
𝐴 Fn 𝐵  Yes  ffn 6291, fndm 6235 
frgp  free group  dffrgp 18507 
(freeGrp‘𝐼)  Yes 
frgpval 18557, frgpadd 18562 
fsupp  finitely supported function 
dffsupp 8564  𝑅 finSupp 𝑍  Yes 
isfsupp 8567, fdmfisuppfi 8572, fsuppco 8595 
fun  function  dffun 6137 
Fun 𝐹  Yes  funrel 6152, ffun 6294 
fv  function value  dffv 6143 
(𝐹‘𝐴)  Yes  fvres 6465, swrdfv 13740 
fz  finite set of sequential integers 
dffz 12644 
(𝑀...𝑁)  Yes  fzval 12645, eluzfz 12654 
fz0  finite set of sequential nonnegative integers 

(0...𝑁)  Yes  nn0fz0 12756, fz0tp 12759 
fzo  halfopen integer range  dffzo 12785 
(𝑀..^𝑁)  Yes 
elfzo 12791, elfzofz 12804 
g  more general (suffix); eliminates "is a set"
hypotheses  
 No  uniexg 7232 
gr  graph  
 No  uhgrf 26410, isumgr 26443, usgrres1 26662 
grp  group  dfgrp 17812 
Grp  Yes  isgrp 17815, tgpgrp 22290 
gsum  group sum  dfgsum 16489 
(𝐺 Σ_{g} 𝐹)  Yes 
gsumval 17657, gsumwrev 18179 
hash  size (of a set)  dfhash 13436 
(♯‘𝐴)  Yes 
hashgval 13438, hashfz1 13451, hashcl 13462 
hb  hypothesis builder (prefix)  
 No  hbxfrbi 1868, hbald 2161, hbequid 35063 
hm  (monoid, group, ring) homomorphism  
 No  ismhm 17723, isghm 18044, isrhm 19110 
i  inference (suffix)  
 No  eleq1i 2850, tcsni 8916 
i  implication (suffix)  
 No  brwdomi 8762, infeq5i 8830 
id  identity  
 No  biid 253 
iedg  indexed edge  dfiedg 26347 
(iEdg‘𝐺)  Yes 
iedgval0 26388, edgiedgb 26402 
idm  idempotent  
 No  anidm 560, tpidm13 4523 
im, imp  implication (label often omitted) 
dfim 14248  (𝐴 → 𝐵)  Yes 
iman 392, imnan 390, impbidd 202 
ima  image  dfima 5368 
(𝐴 “ 𝐵)  Yes  resima 5680, imaundi 5799 
imp  import  
 No  biimpa 470, impcom 398 
in  intersection  dfin 3799 
(𝐴 ∩ 𝐵)  Yes  elin 4019, incom 4028 
inf  infimum  dfinf 8637 
inf(ℝ^{+}, ℝ^{*}, < )  Yes 
fiinfcl 8695, infiso 8702 
is...  is (something a) ...?  
 No  isring 18938 
j  joining, disjoining  
 No  jc 161, jaoi 846 
l  left  
 No  olcd 863, simpl 476 
map  mapping operation or set exponentiation 
dfmap 8142  (𝐴 ↑_{𝑚} 𝐵)  Yes 
mapvalg 8150, elmapex 8161 
mat  matrix  dfmat 20618 
(𝑁 Mat 𝑅)  Yes 
matval 20621, matring 20653 
mdet  determinant (of a square matrix) 
dfmdet 20796  (𝑁 maDet 𝑅)  Yes 
mdetleib 20798, mdetrlin 20813 
mgm  magma  dfmgm 17628 
Magma  Yes 
mgmidmo 17645, mgmlrid 17652, ismgm 17629 
mgp  multiplicative group  dfmgp 18877 
(mulGrp‘𝑅)  Yes 
mgpress 18887, ringmgp 18940 
mnd  monoid  dfmnd 17681 
Mnd  Yes  mndass 17688, mndodcong 18345 
mo  "there exists at most one"  dfmo 2551 
∃*𝑥𝜑  Yes  eumo 2598, moim 2556 
mp  modus ponens  axmp 5 
 No  mpd 15, mpi 20 
mpt  modus ponendo tollens  
 No  mptnan 1812, mptxor 1813 
mpt  mapsto notation for a function 
dfmpt 4966  (𝑥 ∈ 𝐴 ↦ 𝐵)  Yes 
fconstmpt 5411, resmpt 5699 
mpt2  mapsto notation for an operation 
dfmpt2 6927  (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)  Yes 
mpt2mpt 7029, resmpt2 7035 
mul  multiplication (see "t")  dfmul 10284 
(𝐴 · 𝐵)  Yes 
mulcl 10356, divmul 11036, mulcom 10358, mulass 10360 
n, not  not  
¬ 𝜑  Yes 
nan 820, notnotr 128 
ne  not equal  dfne  𝐴 ≠ 𝐵 
Yes  exmidne 2979, neeqtrd 3038 
nel  not element of  dfnel  𝐴 ∉ 𝐵

Yes  neli 3077, nnel 3084 
ne0  not equal to zero (see n0)  
≠ 0  No 
negne0d 10732, ine0 10810, gt0ne0 10840 
nf  "not free in" (prefix)  
 No  nfnd 1903 
ngp  normed group  dfngp 22796 
NrmGrp  Yes  isngp 22808, ngptps 22814 
nm  norm (on a group or ring)  dfnm 22795 
(norm‘𝑊)  Yes 
nmval 22802, subgnm 22845 
nn  positive integers  dfnn 11375 
ℕ  Yes  nnsscn 11379, nncn 11383 
nn0  nonnegative integers  dfn0 11643 
ℕ_{0}  Yes  nnnn0 11650, nn0cn 11653 
n0  not the empty set (see ne0)  
≠ ∅  No  n0i 4148, vn0 4153, ssn0 4202 
OLD  old, obsolete (to be removed soon)  
 No  19.43OLD 1930 
on  ordinal number  dfon 5980 
𝐴 ∈ On  Yes 
elon 5985, 1on 7850 onelon 6001 
op  ordered pair  dfop 4405 
⟨𝐴, 𝐵⟩  Yes  dfopif 4633, opth 5176 
or  or  dfor 837 
(𝜑 ∨ 𝜓)  Yes 
orcom 859, anor 968 
ot  ordered triple  dfot 4407 
⟨𝐴, 𝐵, 𝐶⟩  Yes 
euotd 5210, fnotovb 6971 
ov  operation value  dfov 6925 
(𝐴𝐹𝐵)  Yes
 fnotovb 6971, fnovrn 7086 
p  plus (see "add"), for allconstant
theorems  dfadd 10283 
(3 + 2) = 5  Yes 
3p2e5 11533 
pfx  prefix  dfpfx 13780 
(𝑊 prefix 𝐿)  Yes 
pfxlen 13792, ccatpfx 13810 
pm  Principia Mathematica  
 No  pm2.27 42 
pm  partial mapping (operation)  dfpm 8143 
(𝐴 ↑_{pm} 𝐵)  Yes  elpmi 8159, pmsspw 8175 
pr  pair  dfpr 4401 
{𝐴, 𝐵}  Yes 
elpr 4421, prcom 4499, prid1g 4527, prnz 4543 
prm, prime  prime (number)  dfprm 15791 
ℙ  Yes  1nprm 15797, dvdsprime 15805 
pss  proper subset  dfpss 3808 
𝐴 ⊊ 𝐵  Yes  pssss 3924, sspsstri 3931 
q  rational numbers ("quotients")  dfq 12096 
ℚ  Yes  elq 12097 
r  right  
 No  orcd 862, simprl 761 
rab  restricted class abstraction 
dfrab 3099  {𝑥 ∈ 𝐴 ∣ 𝜑}  Yes 
rabswap 3308, dfoprab 6926 
ral  restricted universal quantification 
dfral 3095  ∀𝑥 ∈ 𝐴𝜑  Yes 
ralnex 3174, ralrnmpt2 7052 
rcl  reverse closure  
 No  ndmfvrcl 6477, nnarcl 7980 
re  real numbers  dfr 10282 
ℝ  Yes  recn 10362, 0re 10378 
rel  relation  dfrel 5362  Rel 𝐴 
Yes  brrelex1 5403, relmpt2opab 7540 
res  restriction  dfres 5367 
(𝐴 ↾ 𝐵)  Yes 
opelres 5648, f1ores 6405 
reu  restricted existential uniqueness 
dfreu 3097  ∃!𝑥 ∈ 𝐴𝜑  Yes 
nfreud 3298, reurex 3356 
rex  restricted existential quantification 
dfrex 3096  ∃𝑥 ∈ 𝐴𝜑  Yes 
rexnal 3176, rexrnmpt2 7053 
rmo  restricted "at most one" 
dfrmo 3098  ∃*𝑥 ∈ 𝐴𝜑  Yes 
nfrmod 3299, nrexrmo 3359 
rn  range  dfrn 5366  ran 𝐴 
Yes  elrng 5559, rncnvcnv 5594 
rng  (unital) ring  dfring 18936 
Ring  Yes 
ringidval 18890, isring 18938, ringgrp 18939 
rot  rotation  
 No  3anrot 1085, 3orrot 1076 
s  eliminates need for syllogism (suffix) 
  No  ancoms 452 
sb  (proper) substitution (of a set) 
dfsb 2012  [𝑦 / 𝑥]𝜑  Yes 
spsbe 2015, sbimi 2017 
sbc  (proper) substitution of a class 
dfsbc 3653  [𝐴 / 𝑥]𝜑  Yes 
sbc2or 3661, sbcth 3667 
sca  scalar  dfsca 16354 
(Scalar‘𝐻)  Yes 
resssca 16423, mgpsca 18883 
simp  simple, simplification  
 No  simpl 476, simp3r3 1339 
sn  singleton  dfsn 4399 
{𝐴}  Yes  eldifsn 4550 
sp  specialization  
 No  spsbe 2015, spei 2359 
ss  subset  dfss 3806 
𝐴 ⊆ 𝐵  Yes  difss 3960 
struct  structure  dfstruct 16257 
Struct  Yes  brstruct 16264, structfn 16272 
sub  subtract  dfsub 10608 
(𝐴 − 𝐵)  Yes 
subval 10613, subaddi 10710 
sup  supremum  dfsup 8636 
sup(𝐴, 𝐵, < )  Yes 
fisupcl 8663, supmo 8646 
supp  support (of a function)  dfsupp 7577 
(𝐹 supp 𝑍)  Yes 
ressuppfi 8589, mptsuppd 7599 
swap  swap (two parts within a theorem) 
  No  rabswap 3308, 2reuswap 3624 
syl  syllogism  syl 17 
 No  3syl 18 
sym  symmetric  
 No  dfsymdif 4067, cnvsym 5765 
symg  symmetric group  dfsymg 18181 
(SymGrp‘𝐴)  Yes 
symghash 18188, pgrpsubgsymg 18211 
t 
times (see "mul"), for allconstant theorems 
dfmul 10284 
(3 · 2) = 6  Yes 
3t2e6 11548 
th  theorem  
 No  nfth 1845, sbcth 3667, weth 9652 
tp  triple  dftp 4403 
{𝐴, 𝐵, 𝐶}  Yes 
eltpi 4456, tpeq1 4509 
tr  transitive  
 No  bitrd 271, biantr 796 
tru  true  dftru 1605 
⊤  Yes  bitru 1611, truanfal 1636 
un  union  dfun 3797 
(𝐴 ∪ 𝐵)  Yes 
uneqri 3978, uncom 3980 
unit  unit (in a ring) 
dfunit 19029  (Unit‘𝑅)  Yes 
isunit 19044, nzrunit 19664 
v  disjoint variable conditions used when
a notfree hypothesis (suffix) 
  No  spimv 2355 
vtx  vertex  dfvtx 26346 
(Vtx‘𝐺)  Yes 
vtxval0 26387, opvtxov 26353 
vv  2 disjoint variables (in a notfree hypothesis)
(suffix)    No  19.23vv 1986 
w  weak (version of a theorem) (suffix)  
 No  ax11w 2124, spnfw 2046 
wrd  word 
dfword 13600  Word 𝑆  Yes 
iswrdb 13605, wrdfn 13614, ffz0iswrd 13629 
xp  cross product (Cartesian product) 
dfxp 5361  (𝐴 × 𝐵)  Yes 
elxp 5378, opelxpi 5392, xpundi 5417 
xr  eXtended reals  dfxr 10415 
ℝ^{*}  Yes  ressxr 10420, rexr 10422, 0xr 10423 
z  integers (from German "Zahlen") 
dfz 11729  ℤ  Yes 
elz 11730, zcn 11733 
zn  ring of integers mod 𝑁  dfzn 20251 
(ℤ/nℤ‘𝑁)  Yes 
znval 20279, zncrng 20288, znhash 20302 
zring  ring of integers  dfzring 20215 
ℤ_{ring}  Yes  zringbas 20220, zringcrng 20216

0, z 
slashed zero (empty set)  dfnul 4142 
∅  Yes 
n0i 4148, vn0 4153; snnz 4542, prnz 4543 
(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 27834 
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 27832 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
quotation mark precedes the punctuation (like at the beginning of this
sentence). We use the double quote as default quotation mark (since the
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,
signaled by "decoration lines" which are 79character long repetitions
of ####, #*#*, ==, and .. (in descending order of sectioning level).
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
comment, which is considered as belonging to that section). The format
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. As with
all revision tags, it should be 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.
 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).
 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 387
introduces the abbreviation "an" for conjunction ("and")), but not
always (e.g., the theorem alim 1854 introduces the abbreviation "al" for
the universal quantifier ("for all")). See conventionslabels 27833 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 27835 
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 2 
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 507 
jca 507, pm3.2i 464 
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 490 
simpld 490, adantr 474 
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 491 
simpr 479, adantl 475 
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 403  ex 403 
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 677, imp 397 
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 863 
olc 857, olci 855, olcd 863 
Definition ∨I in [Pfenning] p. 18,
definition I∨n(1) in [Clemente] p. 12 
∨I_{R}  Γ⊢ 𝜒 =>
Γ⊢ 𝜓 ∨ 𝜒 
orcd 862 
orc 856, orci 854, orcd 862 
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 944 
mpjaodan 944, jaodan 943, jaod 848 
Definition ∨E in [Pfenning] p. 18,
definition E∨m,n,p in [Clemente] p. 12. 
¬I  Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓 
inegd 1622  pm2.01d 182 

¬I  Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 =>
Γ⊢ ¬ 𝜓 
mtand 806  mtand 806 
definition I¬m,n,p in [Clemente] p. 13. 
¬I  Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 =>
Γ⊢ ¬ 𝜓 
pm2.65da 807  pm2.65da 807 
Contradiction. 
¬I 
Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓 
pm2.01da 789  pm2.01d 182, pm2.65da 807, pm2.65d 188 
For an alternative falsumfree natural deduction ruleset 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥ 
pm2.21fal 1624 
pm2.21dd 187  
¬E 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 

pm2.21dd 187 
definition →E in [Indrzejczak] p. 33. 
¬E 
Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃 
pm2.21dd 187  pm2.21dd 187, pm2.21d 119, pm2.21 121 
For an alternative falsumfree natural deduction ruleset.
Definition ¬E in [Pfenning] p. 18. 
⊤I  Γ⊢ ⊤ 
trud 1612  tru 1606, trud 1612, mptru 1609 
Definition ⊤I in [Pfenning] p. 18. 
⊥E  Γ, ⊥⊢ 𝜃 
falimd 1620  falim 1619 
Definition ⊥E in [Pfenning] p. 18. 
∀I 
Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓 
alrimiv 1970  alrimiv 1970, ralrimiva 3148 
Definition ∀I^{a} in [Pfenning] p. 18,
definition I∀n in [Clemente] p. 32. 
∀E 
Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓 
spsbcd 3666  spcv 3501, rspcv 3507 
Definition ∀E in [Pfenning] p. 18,
definition E∀n,t in [Clemente] p. 32. 
∃I 
Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓 
spesbcd 3739  spcev 3502, rspcev 3511 
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 1978  exlimddv 1978, exlimdd 2205,
exlimdv 1976, rexlimdva 3213 
Definition ∃E^{a,u} in [Pfenning] p. 18,
definition E∃m,n,p,a in [Clemente] p. 32. 
⊥C 
Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓 
efald 1623  efald 1623 
Proof by contradiction (classical logic),
definition ⊥C in [Pfenning] p. 17. 
⊥C 
Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓 
pm2.18da 790  pm2.18da 790, pm2.18d 127, pm2.18 125 
For an alternative falsumfree natural deduction ruleset 
¬ ¬C 
Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓 
notnotrd 131  notnotrd 131, notnotr 128 
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. 
EM  Γ⊢ 𝜓 ∨ ¬ 𝜓 
exmidd 882  exmid 881 
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. 
=I  Γ⊢ 𝐴 = 𝐴 
eqidd 2779  eqid 2778, eqidd 2779 
Introduce equality,
definition =I in [Pfenning] p. 127. 
=E  Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 =>
Γ⊢ [𝐵 / 𝑥]𝜓 
sbceq1dd 3658  sbceq1d 3657, 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 27836, exnatded5.3 27839,
exnatded5.5 27842, exnatded5.7 27843, exnatded5.8 27845, exnatded5.13 27847,
exnatded9.20 27849, and exnatded9.26 27851.
(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 27835 and mmnatded.html 27835. 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 27836 
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 507, 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 27837.
A proof without context is shown in exnatded5.2i 27838.
(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 27837 
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
exnatded5.2 27836 and exnatded5.2i 27838. (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 27838 
The same as exnatded5.2 27836 and exnatded5.22 27837 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 27839 
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 27840.
A proof without context is shown in exnatded5.3i 27841.
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 474 to move it into the ND hypothesis 
2  5;6  (𝜒 → 𝜃) 
(𝜑 → (𝜒 → 𝜃)) 
Given 
$e; adantr 474 to move it into the ND hypothesis 
3  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 479, to access the new assumption 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15, the MPE equivalent of →E, 1.3.
adantr 474 was used to transform its dependency
(we could also use imp 397 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 474 was used to transform its dependency 
6  8  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) 
∧I 4,5 
jca 507, the MPE equivalent of ∧I, 4,7 
7  9  (𝜓 → (𝜒 ∧ 𝜃)) 
(𝜑 → (𝜓 → (𝜒 ∧ 𝜃))) 
→I 3,6 
ex 403, 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 27840 
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
exnatded5.3 27839 and exnatded5.3i 27841. (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 27841 
The same as exnatded5.3 27839 and exnatded5.32 27840 but with no context.
Identical to jccir 517, 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 27842 
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 474 to move it into the ND hypothesis 
2  5  ¬ 𝜒 
(𝜑 → ¬ 𝜒)  Given 
$e; we'll use adantr 474 to move it into the ND hypothesis 
3  1 
... 𝜓  ((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 479 
4  4  ... 𝜒 
((𝜑 ∧ 𝜓) → 𝜒) 
→E 1,3 
mpd 15 1,3 
5  6  ... ¬ 𝜒 
((𝜑 ∧ 𝜓) → ¬ 𝜒) 
IT 2 
adantr 474 5 
6  7  ¬ 𝜓 
(𝜑 → ¬ 𝜓) 
∧I 3,4,5 
pm2.65da 807 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 474; simpr 479 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 190;
a proof without context is shown in mto 189.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded5.7 27843 
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 27844.
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 474 because we do not move this
into an ND hypothesis 
2  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption (new scope) 
simpr 479 
3  2  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒)) 
∨I_{L} 2 
orcd 862, the MPE equivalent of ∨I_{L}, 1 
4  3  ... (𝜒 ∧ 𝜃) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜒 ∧ 𝜃)) 
ND hypothesis assumption (new scope) 
simpr 479 
5  4  ... 𝜒 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜒) 
∧E_{L} 4 
simpld 490, the MPE equivalent of ∧E_{L}, 3 
6  6  ... (𝜓 ∨ 𝜒) 
((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒)) 
∨I_{R} 5 
olcd 863, the MPE equivalent of ∨I_{R}, 4 
7  7  (𝜓 ∨ 𝜒) 
(𝜑 → (𝜓 ∨ 𝜒)) 
∨E 1,3,6 
mpjaodan 944, 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 27844 
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
exnatded5.7 27843. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded5.8 27845 
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 474 to move it into the ND hypothesis 
2  3;4  (𝜏 → 𝜃) 
(𝜑 → (𝜏 → 𝜃))  Given 
$e; adantr 474 to move it into the ND hypothesis 
3  7;8 
𝜒  (𝜑 → 𝜒) 
Given 
$e; adantr 474 to move it into the ND hypothesis 
4  1;2  𝜏  (𝜑 → 𝜏) 
Given 
$e. adantr 474 to move it into the ND hypothesis 
5  6  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND Hypothesis/Assumption 
simpr 479. 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 507 (∧I), 6,8 (adantr 474 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 807 (¬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 474; simpr 479 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 27846.
(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 27846 
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
linebyline translation, see exnatded5.8 27845. (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 27847 
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 27848.
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 474 to move it into the ND hypothesis 
3  9  (¬ 𝜏 → ¬ 𝜒) 
(𝜑 → (¬ 𝜏 → ¬ 𝜒)) 
Given 
$e. ad2antrr 716 to move it into the ND subhypothesis 
4  1  ... 𝜓 
((𝜑 ∧ 𝜓) → 𝜓) 
ND hypothesis assumption 
simpr 479 
5  4  ... 𝜃 
((𝜑 ∧ 𝜓) → 𝜃) 
→E 2,4 
mpd 15 1,3 
6  5  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏)) 
∨I 5 
orcd 862 4 
7  6  ... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 479 
8  8  ... ... ¬ 𝜏 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) 
(sub) ND hypothesis assumption 
simpr 479 
9  11  ... ... ¬ 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) 
→E 3,8 
mpd 15 8,10 
10  7  ... ... 𝜒 
(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒) 
IT 7 
adantr 474 6 
11  12  ... ¬ ¬ 𝜏 
((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏) 
¬I 8,9,10 
pm2.65da 807 7,11 
12  13  ... 𝜏 
((𝜑 ∧ 𝜒) → 𝜏) 
¬E 11 
notnotrd 131 12 
13  14  ... (𝜃 ∨ 𝜏) 
((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏)) 
∨I 12 
olcd 863 13 
14  16  (𝜃 ∨ 𝜏) 
(𝜑 → (𝜃 ∨ 𝜏)) 
∨E 1,6,13 
mpjaodan 944 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 474; simpr 479 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 27848 
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with exnatded5.13 27847. (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 27849 
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 490 1 
3  11 
(𝜒 ∨ 𝜃) 
(𝜑 → (𝜒 ∨ 𝜃)) 
∧E_{R} 1 
simprd 491 1 
4  4 
... 𝜒 
((𝜑 ∧ 𝜒) → 𝜒) 
ND hypothesis assumption 
simpr 479 
5  5 
... (𝜓 ∧ 𝜒) 
((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)) 
∧I 2,4 
jca 507 3,4 
6  6 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜒) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{R} 5 
orcd 862 5 
7  8 
... 𝜃 
((𝜑 ∧ 𝜃) → 𝜃) 
ND hypothesis assumption 
simpr 479 
8  9 
... (𝜓 ∧ 𝜃) 
((𝜑 ∧ 𝜃) → (𝜓 ∧ 𝜃)) 
∧I 2,7 
jca 507 7,8 
9  10 
... ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
((𝜑 ∧ 𝜃) → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨I_{L} 8 
olcd 863 9 
10  12 
((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃)) 
(𝜑 → ((𝜓 ∧ 𝜒) ∨ (𝜓 ∧ 𝜃))) 
∨E 3,6,9 
mpjaodan 944 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 474; simpr 479 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 27850.
(Contributed by David A. Wheeler, 19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded9.202 27850 
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with exnatded9.20 27849. (Contributed by David A. Wheeler,
19Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem  exnatded9.26 27851* 
Theorem 9.26 of [Clemente] p. 45, translated line by line using an
interpretation of natural deduction in Metamath. This proof has some
additional complications due to the fact that Metamath's existential
elimination rule does not change bound variables, so we need to verify
that 𝑥 is bound in the conclusion.
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  3  ∃𝑥∀𝑦𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥∀𝑦𝜓) 
Given 
$e. 
2  6  ... ∀𝑦𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) 
ND hypothesis assumption 
simpr 479. Later statements will have this scope. 
3  7;5,4  ... 𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → 𝜓) 
∀E 2,y 
spsbcd 3666 (∀E), 5,6. To use it we need a1i 11 and vex 3401.
This could be immediately done with 19.21bi 2173, but we want to show
the general approach for substitution.

4  12;8,9,10,11  ... ∃𝑥𝜓(𝑥, 𝑦) 
((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) 
∃I 3,a 
spesbcd 3739 (∃I), 11.
To use it we need sylibr 226, which in turn requires sylib 210 and
two uses of sbcid 3669.
This could be more immediately done using 19.8a 2166, but we want to show
the general approach for substitution.

5  13;1,2  ∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∃𝑥𝜓)  ∃E 1,2,4,a 
exlimdd 2205 (∃E), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1957 and nfe1 2144 (MPE# 1,2) 
6  14  ∀𝑦∃𝑥𝜓(𝑥, 𝑦) 
(𝜑 → ∀𝑦∃𝑥𝜓) 
∀I 5 
alrimiv 1970 (∀I), 13 
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).
Note that in the original proof, 𝜓(𝑥, 𝑦) has explicit
parameters. In Metamath, these parameters are always implicit, and the
parameters upon which a wff variable can depend are recorded in the
"allowed substitution hints" below.
A much more efficient proof, using more of Metamath and MPE's
capabilities, is shown in exnatded9.262 27852.
(Contributed by Mario Carneiro, 9Feb2017.)
(Revised by David A. Wheeler, 18Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

Theorem  exnatded9.262 27852* 
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with exnatded9.26 27851. (Contributed by Mario Carneiro,
9Feb2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝜑 → ∃𝑥∀𝑦𝜓) ⇒ ⊢ (𝜑 → ∀𝑦∃𝑥𝜓) 

17.1.4 Definitional examples


Theorem  exor 27853 
Example for dfor 837. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 3 ∨ 4 = 4) 

Theorem  exan 27854 
Example for dfan 387. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9May2015.)

⊢ (2 = 2 ∧ 3 = 3) 

Theorem  exdif 27855 
Example for dfdif 3795. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ({1, 3} ∖ {1, 8}) =
{3} 

Theorem  exun 27856 
Example for dfun 3797. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∪ {1, 8}) = {1, 3,
8} 

Theorem  exin 27857 
Example for dfin 3799. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ ({1, 3} ∩ {1, 8}) = {1} 

Theorem  exuni 27858 
Example for dfuni 4672. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2Jul2016.)

⊢ ∪ {{1, 3}, {1,
8}} = {1, 3, 8} 

Theorem  exss 27859 
Example for dfss 3806. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ {1, 2} ⊆ {1, 2, 3} 

Theorem  expss 27860 
Example for dfpss 3808. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ {1, 2} ⊊ {1, 2, 3} 

Theorem  expw 27861 
Example for dfpw 4381. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2Jul2016.)

⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5},
{7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))) 

Theorem  expr 27862 
Example for dfpr 4401. (Contributed by Mario Carneiro,
7May2015.)

⊢ (𝐴 ∈ {1, 1} → (𝐴↑2) = 1) 

Theorem  exbr 27863 
Example for dfbr 4887. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6May2015.)

⊢ (𝑅 = {⟨2, 6⟩, ⟨3, 9⟩}
→ 3𝑅9) 

Theorem  exopab 27864* 
Example for dfopab 4949. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4) 

Theorem  exeprel 27865 
Example for dfeprel 5266. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ 5 E {1, 5} 

Theorem  exid 27866 
Example for dfid 5261. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ (5 I 5 ∧ ¬ 4 I 5) 

Theorem  expo 27867 
Example for dfpo 5274. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

⊢ ( < Po ℝ ∧ ¬ ≤ Po
ℝ) 

Theorem  exxp 27868 
Example for dfxp 5361. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ({1, 5} × {2, 7}) = ({⟨1,
2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5,
7⟩}) 

Theorem  excnv 27869 
Example for dfcnv 5363. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6May2015.)

⊢ ^{◡}{⟨2, 6⟩, ⟨3, 9⟩} =
{⟨6, 2⟩, ⟨9, 3⟩} 

Theorem  exco 27870 
Example for dfco 5364. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ ((exp ∘ cos)‘0) =
e 

Theorem  exdm 27871 
Example for dfdm 5365. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ dom 𝐹 = {2,
3}) 

Theorem  exrn 27872 
Example for dfrn 5366. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ ran 𝐹 = {6,
9}) 

Theorem  exres 27873 
Example for dfres 5367. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 ↾ 𝐵) = {⟨2,
6⟩}) 

Theorem  exima 27874 
Example for dfima 5368. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7May2015.)

⊢ ((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
∧ 𝐵 = {1, 2}) →
(𝐹 “ 𝐵) = {6}) 

Theorem  exfv 27875 
Example for dffv 6143. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7May2015.)

⊢ (𝐹 = {⟨2, 6⟩, ⟨3, 9⟩}
→ (𝐹‘3) =
9) 

Theorem  ex1st 27876 
Example for df1st 7445. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (1^{st} ‘⟨3, 4⟩) =
3 

Theorem  ex2nd 27877 
Example for df2nd 7446. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18Jun2015.)

⊢ (2^{nd} ‘⟨3, 4⟩) =
4 

Theorem  1kp2ke3k 27878 
Example for dfdec 11846, 1000 + 2000 = 3000.
This proof disproves (by counterexample) the assertion of Hao Wang, who
stated, "There is a theorem in the primitive notation of set theory
that
corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula
would be forbiddingly long... even if (one) knows the definitions and is
asked to simplify the long formula according to them, chances are he will
make errors and arrive at some incorrect result." (Hao Wang,
"Theory and
practice in mathematics" , In Thomas Tymoczko, editor, New
Directions in
the Philosophy of Mathematics, pp 129152, Birkauser Boston, Inc.,
Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)
This is noted in Metamath: A Computer Language for Pure
Mathematics by
Norman Megill (2007) section 1.1.3. Megill then states, "A number of
writers have conveyed the impression that the kind of absolute rigor
provided by Metamath is an impossible dream, suggesting that a complete,
formal verification of a typical theorem would take millions of steps in
untold volumes of books... These writers assume, however, that in order
to achieve the kind of complete formal verification they desire one must
break down a proof into individual primitive steps that make direct
reference to the axioms. This is not necessary. There is no reason not
to make use of previously proved theorems rather than proving them over
and over... A hierarchy of theorems and definitions permits an
exponential growth in the formula sizes and primitive proof steps to be
described with only a linear growth in the number of symbols used. Of
course, this is how ordinary informal mathematics is normally done anyway,
but with Metamath it can be done with absolute rigor and precision."
The proof here starts with (2 + 1) = 3, commutes
it, and repeatedly
multiplies both sides by ten. This is certainly longer than traditional
mathematical proofs, e.g., there are a number of steps explicitly shown
here to show that we're allowed to do operations such as multiplication.
However, while longer, the proof is clearly a manageable size  even
though every step is rigorously derived all the way back to the primitive
notions of set theory and logic. And while there's a risk of making
errors, the many independent verifiers make it much less likely that an
incorrect result will be accepted.
This proof heavily relies on the decimal constructor dfdec 11846 developed by
Mario Carneiro in 2015. The underlying Metamath language has an
intentionally very small set of primitives; it doesn't even have a
builtin construct for numbers. Instead, the digits are defined using
these primitives, and the decimal constructor is used to make it easy to
express larger numbers as combinations of digits.
(Contributed by David A. Wheeler, 29Jun2016.) (Shortened by Mario
Carneiro using the arithmetic algorithm in mmj2, 30Jun2016.)

⊢ (;;;1000 + ;;;2000) = ;;;3000 

Theorem  exfl 27879 
Example for dffl 12912. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18Jun2015.)

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

Theorem  exceil 27880 
Example for dfceil 12913. (Contributed by AV, 4Sep2021.)

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

Theorem  exmod 27881 
Example for dfmod 12988. (Contributed by AV, 3Sep2021.)

⊢ ((5 mod 3) = 2 ∧ (7 mod 2) =
1) 

Theorem  exexp 27882 
Example for dfexp 13179. (Contributed by AV, 4Sep2021.)

⊢ ((5↑2) = ;25 ∧ (3↑2) = (1 / 9)) 

Theorem  exfac 27883 
Example for dffac 13379. (Contributed by AV, 4Sep2021.)

⊢ (!‘5) = ;;120 

Theorem  exbc 27884 
Example for dfbc 13408. (Contributed by AV, 4Sep2021.)

⊢ (5C3) = ;10 

Theorem  exhash 27885 
Example for dfhash 13436. (Contributed by AV, 4Sep2021.)

⊢ (♯‘{0, 1, 2}) =
3 

Theorem  exsqrt 27886 
Example for dfsqrt 14382. (Contributed by AV, 4Sep2021.)

⊢ (√‘;25) = 5 

Theorem  exabs 27887 
Example for dfabs 14383. (Contributed by AV, 4Sep2021.)

⊢ (abs‘2) = 2 

Theorem  exdvds 27888 
Example for dfdvds 15388: 3 divides into 6. (Contributed by David A.
Wheeler, 19May2015.)

⊢ 3 ∥ 6 

Theorem  exgcd 27889 
Example for dfgcd 15623. (Contributed by AV, 5Sep2021.)

⊢ (6 gcd 9) = 3 

Theorem  exlcm 27890 
Example for dflcm 15709. (Contributed by AV, 5Sep2021.)

⊢ (6 lcm 9) = ;18 

Theorem  exprmo 27891 
Example for dfprmo 16140: (#_{p}‘10) =
2 · 3 · 5 · 7.
(Contributed by AV, 6Sep2021.)

⊢ (#_{p}‘;10) = ;;210 

17.1.5 Other examples


Theorem  aevdemo 27892* 
Proof illustrating the comment of aev2 2105. (Contributed by BJ,
30Mar2021.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (∀𝑥 𝑥 = 𝑦 → ((∃𝑎∀𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀ℎ(𝑖 = 𝑗 → 𝑘 = 𝑙))) 

Theorem  exinddvds 27893 
Example of a proof by induction (divisibility result). (Contributed by
Stanislas Polu, 9Mar2020.) (Revised by BJ, 24Mar2020.)

⊢ (𝑁 ∈ ℕ_{0} → 3
∥ ((4↑𝑁) +
2)) 

17.2 Humor


17.2.1 April Fool's theorem


Theorem  avril1 27894 
Poisson d'Avril's Theorem. This theorem is noted for its
Selbstdokumentieren property, which means, literally,
"selfdocumenting" and recalls the principle of quidquid
german dictum
sit, altum viditur, often used in set theory. Starting with the
seemingly simple yet profound fact that any object 𝑥 equals
itself
(proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we
demonstrate that the power set of the real numbers, as a relation on the
value of the imaginary unit, does not conjoin with an empty relation on
the product of the additive and multiplicative identity elements,
leading to this startling conclusion that has left even seasoned
professional mathematicians scratching their heads. (Contributed by
Prof. Loof Lirpa, 1Apr2005.) (Proof modification is discouraged.)
(New usage is discouraged.)
A reply to skeptics can be found at mmnotes.txt, under the
1Apr2006 entry.

⊢ ¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 ·
1)) 

Theorem  2bornot2b 27895 
The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet,
Prince of Denmark (1602). Its author leaves its proof as an exercise
for
the reader  "To be, or not to be: that is the question" 
starting a
trend that has become standard in modernday textbooks, serving to make
the frustrated reader feel inferior, or in some cases to mask the fact
that the author does not know its solution. (Contributed by Prof. Loof
Lirpa, 1Apr2006.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ (2 · 𝐵 ∨ ¬ 2 · 𝐵) 

Theorem  helloworld 27896 
The classic "Hello world" benchmark has been translated into 314
computer
programming languages  see http://www.roeslerac.de/wolfram/hello.htm.
However, for many years it eluded a proof that it is more than just a
conjecture, even though a wily mathematician once claimed, "I have
discovered a truly marvelous proof of this, which this margin is too
narrow to contain." Using an IBM 709 mainframe, a team of
mathematicians
led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able
put it rest with a remarkably short proof only 4 lines long. (Contributed
by Prof. Loof Lirpa, 1Apr2007.) (Proof modification is discouraged.)
(New usage is discouraged.)

⊢ ¬ (ℎ ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑)) 

Theorem  1p1e2apr1 27897 
One plus one equals two. Using proofshortening techniques pioneered by
Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof.
Loof Lirpa and colleagues were able to shorten Whitehead and Russell's
360page proof that 1+1=2 in Principia Mathematica to this
remarkable
proof only two steps long, thus establishing a new world's record for this
famous theorem. (Contributed by Prof. Loof Lirpa, 1Apr2008.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (1 + 1) = 2 

Theorem  eqid1 27898 
Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine]
p. 41.
This law is thought to have originated with Aristotle
(Metaphysics,
Book VII, Part 17). It is one of the three axioms of Ayn Rand's
philosophy (Atlas Shrugged, Part Three, Chapter VII). While some
have
proposed extending Rand's axiomatization to include Compassion and
Kindness, others fear that such an extension may flirt with logical
inconsistency. (Contributed by Stefan Allan, 1Apr2009.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ 𝐴 = 𝐴 

Theorem  1div0apr 27899 
Division by zero is forbidden! If we try, we encounter the DO NOT ENTER
sign, which in mathematics means it is foolhardy to venture any further,
possibly putting the underlying fabric of reality at risk. Based on a
dare by David A. Wheeler. (Contributed by Mario Carneiro, 1Apr2014.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (1 / 0) = ∅ 

Theorem  topnfbey 27900 
Nothing seems to be impossible to Prof. Lirpa. After years of intensive
research, he managed to find a proof that when given a chance to reach
infinity, one could indeed go beyond, thus giving formal soundness to Buzz
Lightyear's motto "To infinity... and beyond!" (Contributed by
Prof.
Loof Lirpa, 1Apr2020.) (Revised by Thierry Arnoux, 2Aug2020.)
(Proof modification is discouraged.) (New usage is discouraged.)

⊢ (𝐵 ∈ (0...+∞) → +∞ <
𝐵) 