Type | Label | Description |
Statement |
|
Theorem | numclwlk1 29101* |
Statement 9 in [Huneke] p. 2: "If n >
1, then the number of closed
n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v
is kf(n-2)". Since πΊ is k-regular, the vertex v(n-2) = v
has k
neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via
each of v's neighbors) completing each of the f(n-2) walks from v=v(0)
to v(n-2)=v. This theorem holds even for k=0. (Contributed by AV,
23-May-2022.)
|
β’ π = (VtxβπΊ)
& β’ πΆ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st
βπ€)) = π β§ ((2nd
βπ€)β0) = π β§ ((2nd
βπ€)β(π β 2)) = π)} & β’ πΉ = {π€ β (ClWalksβπΊ) β£ ((β―β(1st
βπ€)) = (π β 2) β§
((2nd βπ€)β0) = π)} β β’ (((π β Fin β§ πΊ RegUSGraph πΎ) β§ (π β π β§ π β (β€β₯β2)))
β (β―βπΆ) =
(πΎ Β·
(β―βπΉ))) |
|
Theorem | numclwwlkovh0 29102* |
Value of operation π», mapping a vertex π£ and an
integer π
greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n)
from v
= v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in
[Huneke] p. 2. (Contributed by AV,
1-May-2022.)
|
β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ ((π β π β§ π β (β€β₯β2))
β (ππ»π) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π}) |
|
Theorem | numclwwlkovh 29103* |
Value of operation π», mapping a vertex π£ and an
integer π
greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n)
from v
= v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in
[Huneke] p. 2. Definition of ClWWalksNOn resolved. (Contributed by
Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-May-2021.)
(Revised by AV, 1-May-2022.)
|
β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ ((π β π β§ π β (β€β₯β2))
β (ππ»π) = {π€ β (π ClWWalksN πΊ) β£ ((π€β0) = π β§ (π€β(π β 2)) β (π€β0))}) |
|
Theorem | numclwwlkovq 29104* |
Value of operation π, mapping a vertex π£ and a
positive integer
π to the not closed walks v(0) ... v(n)
of length π from a fixed
vertex π£ = v(0). "Not closed" means
v(n) =/= v(0). Remark:
π
β β0 would not be useful: numclwwlkqhash 29105 would not hold,
because (πΎβ0) = 1! (Contributed by
Alexander van der Vekens,
27-Sep-2018.) (Revised by AV, 30-May-2021.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) β β’ ((π β π β§ π β β) β (πππ) = {π€ β (π WWalksN πΊ) β£ ((π€β0) = π β§ (lastSβπ€) β π)}) |
|
Theorem | numclwwlkqhash 29105* |
In a πΎ-regular graph, the size of the set
of walks of length π
starting with a fixed vertex π and ending not at this vertex is the
difference between πΎ to the power of π and the
size of the set
of closed walks of length π on vertex π. (Contributed by
Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.)
(Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 7-Jul-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) β β’ (((πΊ RegUSGraph πΎ β§ π β Fin) β§ (π β π β§ π β β)) β
(β―β(πππ)) = ((πΎβπ) β (β―β(π(ClWWalksNOnβπΊ)π)))) |
|
Theorem | numclwwlk2lem1 29106* |
In a friendship graph, for each walk of length π starting at a fixed
vertex π£ 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 π».
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 π», 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, 3-Oct-2018.) (Revised by AV,
30-May-2021.) (Revised by AV, 1-May-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β (π β (πππ) β β!π£ β π (π ++ β¨βπ£ββ©) β (ππ»(π + 2)))) |
|
Theorem | numclwlk2lem2f 29107* |
π
is a function mapping the "closed (n+2)-walks v(0) ... v(n-2)
v(n-1) v(n) v(n+1) v(n+2) starting at π = v(0) = v(n+2) with
v(n)
=/= X" to the words representing the prefix v(0) ... v(n-2)
v(n-1)
v(n) of the walk. (Contributed by Alexander van der Vekens,
5-Oct-2018.) (Revised by AV, 31-May-2021.) (Proof shortened by AV,
23-Mar-2022.) (Revised by AV, 1-Nov-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£})
& β’ π
= (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) β β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β π
:(ππ»(π + 2))βΆ(πππ)) |
|
Theorem | numclwlk2lem2fv 29108* |
Value of the function π
. (Contributed by Alexander van der
Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV,
1-Nov-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£})
& β’ π
= (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) β β’ ((π β π β§ π β β) β (π β (ππ»(π + 2)) β (π
βπ) = (π prefix (π + 1)))) |
|
Theorem | numclwlk2lem2f1o 29109* |
π
is a 1-1 onto function. (Contributed by Alexander van der
Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Proof shortened
by AV, 17-Mar-2022.) (Revised by AV, 1-Nov-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£})
& β’ π
= (π₯ β (ππ»(π + 2)) β¦ (π₯ prefix (π + 1))) β β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β π
:(ππ»(π + 2))β1-1-ontoβ(πππ)) |
|
Theorem | numclwwlk2lem3 29110* |
In a friendship graph, the size of the set of walks of length π
starting with a fixed vertex π and ending not at this vertex equals
the size of the set of all closed walks of length (π + 2)
starting
at this vertex π and not having this vertex as last
but 2 vertex.
(Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV,
31-May-2021.) (Proof shortened by AV, 3-Nov-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ ((πΊ β FriendGraph β§ π β π β§ π β β) β
(β―β(πππ)) = (β―β(ππ»(π + 2)))) |
|
Theorem | numclwwlk2 29111* |
Statement 10 in [Huneke] p. 2: "If n >
1, then the number of closed
n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2)
=/= v is k^(n-2) - f(n-2)." According to rusgrnumwlkg 28708, we have
k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this
number, the number of closed walks of length (n-2), which is f(n-2) per
definition, must be subtracted, because for these walks v(n-2) =/= v(0)
= v would hold. Because of the friendship condition, there is exactly
one vertex v(n-1) which is a neighbor of v(n-2) as well as of
v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of
walks v(0) ... v(n-2) is identical with the number of walks v(0) ...
v(n), that means each (not closed) walk v(0) ... v(n-2) 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, 6-Oct-2018.) (Revised by AV,
31-May-2021.) (Revised by AV, 1-May-2022.)
|
β’ π = (VtxβπΊ)
& β’ π = (π£ β π, π β β β¦ {π€ β (π WWalksN πΊ) β£ ((π€β0) = π£ β§ (lastSβπ€) β π£)}) & β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3)))
β (β―β(ππ»π)) = ((πΎβ(π β 2)) β (β―β(π(ClWWalksNOnβπΊ)(π β 2))))) |
|
Theorem | numclwwlk3lem1 29112 |
Lemma 2 for numclwwlk3 29115. (Contributed by Alexander van der Vekens,
26-Aug-2018.) (Proof shortened by AV, 23-Jan-2022.)
|
β’ ((πΎ β β β§ π β β β§ π β (β€β₯β2))
β (((πΎβ(π β 2)) β π) + (πΎ Β· π)) = (((πΎ β 1) Β· π) + (πΎβ(π β 2)))) |
|
Theorem | numclwwlk3lem2lem 29113* |
Lemma for numclwwlk3lem2 29114: The set of closed vertices of a fixed
length π on a fixed vertex π is the
union of the set of closed
walks of length π at π with the last but one
vertex being π
and the set of closed walks of length π at π with the last but
one vertex not being π. (Contributed by AV, 1-May-2022.)
|
β’ πΆ = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£})
& β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ ((π β π β§ π β (β€β₯β2))
β (π(ClWWalksNOnβπΊ)π) = ((ππ»π) βͺ (ππΆπ))) |
|
Theorem | numclwwlk3lem2 29114* |
Lemma 1 for numclwwlk3 29115: The number of closed vertices of a fixed
length π on a fixed vertex π is the
sum of the number of closed
walks of length π at π with the last but one
vertex being π
and the set of closed walks of length π at π with the last but
one vertex not being π. (Contributed by Alexander van der
Vekens,
6-Oct-2018.) (Revised by AV, 1-Jun-2021.) (Revised by AV,
1-May-2022.)
|
β’ πΆ = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£})
& β’ π» = (π£ β π, π β (β€β₯β2)
β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) β π£}) β β’ (((πΊ β FinUSGraph β§ π β π) β§ π β (β€β₯β2))
β (β―β(π(ClWWalksNOnβπΊ)π)) = ((β―β(ππ»π)) + (β―β(ππΆπ)))) |
|
Theorem | numclwwlk3 29115 |
Statement 12 in [Huneke] p. 2: "Thus f(n)
= (k - 1)f(n - 2) + k^(n-2)."
- the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) is the sum
of the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) with
v(n-2) = v(n) (see numclwwlk1 29091) and with v(n-2) =/= v(n) (see
numclwwlk2 29111): f(n) = kf(n-2) + k^(n-2) - f(n-2) =
(k-1)f(n-2) +
k^(n-2). (Contributed by Alexander van der Vekens, 26-Aug-2018.)
(Revised by AV, 6-Mar-2022.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β Fin β§ π β π β§ π β (β€β₯β3)))
β (β―β(π(ClWWalksNOnβπΊ)π)) = (((πΎ β 1) Β· (β―β(π(ClWWalksNOnβπΊ)(π β 2)))) + (πΎβ(π β 2)))) |
|
Theorem | numclwwlk4 29116* |
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, 7-Oct-2018.) (Revised by AV,
2-Jun-2021.) (Revised by AV, 7-Mar-2022.) (Proof shortened by AV,
28-Mar-2022.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FinUSGraph β§ π β β) β
(β―β(π
ClWWalksN πΊ)) =
Ξ£π₯ β π (β―β(π₯(ClWWalksNOnβπΊ)π))) |
|
Theorem | numclwwlk5lem 29117 |
Lemma for numclwwlk5 29118. (Contributed by Alexander van der Vekens,
7-Oct-2018.) (Revised by AV, 2-Jun-2021.) (Revised by AV,
7-Mar-2022.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ RegUSGraph πΎ β§ π β π β§ πΎ β β0) β (2
β₯ (πΎ β 1)
β ((β―β(π(ClWWalksNOnβπΊ)2)) mod 2) = 1)) |
|
Theorem | numclwwlk5 29118 |
Statement 13 in [Huneke] p. 2: "Let p be
a prime divisor of k-1; then
f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander
van
der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.) (Revised by AV,
7-Mar-2022.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph β§ π β Fin) β§ (π β π β§ π β β β§ π β₯ (πΎ β 1))) β ((β―β(π(ClWWalksNOnβπΊ)π)) mod π) = 1) |
|
Theorem | numclwwlk7lem 29119 |
Lemma for numclwwlk7 29121, frgrreggt1 29123 and frgrreg 29124: If a finite,
nonempty friendship graph is πΎ-regular, the πΎ is a nonnegative
integer. (Contributed by AV, 3-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β β
β§ π β Fin)) β πΎ β
β0) |
|
Theorem | numclwwlk6 29120 |
For a prime divisor π of πΎ β 1, the total
number of closed
walks of length π in a πΎ-regular friendship graph
is equal
modulo π to the number of vertices.
(Contributed by Alexander van
der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) (Proof shortened
by AV, 7-Mar-2022.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph β§ π β Fin) β§ (π β β β§ π β₯ (πΎ β 1))) β ((β―β(π ClWWalksN πΊ)) mod π) = ((β―βπ) mod π)) |
|
Theorem | numclwwlk7 29121 |
Statement 14 in [Huneke] p. 2: "The total
number of closed walks of
length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)",
since the
number of vertices in a friendship graph is (k(k-1)+1), see
frrusgrord0 29070 or frrusgrord 29071, and p divides (k-1), i.e., (k-1) mod p =
0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the null graph
is a
friendship graph, see frgr0 28995, as well as k-regular (for any k), see
0vtxrgr 28310, but has no closed walk, see 0clwlk0 28862, this theorem would
be false for a null graph: ((β―β(π ClWWalksN πΊ)) mod π) = 0
β 1, so this case must be excluded (by
assuming π
β β
).
(Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV,
3-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ RegUSGraph πΎ β§ πΊ β FriendGraph ) β§ (π β β
β§ π β Fin) β§ (π β β β§ π β₯ (πΎ β 1))) β ((β―β(π ClWWalksN πΊ)) mod π) = 1) |
|
Theorem | numclwwlk8 29122 |
The size of the set of closed walks of length π, π prime, is
divisible by π. 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 28816. (Contributed by
Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)
(Proof shortened by AV, 2-Mar-2022.)
|
β’ ((πΊ β FinUSGraph β§ π β β) β
((β―β(π
ClWWalksN πΊ)) mod π) = 0) |
|
Theorem | frgrreggt1 29123 |
If a finite nonempty friendship graph is πΎ-regular with πΎ > 1,
then πΎ must be 2.
(Contributed by Alexander van der Vekens,
7-Oct-2018.) (Revised by AV, 3-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FriendGraph β§ π β Fin β§ π β β
) β ((πΊ RegUSGraph πΎ β§ 1 < πΎ) β πΎ = 2)) |
|
Theorem | frgrreg 29124 |
If a finite nonempty friendship graph is πΎ-regular, then πΎ must
be 2 (or 0).
(Contributed by Alexander van der Vekens,
9-Oct-2018.) (Revised by AV, 3-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ ((π β Fin β§ π β β
) β ((πΊ β FriendGraph β§ πΊ RegUSGraph πΎ) β (πΎ = 0 β¨ πΎ = 2))) |
|
Theorem | frgrregord013 29125 |
If a finite friendship graph is πΎ-regular, then it must have order
0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
(Revised by AV, 4-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FriendGraph β§ π β Fin β§ πΊ RegUSGraph πΎ) β ((β―βπ) = 0 β¨ (β―βπ) = 1 β¨ (β―βπ) = 3)) |
|
Theorem | frgrregord13 29126 |
If a nonempty finite friendship graph is πΎ-regular, then it must
have order 1 or 3. Special case of frgrregord013 29125. (Contributed by
Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ (((πΊ β FriendGraph β§ π β Fin β§ π β β
) β§ πΊ RegUSGraph πΎ) β ((β―βπ) = 1 β¨ (β―βπ) = 3)) |
|
Theorem | frgrogt3nreg 29127* |
If a finite friendship graph has an order greater than 3, it cannot be
π-regular for any π.
(Contributed by Alexander van der Vekens,
9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FriendGraph β§ π β Fin β§ 3 <
(β―βπ)) β
βπ β
β0 Β¬ πΊ RegUSGraph π) |
|
Theorem | friendshipgt3 29128* |
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, 9-Oct-2018.)
(Revised by AV, 4-Jun-2021.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FriendGraph β§ π β Fin β§ 3 <
(β―βπ)) β
βπ£ β π βπ€ β (π β {π£}){π£, π€} β (EdgβπΊ)) |
|
Theorem | friendship 29129* |
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,
9-Oct-2018.)
|
β’ π = (VtxβπΊ) β β’ ((πΊ β FriendGraph β§ π β β
β§ π β Fin) β βπ£ β π βπ€ β (π β {π£}){π£, π€} β (EdgβπΊ)) |
|
PART 18 GUIDES AND
MISCELLANEA
|
|
18.1 Guides (conventions, explanations, and
examples)
|
|
18.1.1 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 29130 |
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 (left-to-right). For example, summation is written in the
form Ξ£π β π΄π΅ (df-sum 15506) which denotes that index
variable π ranges over π΄ when evaluating π΅. Thus,
Ξ£π β β(1 / (2βπ)) = 1 means 1/2 + 1/4 + 1/8 + ...
= 1 (geoihalfsum 15702).
The notation is usually explained in more detail when first introduced.
Axiomatic assertions ($a).
All axiomatic assertions ($a statements)
starting with " β’ " 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 four of them (df-bi 206, df-clab 2716, df-cleq 2730, df-clel 2816)
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 2816. When we wish
to use a previously-proven assertion as an axiom, our convention
is that we use the
regular "ax-NAME" label naming convention to define the axiom,
but we precede it with a proof of the same statement with the label
"axNAME" . An example is the complex arithmetic axiom ax-1cn 11043,
proven by the preceding theorem ax1cn 11019.
The Metamath program will warn if an axiom does not match the preceding
theorem that justifies it if the names match in this way.
Definitions (df-...).
We encourage definitions to include hypertext links to proven examples.
Statements with hypotheses.
Many theorems and some axioms, such as ax-mp 5, have hypotheses that
must be satisfied in order for the conclusion to hold, in this case min
and maj. When displayed in summarized form such as in the "Theorem
List" page (to get to it, click on "Nearby theorems" on the ax-mp 5
page), the hypotheses are connected with an ampersand and separated from
the conclusion with a double right arrow, such as in
" β’ π & β’ (π β π) β β’ π". 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 ax-1cn 11043 (not ax1cn 11019) and ax-1ne0 11054 (not ax1ne0 11030), as these
are proven axioms for complex arithmetic. Thus, both
ax1cn 11019 and ax1ne0 11030 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 β€ for inequality expressions, and
use ((sinβ(i Β· π΄)) / i) instead of (sinhβπ΄)
for the hyperbolic sine.
Minimizing axiom dependencies.
We prefer proofs that depend on fewer and/or weaker axioms, even if
the proofs are longer. In particular, because of the non-constructive
nature of the axiom of choice df-ac 9986, we prefer proofs that do not use
it, or use weaker versions like countable choice ax-cc 10305 or dependent
choice ax-dc 10316. An example is our proof of the Schroeder-Bernstein
Theorem sbth 8971, which does not use the axiom of choice. Similarly,
any theorem in first-order logic (FOL) that contains only setvar
variables that are all mutually distinct, and has no wff variables, can
be proved without using ax-10 2138 through ax-13 2372, by using ax10w 2126
through ax13w 2133 instead.
We do not try to similarly reduce dependencies on definitions, since
definitions are conservative (they do not increase the proving power of
a deductive system), and are introduced in order to be used to increase
readability). An exception is made for Definitions df-clab 2716,
df-cleq 2730, and df-clel 2816, since they can be considered as axioms under
some definitions of what a definition is exactly (see their comments).
Alternate proofs (ALT).
If a different proof is shorter or clearer but uses more or stronger
axioms, we make that proof an "alternate" proof (marked with an ALT
label suffix), even if this alternate proof was formalized first.
We then make the proof that requires fewer axioms the main proof.
Alternate proofs can also occur in other cases when an alternate proof
gives some particular insight. Their comment should begin with
"Alternate proof of ~ xxx " followed by a description of the
specificity of that alternate proof. There can be multiple alternates.
Alternate (*ALT) theorems should have "(Proof modification is
discouraged.) (New usage is discouraged.)" in their comment and should
follow the main statement, so that people reading the text in order will
see the main statement first. The alternate and main statement comments
should use hyperlinks to refer to each other.
Alternate versions (ALTV).
The suffix ALTV is reserved for theorems (or definitions) which are
alternate versions, or variants, of an existing theorem. This is
reserved to statements in mathboxes and is typically used temporarily,
when it is not clear yet which variant to use. If it is decided that
both variants should be kept and moved to the main part of set.mm, then
a label for the variant should be found with a more explicit suffix
indicating how it is a variant (e.g., commutation of some subformula,
antecedent replaced with hypothesis, (un)curried variant, biconditional
instead of implication, etc.). There is no requirement to add
discouragement tags, but their comment should have a link to the main
version of the statement and describe how it is a variant of it.
Old (OLD) versions or proofs.
If a proof, definition, axiom, or theorem is going to be removed, we
often stage that change by first renaming its label with an OLD suffix
(to make it clear that it is going to be removed). Old (*OLD)
statements should have
"(Proof modification is discouraged.) (New usage is discouraged.)" and
"Obsolete version of ~ xxx as of dd-Mmm-yyyy." (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 "MM-PA> 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 well-formed formulas or wffs) are
represented with lowercase Greek letters and are generally used
in this order:
π = phi, π = psi, π = chi, π = theta,
π = tau, π = eta, π = zeta, and π = sigma.
Individual setvar variables are represented with lowercase Latin letters
and are generally used in this order:
π₯, π¦, π§, π€, π£, π’, and π‘.
Variables that represent classes are often represented by
uppercase Latin letters:
π΄, π΅, πΆ, π·, πΈ, 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 9274)
and connective symbols such as + for some group addition operation
(e.g., grprinvd ).
Class variables are selected in alphabetical order starting
from π΄ 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.
"β’ ", 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 Β¬ π".
Biconditional (β).
There are basically two ways to maximize the effectiveness of
biconditionals (β):
you can either have one-directional simplifications of all theorems
that produce biconditionals, or you can have one-directional
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
ax-mp 5, there is a corresponding version with a biconditional or a
reversed biconditional, like mpbi 229 or mpbir 230. 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 215 are available, in most cases there is already a theorem that
combines it with your theorem of choice, like mpbir2an 710, sylbir 234,
or 3imtr4i 292.
Quantifiers.
The quantifiers are named as follows:
- β: universal quantifier (wal 1540);
- β: existential quantifier (df-ex 1783);
- β*: at-most-one quantifier (df-mo 2540);
- β!: unique existential quantifier (df-eu 2569).
The phrase "uniqueness quantifier" is avoided since it is ambiguous:
it can be understood as claiming either uniqueness (β*) or unique
existence (β!).
Substitution.
The expression "[π¦ / π₯]π" should be read "the formula that
results from the proper substitution of π¦ for π₯ in the formula
π". See df-sb 2069 and the related df-sbc 3739 and df-csb 3855.
Is-a-set.
" π΄ β V" should be read "Class π΄ is a set (i.e., exists)."
This is a convention based on Definition 2.9 of [Quine] p. 19.
See df-v 3446 and isset 3457.
However, instead of using πΌ β V in the antecedent of a theorem for
some variable πΌ, we now prefer to use πΌ β π (or another
variable if π is not available) to make it more general. That way
we can often avoid extra uses of elex 3462 and syl 17 in the common case
where πΌ is already a member of something. For hypotheses
($e statement) of theorems (mostly in inference form), however,
β’ π΄ β V is used rather than β’ π΄ β π (e.g., difexi 5284).
This is because π΄ β V is almost always satisfied using an
existence theorem stating " ... β V", and a hard-coded V in
the $e statement saves a couple of syntax building steps that substitute
V into π. Notice that this does not hold for hypotheses of
theorems in deduction form: Here still β’ (π β π΄ β π) should be
used rather than β’ (π β π΄ β V).
Converse.
The symbol " β‘" denotes the converse of a relation, so
" β‘π
" denotes the converse of the class π
, which is
typically a relation in that context (see df-cnv 5639). The converse of a
relation π
is sometimes denoted by R-1 in textbooks,
especially when π
is a function, but we avoid this notation since it
is generally not a genuine inverse (see f1cocnv1 6810 and funcocnv2 6805 for
cases where it is a left or right-inverse). This can be used to define
a subset, e.g., df-tan 15889 notates "the set of values whose cosine is a
nonzero complex number" as (β‘cos β (β β {0})).
Function application.
The symbols "(πΉβπ₯)" should be read "the value
of (function) πΉ at π₯" and has the same meaning as the more
familiar but ambiguous notation F(x). For example,
(cosβ0) = 1 (see cos0 15967). 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 df-fv 6500.
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 df-ov 7353).
For example, the + in (2 + 2); see 2p2e4 12222.
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
(π β π), (π β¨ π), (π β§ π), and
(π β π)
(see wi 4, df-or 847, df-an 398, and df-bi 206 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 π΄ β π΅ (see wel 2108),
equality π΄ = π΅ (see df-cleq 2730),
subset π΄ β π΅ (see df-ss 3926), and
less-than π΄ < π΅ (see df-lt 10998). For the general definition
of a binary relation in the form π΄π
π΅, see df-br 5105.
For example, 0 < 1 (see 0lt1 11611) 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 11320.
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 15882).
The function is then defined labeled df-NAME; definitions
are typically given using the maps-to notation (e.g., df-cos 15888).
Typically, there are other proofs such as its
closure labeled NAMEcl (e.g., coscl 15944), its
function application form labeled NAMEval (e.g., cosval 15940),
and at least one simple value (e.g., cos0 15967).
Another way to define functions is to use recursion (for more details
about recursion see below). For an example of how to define functions
that aren't primitive recursive using recursion, see the Ackermann
function definition df-ack 46446 (which is based on the sequence builder
seq, see df-seq 13836).
Factorial.
The factorial function is traditionally a postfix operation,
but we treat it as a normal function applied in prefix form, e.g.,
(!β4) = ;24 (df-fac 14102 and fac4 14109).
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 (df-0 10992), while
"0g" is the group identity element (df-0g 17258),
"0." is the poset zero (df-p0 18249),
"0π" is the zero polynomial (df-0p 24956),
"0vec" is the zero vector in a normed subcomplex vector space
(df-0v 29326), and
"0" is a class variable for use as a connective symbol
(this is used, for example, in p0val 18251).
There are other class variables used as connective symbols
where traditional notation would use ambiguous symbols, including
"1", "+", "β", and "β₯".
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 β,
are often represented by that letter followed by a period.
For example, "A." is used to represent β,
"e." is used to represent β, and
"E." is used to represent β.
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 π«.
Originally, all setvar and class variables used only single letters
a-z and A-Z, 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 have somewhat followed, that is
also not uncommon in the literature, is to start tokens with a
capital letter for collection-like objects and lower case for
function-like 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 collection-like,
e.g., Lim indirectly represents the collection of limit ordinals,
but it cannot 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 collection-like or function-like, 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 Definition df-ov 7353, each operation value can be
written as function value of an ordered pair), the convention for its
primary usage should be used, e.g., (iEdgβπΊ) versus
(πiEdgπΈ) for the edges of a graph πΊ = β¨π, πΈβ©.
LaTeX definitions.
Each token has a "LaTeX definition" which is used by the Metamath
program to output tex files. When writing LaTeX definitions,
contributors should favor simplicity over perfection of the display, and
should only use core LaTeX symbols or symbols from standard packages; if
packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic,
graphicx are needed, this should be discussed. A useful resource is
The Comprehensive LaTeX
Symbol List.
Number construction independence.
There are many ways to model complex numbers.
After deriving the complex number postulates we
reintroduce them as new axioms on top of set theory.
This lets us easily identify which axioms are needed
for a particular complex number proof, without the obfuscation
of the set theory used to derive them.
This also lets us be independent of the specific construction,
which we believe is valuable.
See mmcomplex.html 7353 for details.
Thus, for example, we don't allow the use of β
β β,
as handy as that would be, because that would be
construction-specific. We want proofs about β to be independent
of whether or not β
β β.
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
β). 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
β’ (π΄ < π΅ β π΅ β π΄) without any hypotheses, but we require that
theorems using this result prove that π΄ and π΅ are real numbers,
so that the statement we use is ltnei 11213. Here are the reasons as
discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 11213:
- Having the hypotheses immediately shows the intended domain of
applicability (is it β, β*, Ο, 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 out-of-domain values are dependent on contingent details of
definitions, so hypothesis-free theorems would be non-portable and
"brittle".
- Only a few theorems can have their hypotheses removed in this
fashion, due to coincidences for our particular set-theoretical
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 set-theoretical 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 β (df-nn 12088) for the set of positive integers starting
from 1, and β0 (df-n0 12348) 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 df-dec 12552, e.g., ;;;4001 (see 4001prm 16952
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 4143). 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 $e hypotheses, and the hypotheses and the conclusion
are implications that share the same antecedent. More precisely, the
conclusion is an implication with a wff variable as the antecedent
(usually π), and every hypothesis ($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 π· in
lhop 25302) or a wff variable (this is a wff variable followed by
β); 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 π) or is a conjunction (π β© ...)
including that wff variable (π). E.g., a1d 25, unssd 4145.
Although they are no real deductions, theorems without $e hypotheses,
but in the form (π β ...), 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 2739, 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 4144).
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 154 is the
inference associated with con3 153). The inference associated with a
theorem is easily derivable from that theorem by a simple use of
ax-mp 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 to easily use the theorem in places where (in traditional
textbook formalizations) the standard Deduction Theorem (see below)
would be used. We call this approach "deduction style".
In contrast, we usually avoid theorems in "inference form" when that
would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other
forms of the same theorem (e.g., pm2.43d 53). The labels for inferences
usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems
in "closed form" would have no special suffix (e.g., pm2.43 56) or, if
the non-suffixed label is already used, then we add the suffix "t" (for
"theorem" or "tautology", e.g., ancomst 466 or nfimt 1899). When an
inference with an "is a set" hypothesis (e.g., π΄ β V) is converted
to a theorem (in closed form) by replacing the hypothesis with an
antecedent of the form (π΄ β π β, we sometimes suffix the closed
form with "g" (for "more general") as in uniex 7669 versus uniexg 7668. 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 5941. Its inference form which was
the original "brres", now is brresi 5943. The same holds for the suffix
"t".
Deduction theorem.
The Deduction Theorem is a metalogical theorem that provides an
algorithm for constructing a proof of a theorem from the proof of its
corresponding deduction (its associated inference). See for instance
Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually
carries out the algorithm, because (in its most basic form) it involves
an exponential explosion of the number of proof steps as more hypotheses
are eliminated. Instead, in ordinary mathematics the Deduction Theorem
is invoked simply to claim that something can be done in principle,
without actually doing it. For more details, see mmdeduction.html 5943.
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 4543, which works in certain cases in
set theory. We also sometimes use dedhb 3660. 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 π β 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 3660; a
list of translations for common natural deduction rules is given in
natded 29133.
Recursion.
We define recursive functions using various "recursion constructors".
These allow to define, with compact direct definitions, functions
that are usually defined in textbooks with indirect self-referencing
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(πΉ) in df-recs 8285, rec(πΉ, πΌ) in df-rdg 8324,
seqΟ(πΉ, πΌ) in df-seqom 8362, and seqπ( + , πΉ) in
df-seq 13836. These have characteristic function πΉ and initial value
πΌ. (Ξ£g in df-gsum 17259 isn't really designed for arbitrary
recursion, but you could do it with the right magma.) The logically
primary one is df-recs 8285, but for the "average user" the most useful
one is probably df-seq 13836- 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 df-struct 16954.
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 (β©π₯π) is meant to be "the π₯ such that π"
(where π typically depends on x).
What should that expression produce when there is no such π₯?
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 so-called 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 π₯).
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 π΄ β (πΉβπ΅) β π΅ β dom πΉ,
which is useful when πΉ 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 df-NAME 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 π¦π₯ $. $d π¦π $.
df-iota $a β’ (β!π₯π β (β©π₯π) = βͺ {π₯ β£ π}) $. $}.
This will be rejected by the definition checker, but the bigger
theoretical reason to reject this axiom is that it breaks equality -
the metatheorem (π₯ = π¦ β P(x) = P(y) ) fails
to hold if definitions don't unfold without some assumptions.
(That is, iotabidv 6476 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 6851 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 8961 is the
first lemma for sbth 8971. 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 "MM-PA> 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 hard-and-fast 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 8961 is the first lemma for sbth 8971.
Proof stubs.
It's sometimes useful to record partial proof results, e.g.,
incomplete proofs or proofs that depend on something else not fully
proven.
Some systems (like Lean) support a "sorry" axiom, which lets you assert
anything is true, but this can quickly run into trouble, because
the Metamath tooling is smart and may end up using it
to prove everything.
If you want to create a proof based on some other claim, without
proving that claim, you can choose to define the claim as an axiom.
If you temporarily define a claim as an axiom, we encourage you to
include "Temporarily provided as axiom" in its comment.
Such incomplete work will generally only be accepted in a mathbox
until the rest of the work is complete.
When you're working on your personal copy of the database
you can use "?" in proofs to indicate an unknown step.
However, since proofs with "?" will (obviously) fail
verification, we don't accept proofs with unknown steps in
the public database.
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
space-separated tilde (e.g., " ~ df-prm " results in " df-prm 16483").
When the Metamath program is used to generate HTML, it automatically
inserts hypertext links for syntax used (e.g., every symbol used), every
axiom and definition depended on, the justification for each step in a
proof, and to both the next and previous assertions.
Hypertext links to section headers.
Some section headers have text under them that describes or explains the
section. However, they are not part of the description of axioms or
theorems, and there is no way to link to them directly. To provide for
this, section headers with accompanying text (indicated with "*"
prefixed to mmtheorems.html#mmdtoc 16483 entries) have an anchor in
mmtheorems.html 16483 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 27723. To reference it we link to the anchor using a
space-separated tilde followed by the space-separated link
mmtheorems.html#cedgf, which will become the hyperlink
mmtheorems.html#cedgf 27723. 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 27723.
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 case-sensitive,
must be one of the following: Axiom, Chapter, Compare, Condition,
Corollary, Definition, Equation, Example, Exercise, Figure, Item,
Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem,
Property, Proposition, Remark, Rule, Scheme, Section, or Theorem.
The IDENTIFIER is optional, as in for example
"Remark in [Monk1] p. 22".
The NOISEWORDS(S) are zero or more from the list: from, in, of, on.
The AUTHOR(S) must be present in the file identified with the
htmlbibliography assignment (e.g., mmset.html) as a named anchor
(NAME=). If there is more than one document by the same author(s),
add a numeric suffix (as shown here).
The NUMBER is a page number, and may be any alphanumeric string such as
an integer or Roman numeral.
Note that we _require_ page numbers in comments for individual
$a or $p statements. We allow names in square brackets without
page numbers (a reference to an entire document) in
heading comments.
If this is a new reference, please also add it to the
"Bibliography" section of mmset.html.
(The file mmbiblio.html is automatically rebuilt, e.g.,
using the Metamath program "MM> WRITE BIBLIOGRAPHY" command.)
Acceptable shorter proofs.
Shorter proofs are welcome, and any shorter proof we accept
will be acknowledged in the theorem description. However,
in some cases a proof may be "shorter" or not depending on
how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1)
shorten the set.mm file (with compressed proofs), (2) reduce
the size of the HTML file generated with SHOW STATEMENT xx
/ HTML, (3) use only existing, unmodified theorems in the
database (the order of theorems may be changed, though), and
(4) use no additional axioms.
Usually we will also automatically accept a _new_ theorem
that is used to shorten multiple proofs, if the total size
of set.mm (including the comment of the new theorem, not
including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on
the number of compressed proof steps and less on the length
of the label section (since the names are in principle
arbitrary). If two proofs have the same number of compressed
proof steps, we will typically give preference to the one
with the smaller number of different labels, or if these
numbers are the same, the proof with the fewest number of
characters that the proofs happen to have by chance when
label lengths are included.
A few theorems have a longer proof than necessary in order
to avoid the use of certain axioms, for pedagogical purposes,
and for other reasons. These theorems will (or should) have
a "(Proof modification is discouraged.)" tag in their
description. For example, idALT 23 shows a proof directly from
axioms. Shorter proofs for such cases won't be accepted,
of course, unless the criteria described continues to be
satisfied.
Information on syntax, axioms, and definitions.
For a hyperlinked list of syntax, axioms, and definitions, see
mmdefinitions.html 23.
If you have questions about a specific symbol or axiom, it is best
to go directly to its definition to learn more about it.
The generated HTML for each theorem and axiom includes hypertext
links to each symbol's definition.
Reserved symbols: 'LETTER.
Some symbols are reserved for potential future use.
Symbols with the pattern 'LETTER are reserved for possibly
representing characters (this is somewhat similar to Lisp).
We would expect '\n to represent newline, 'sp for space, and perhaps
'\x24 for the dollar character.
The challenge of varying mathematical conventions
We try to follow mathematical conventions, but in many cases
different texts use different conventions.
In those cases we pick some reasonably common convention and stick to
it.
We have already mentioned that the term "natural number" has
varying definitions (some start from 0, others start from 1), but
that is not the only such case.
A useful example is the set of metavariables used to represent
arbitrary well-formed formulas (wffs).
We use an open phi, Ο, 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 Ο (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 Ο (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 Ξ±, Ξ², Ξ³:
Duffy's Principles of Automated Theorem Proving (1991).
- Roman A, B, C:
Kleene's Introduction to Metamathematics (1974),
Smullyan's First-Order 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:
- β²π₯π is read " π₯ is not free in (wff) π";
see df-nf 1787 (whose description has some important technical
details). Similarly, β²π₯π΄ is read π₯ is not free in (class)
π΄, see df-nfc 2888.
- "$d π₯π¦ $." should be read "Assume π₯ and π¦ are distinct
variables."
- "$d ππ₯ $." should be read "Assume π₯ does not occur in
Ο." Sometimes a theorem is proved using β²π₯π (df-nf 1787)
in place of "$d ππ₯ $." when a more general result is desired;
ax-5 1914 can be used to derive the $d version. For an example of
how to get from the $d version back to the $e version, see the
proof of euf 2576 from eu6 2574.
- "$d π΄π₯ $." should be read "Assume π₯ is not a variable
occurring in class π΄."
- "$d π΄π₯ $. $d ππ₯ $.
$e |- (π₯ = π΄ β (π β π)) $." is an idiom often used instead
of explicit substitution, meaning "Assume Ο results from the
proper substitution of π΄ for π₯ in Ο." Therefore, we often
use the term "implicit substitution" for such a hypothesis.
- Class and wff variables should appear at the beginning of distinct
variable conditions, and setvars should be in alphabetical order.
E.g., "$d ππ₯π¦ $.", "$d πππ₯ $.". This convention should
be applied for new theorems (formerly, the class and wff variables
mostly appear at the end) and will be assured by a formatter in the
future.
- " β’ (Β¬ βπ₯π₯ = π¦ β ...)" 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
β²π₯π΄ or β²π₯π; here it is.
β’ T[x, A] where $d π₯π΄,
and you wish to prove β’ β²π₯π΄ β β’ T[x, A].
You apply the theorem substituting π¦ for π₯ and π΄ for π΄,
where π¦ is a new dummy variable, so that
$d y A is satisfied.
You obtain β’ T[y, A], and apply chvar to obtain β’
T[x, A] (or just use mpbir 230 if T[x, A] binds π₯).
The side goal is β’ (π₯ = π¦ β ( T[y, A] β T[x, A] )),
where you can use equality theorems, except
that when you get to a bound variable you use a non-dv bound variable
renamer theorem like cbval 2398. The section
mmtheorems32.html#mm3146s 2398 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 (df-bi 206, df-clab 2716, df-cleq 2730, and df-clel 2816);
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 $a-statement with a given label and
typecode β’ passes the test if and only if it
respects the following rules (these rules require that we have
an unambiguous tree parse, which is checked separately):
The expression must be a biconditional or an equality (i.e. its
root-symbol must be β or =).
If the proposed definition passes this first rule, we then
define its definiendum as its left hand side (LHS) and
its definiens as its right hand side (RHS).
We define the *defined symbol* as the root-symbol of the LHS.
We define a *dummy variable* as a variable occurring
in the RHS but not in the LHS.
Note that the "root-symbol" is the root of the considered tree;
it need not correspond to a single token in the database
(e.g., see w3o 1087 or wsb 2068).
The defined expression must not appear in any statement
between its syntax axiom ($a wff ) and its definition,
and the defined expression must not be used in its definiens.
See df-3an 1090 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 non-setvar dummy variables, or
(b) there must be a justification theorem.
The justification theorem must be of form
β’ ( definiens root-symbol definiens' )
where definiens' is definiens but the dummy variables are all
replaced with other unused dummy variables of the same type.
Note that root-symbol is β 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
(π β βπ₯π) for each setvar dummy variable π₯
where π is the definiens.
We use two different tests for nonfreeness; one must succeed
for each setvar dummy variable π₯.
The first test requires that the setvar dummy variable π₯
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 directly-stated proof of (π β βπ₯π)
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 $. df-foo $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 3445.
Definition df-v 3446 β’ V = {π₯ β£ π₯ = π₯} is justified
by the justification theorem vjust 3445
β’ {π₯ β£ π₯ = π₯} = {π¦ β£ π¦ = π¦}.
Another example of a justification theorem is trujust 1544;
Definition df-tru 1545 β’ (β€ β (βπ₯π₯ = π₯ β βπ₯π₯ = π₯))
is justified by trujust 1544 β’ ((βπ₯π₯ = π₯ β βπ₯π₯ = π₯) β (βπ¦π¦ = π¦ β βπ¦π¦ = π¦)).
Here is more information about our processes for checking and
contributing to this work:
Multiple verifiers.
This entire file is verified by multiple independently-implemented
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 1544.
Community.
We encourage anyone interested in Metamath to join our mailing list:
https://groups.google.com/g/metamath 1544.
(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision.
(Revised by the Metamath team, 22-Sep-2022.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ π β β’ π |
|
Theorem | conventions-labels 29131 |
The following gives conventions used in the Metamath Proof Explorer
(MPE, set.mm) regarding labels.
For other conventions, see conventions 29130 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
easy-to-remember hints about their contents.
Labels are not a 1-to-1 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 "ax-NAME",
proofs of proven axioms are named "axNAME", and
definitions are named "df-NAME".
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 3065"rgen.1 $e |- ( x e. A -> ph ) $."
or letters corresponding to the (main) class variable used in the
hypothesis, e.g., for mdet0 21877: "mdet0.d $e |- D = ( N maDet R ) $.").
- Common names.
If a theorem has a well-known name, that name (or a short version of it)
is sometimes used directly. Examples include
barbara 2664 and stirling 44040.
- 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 1842, and the restricted quantifier version of Theorem 21 from
Section 19 of [Margaris] p. 90 is labeled r19.21 3236.
- 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... 15701. 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 "df-NAME", and normally NAME is the syntax label fragment. For
example, the class difference construct (π΄ β π΅) is defined in
df-dif 3912, and thus its syntax label fragment is "dif". Similarly, the
subclass relation π΄ β π΅ has syntax label fragment "ss"
because it is defined in df-ss 3926. Most theorem names follow from
these fragments, for example, the theorem proving (π΄ β π΅) β π΄
involves a class difference ("dif") of a subset ("ss"), and thus is
labeled difss 4090. There are many other syntax label fragments, e.g.,
singleton construct {π΄} has syntax label fragment "sn" (because it
is defined in df-sn 4586), and the pair construct {π΄, π΅} has
fragment "pr" ( from df-pr 4588). 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 is-element-of
(is member of) construct π΄ β π΅ does not have a df-NAME definition;
in this case its syntax label fragment is "el". Thus, because the
theorem beginning with (π΄ β (π΅ β {πΆ}) uses is-element-of
("el") of a class difference ("dif") of a singleton ("sn"), it is
labeled eldifsn 4746. An "n" is often used for negation (Β¬), e.g.,
nan 829.
- Exceptions.
Sometimes there is a definition df-NAME 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 β (even though its definition is in
df-c 10991) and "re" represents real numbers β (Definition df-r 10995).
The empty set β
often uses fragment 0, even though it is defined
in df-nul 4282. The syntax construct (π΄ + π΅) usually uses the
fragment "add" (which is consistent with df-add 10996), but "p" is used as
the fragment for constant theorems. Equality (π΄ = π΅) often uses
"e" as the fragment. As a result, "two plus two equals four" is labeled
2p2e4 12222.
- 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 15967 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 df-NAME is defined, there is typically a
proof of its value labeled "NAMEval" and of its closure labeled
"NAMEcl". E.g., for cosine (df-cos 15888) we have value cosval 15940 and
closure coscl 15944.
- Special cases.
Sometimes, syntax and related markings are insufficient to distinguish
different theorems. For example, there are over a hundred different
implication-only theorems. They are grouped in a more ad-hoc 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 ax-mp 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 29133 for a list), and that is
about all you need for most things. As for the rest, you can just
assume that if it involves at most three connectives, then it is
probably already proved in set.mm, and searching for it will give you
the label.
- Suffixes.
Suffixes are used to indicate the form of a theorem (inference,
deduction, or closed form, see above).
Additionally, we sometimes suffix with "v" the label of a theorem adding
a disjoint variable condition, as in 19.21v 1943 versus 19.21 2201. This
often permits to prove the result using fewer axioms, and/or to
eliminate a nonfreeness hypothesis (such as β²π₯π in 19.21 2201).
If no constraint is put on axiom use, then the v-version can be proved
from the original theorem using nfv 1918. If two (resp. three) such
disjoint variable conditions are added, then the suffix "vv" (resp.
"vvv") is used, e.g., exlimivv 1936.
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 2576 derived from eu6 2574. The "f" stands
for "not free in" which is less restrictive than "does not occur in."
The suffix "b" often means "biconditional" (β, "iff" , "if and
only if"), e.g., sspwb 5405.
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 less-preferred 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 2409 (cbval 2398 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 3509.
Here is a non-exhaustive list of common suffixes:
- a : theorem having a conjunction as antecedent
- b : theorem expressing a logical equivalence
- c : contraction (e.g., sylc 65, syl2anc 585), commutes
(e.g., biimpac 480)
- d : theorem in deduction form
- f : theorem with a hypothesis such as β²π₯π
- g : theorem in closed form having an "is a set" antecedent
- i : theorem in inference form
- l : theorem concerning something at the left
- r : theorem concerning something at the right
- r : theorem with something reversed (e.g., a biconditional)
- s : inference that manipulates an antecedent ("s" refers to an
application of syl 17 that is eliminated)
- t : theorem in closed form (not having an "is a set" antecedent)
- v : theorem with one (main) disjoint variable condition
- vv : theorem with two (main) disjoint variable conditions
- w : weak(er) form of a theorem
- ALT : alternate proof of a theorem
- ALTV : alternate version of a theorem or definition (mathbox
only)
- OLD : old/obsolete version of a theorem (or proof) or definition
- Reuse.
When creating a new theorem or axiom, try to reuse abbreviations used
elsewhere. A comment should explain the first use of an abbreviation.
The following table shows some commonly used abbreviations in labels, in
alphabetical order. For each abbreviation we provide a mnenomic, the
source theorem or the assumption defining it, an expression showing what
it looks like, whether or not it is a "syntax fragment" (an abbreviation
that indicates a particular kind of syntax), and hyperlinks to label
examples that use the abbreviation. The abbreviation is bolded if there
is a df-NAME 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 478, rexlimiva 3143 |
abl | Abelian group | df-abl 19494 |
Abel | Yes | ablgrp 19496, zringabl 20796 |
abs | absorption | | | No |
ressabs 17065 |
abs | absolute value (of a complex number) |
df-abs 15055 | (absβπ΄) | Yes |
absval 15057, absneg 15097, abs1 15117 |
ad | adding | |
| No | adantr 482, ad2antlr 726 |
add | add (see "p") | df-add 10996 |
(π΄ + π΅) | Yes |
addcl 11067, addcom 11275, addass 11072 |
al | "for all" | |
βπ₯π | No | alim 1813, alex 1829 |
ALT | alternative/less preferred (suffix) | |
| No | idALT 23 |
an | and | df-an 398 |
(π β§ π) | Yes |
anor 982, iman 403, imnan 401 |
ant | antecedent | |
| No | adantr 482 |
ass | associative | |
| No | biass 386, orass 921, mulass 11073 |
asym | asymmetric, antisymmetric | |
| No | intasym 6066, asymref 6067, posasymb 18143 |
ax | axiom | |
| No | ax6dgen 2125, ax1cn 11019 |
bas, base |
base (set of an extensible structure) | df-base 17019 |
(Baseβπ) | Yes |
baseval 17020, ressbas 17053, cnfldbas 20723 |
b, bi | biconditional ("iff", "if and only if")
| df-bi 206 | (π β π) | Yes |
impbid 211, sspwb 5405 |
br | binary relation | df-br 5105 |
π΄π
π΅ | Yes | brab1 5152, brun 5155 |
cbv | change bound variable | | |
No | cbvalivw 2011, cbvrex 3335 |
cdm | codomain | |
| No | ffvelcdm 7028, focdmex 7879 |
cl | closure | | | No |
ifclda 4520, ovrcl 7391, zaddcl 12474 |
cn | complex numbers | df-c 10991 |
β | Yes | nnsscn 12092, nncn 12095 |
cnfld | field of complex numbers | df-cnfld 20720 |
βfld | Yes | cnfldbas 20723, cnfldinv 20751 |
cntz | centralizer | df-cntz 19029 |
(Cntzβπ) | Yes |
cntzfval 19032, dprdfcntz 19723 |
cnv | converse | df-cnv 5639 |
β‘π΄ | Yes | opelcnvg 5833, f1ocnv 6792 |
co | composition | df-co 5640 |
(π΄ β π΅) | Yes | cnvco 5838, fmptco 7070 |
com | commutative | |
| No | orcom 869, bicomi 223, eqcomi 2747 |
con | contradiction, contraposition | |
| No | condan 817, con2d 134 |
csb | class substitution | df-csb 3855 |
β¦π΄ / π₯β¦π΅ | Yes |
csbid 3867, csbie2g 3897 |
cyg | cyclic group | df-cyg 19584 |
CycGrp | Yes |
iscyg 19585, zringcyg 20813 |
d | deduction form (suffix) | |
| No | idd 24, impbid 211 |
df | (alternate) definition (prefix) | |
| No | dfrel2 6138, dffn2 6666 |
di, distr | distributive | |
| No |
andi 1007, imdi 391, ordi 1005, difindi 4240, ndmovdistr 7536 |
dif | class difference | df-dif 3912 |
(π΄ β π΅) | Yes |
difss 4090, difindi 4240 |
div | division | df-div 11747 |
(π΄ / π΅) | Yes |
divcl 11753, divval 11749, divmul 11750 |
dm | domain | df-dm 5641 |
dom π΄ | Yes | dmmpt 6189, iswrddm0 14354 |
e, eq, equ | equals (equ for setvars, eq for
classes) | df-cleq 2730 |
π΄ = π΅ | Yes |
2p2e4 12222, uneqri 4110, equtr 2025 |
edg | edge | df-edg 27785 |
(EdgβπΊ) | Yes |
edgopval 27788, usgredgppr 27930 |
el | element of | |
π΄ β π΅ | Yes |
eldif 3919, eldifsn 4746, elssuni 4897 |
en | equinumerous | df-en |
π΄ β π΅ | Yes | domen 8835, enfi 9068 |
eu | "there exists exactly one" | eu6 2574 |
β!π₯π | Yes | euex 2577, euabsn 4686 |
ex | exists (i.e. is a set) | |
β V | No | brrelex1 5682, 0ex 5263 |
ex, e | "there exists (at least one)" |
df-ex 1783 |
βπ₯π | Yes | exim 1837, alex 1829 |
exp | export | |
| No | expt 177, expcom 415 |
f | "not free in" (suffix) | |
| No | equs45f 2459, sbf 2264 |
f | function | df-f 6496 |
πΉ:π΄βΆπ΅ | Yes | fssxp 6692, opelf 6699 |
fal | false | df-fal 1555 |
β₯ | Yes | bifal 1558, falantru 1577 |
fi | finite intersection | df-fi 9281 |
(fiβπ΅) | Yes | fival 9282, inelfi 9288 |
fi, fin | finite | df-fin 8821 |
Fin | Yes |
isfi 8850, snfi 8922, onfin 9108 |
fld | field (Note: there is an alternative
definition Fld of a field, see df-fld 36337) | df-field 20111 |
Field | Yes | isfld 20119, fldidom 20698 |
fn | function with domain | df-fn 6495 |
π΄ Fn π΅ | Yes | ffn 6664, fndm 6601 |
frgp | free group | df-frgp 19421 |
(freeGrpβπΌ) | Yes |
frgpval 19469, frgpadd 19474 |
fsupp | finitely supported function |
df-fsupp 9240 | π
finSupp π | Yes |
isfsupp 9243, fdmfisuppfi 9248, fsuppco 9272 |
fun | function | df-fun 6494 |
Fun πΉ | Yes | funrel 6514, ffun 6667 |
fv | function value | df-fv 6500 |
(πΉβπ΄) | Yes | fvres 6857, swrdfv 14468 |
fz | finite set of sequential integers |
df-fz 13354 |
(π...π) | Yes | fzval 13355, eluzfz 13365 |
fz0 | finite set of sequential nonnegative integers |
|
(0...π) | Yes | nn0fz0 13468, fz0tp 13471 |
fzo | half-open integer range | df-fzo 13497 |
(π..^π) | Yes |
elfzo 13503, elfzofz 13517 |
g | more general (suffix); eliminates "is a set"
hypotheses | |
| No | uniexg 7668 |
gr | graph | |
| No | uhgrf 27799, isumgr 27832, usgrres1 28049 |
grp | group | df-grp 18686 |
Grp | Yes | isgrp 18689, tgpgrp 23351 |
gsum | group sum | df-gsum 17259 |
(πΊ Ξ£g πΉ) | Yes |
gsumval 18467, gsumwrev 19079 |
hash | size (of a set) | df-hash 14159 |
(β―βπ΄) | Yes |
hashgval 14161, hashfz1 14174, hashcl 14184 |
hb | hypothesis builder (prefix) | |
| No | hbxfrbi 1828, hbald 2169, hbequid 37257 |
hm | (monoid, group, ring) homomorphism | |
| No | ismhm 18538, isghm 18940, isrhm 20075 |
i | inference (suffix) | |
| No | eleq1i 2829, tcsni 9613 |
i | implication (suffix) | |
| No | brwdomi 9438, infeq5i 9506 |
id | identity | |
| No | biid 261 |
iedg | indexed edge | df-iedg 27736 |
(iEdgβπΊ) | Yes |
iedgval0 27777, edgiedgb 27791 |
idm | idempotent | |
| No | anidm 566, tpidm13 4716 |
im, imp | implication (label often omitted) |
df-im 14920 | (π΄ β π΅) | Yes |
iman 403, imnan 401, impbidd 209 |
ima | image | df-ima 5644 |
(π΄ β π΅) | Yes | resima 5968, imaundi 6099 |
imp | import | |
| No | biimpa 478, impcom 409 |
in | intersection | df-in 3916 |
(π΄ β© π΅) | Yes | elin 3925, incom 4160 |
inf | infimum | df-inf 9313 |
inf(β+, β*, < ) | Yes |
fiinfcl 9371, infiso 9378 |
is... | is (something a) ...? | |
| No | isring 19892 |
j | joining, disjoining | |
| No | jc 161, jaoi 856 |
l | left | |
| No | olcd 873, simpl 484 |
map | mapping operation or set exponentiation |
df-map 8701 | (π΄ βm π΅) | Yes |
mapvalg 8709, elmapex 8720 |
mat | matrix | df-mat 21677 |
(π Mat π
) | Yes |
matval 21680, matring 21714 |
mdet | determinant (of a square matrix) |
df-mdet 21856 | (π maDet π
) | Yes |
mdetleib 21858, mdetrlin 21873 |
mgm | magma | df-mgm 18432 |
Magma | Yes |
mgmidmo 18450, mgmlrid 18457, ismgm 18433 |
mgp | multiplicative group | df-mgp 19826 |
(mulGrpβπ
) | Yes |
mgpress 19840, ringmgp 19894 |
mnd | monoid | df-mnd 18492 |
Mnd | Yes | mndass 18500, mndodcong 19256 |
mo | "there exists at most one" | df-mo 2540 |
β*π₯π | Yes | eumo 2578, moim 2544 |
mp | modus ponens | ax-mp 5 |
| No | mpd 15, mpi 20 |
mpo | maps-to notation for an operation |
df-mpo 7355 | (π₯ β π΄, π¦ β π΅ β¦ πΆ) | Yes |
mpompt 7463, resmpo 7469 |
mpt | modus ponendo tollens | |
| No | mptnan 1771, mptxor 1772 |
mpt | maps-to notation for a function |
df-mpt 5188 | (π₯ β π΄ β¦ π΅) | Yes |
fconstmpt 5691, resmpt 5988 |
mpt2 | maps-to notation for an operation (deprecated).
We are in the process of replacing mpt2 with mpo in labels. |
df-mpo 7355 | (π₯ β π΄, π¦ β π΅ β¦ πΆ) | Yes |
mpompt 7463, resmpo 7469 |
mul | multiplication (see "t") | df-mul 10997 |
(π΄ Β· π΅) | Yes |
mulcl 11069, divmul 11750, mulcom 11071, mulass 11073 |
n, not | not | |
Β¬ π | Yes |
nan 829, notnotr 130 |
ne | not equal | df-ne | π΄ β π΅ |
Yes | exmidne 2952, neeqtrd 3012 |
nel | not element of | df-nel | π΄ β π΅
|
Yes | neli 3050, nnel 3057 |
ne0 | not equal to zero (see n0) | |
β 0 | No |
negne0d 11444, ine0 11524, gt0ne0 11554 |
nf | "not free in" (prefix) | |
| No | nfnd 1862 |
ngp | normed group | df-ngp 23861 |
NrmGrp | Yes | isngp 23874, ngptps 23880 |
nm | norm (on a group or ring) | df-nm 23860 |
(normβπ) | Yes |
nmval 23867, subgnm 23911 |
nn | positive integers | df-nn 12088 |
β | Yes | nnsscn 12092, nncn 12095 |
nn0 | nonnegative integers | df-n0 12348 |
β0 | Yes | nnnn0 12354, nn0cn 12357 |
n0 | not the empty set (see ne0) | |
β β
| No | n0i 4292, vn0 4297, ssn0 4359 |
OLD | old, obsolete (to be removed soon) | |
| No | 19.43OLD 1887 |
on | ordinal number | df-on 6318 |
π΄ β On | Yes |
elon 6323, 1on 8392 onelon 6339 |
op | ordered pair | df-op 4592 |
β¨π΄, π΅β© | Yes | dfopif 4826, opth 5432 |
or | or | df-or 847 |
(π β¨ π) | Yes |
orcom 869, anor 982 |
ot | ordered triple | df-ot 4594 |
β¨π΄, π΅, πΆβ© | Yes |
euotd 5468, fnotovb 7400 |
ov | operation value | df-ov 7353 |
(π΄πΉπ΅) | Yes
| fnotovb 7400, fnovrn 7522 |
p | plus (see "add"), for all-constant
theorems | df-add 10996 |
(3 + 2) = 5 | Yes |
3p2e5 12238 |
pfx | prefix | df-pfx 14491 |
(π prefix πΏ) | Yes |
pfxlen 14503, ccatpfx 14521 |
pm | Principia Mathematica | |
| No | pm2.27 42 |
pm | partial mapping (operation) | df-pm 8702 |
(π΄ βpm π΅) | Yes | elpmi 8718, pmsspw 8749 |
pr | pair | df-pr 4588 |
{π΄, π΅} | Yes |
elpr 4608, prcom 4692, prid1g 4720, prnz 4737 |
prm, prime | prime (number) | df-prm 16483 |
β | Yes | 1nprm 16490, dvdsprime 16498 |
pss | proper subset | df-pss 3928 |
π΄ β π΅ | Yes | pssss 4054, sspsstri 4061 |
q | rational numbers ("quotients") | df-q 12803 |
β | Yes | elq 12804 |
r | right | |
| No | orcd 872, simprl 770 |
rab | restricted class abstraction |
df-rab 3407 | {π₯ β π΄ β£ π} | Yes |
rabswap 3415, df-oprab 7354 |
ral | restricted universal quantification |
df-ral 3064 | βπ₯ β π΄π | Yes |
ralnex 3074, ralrnmpo 7487 |
rcl | reverse closure | |
| No | ndmfvrcl 6874, nnarcl 8531 |
re | real numbers | df-r 10995 |
β | Yes | recn 11075, 0re 11091 |
rel | relation | df-rel 5638 | Rel π΄ |
Yes | brrelex1 5682, relmpoopab 8015 |
res | restriction | df-res 5643 |
(π΄ βΎ π΅) | Yes |
opelres 5940, f1ores 6794 |
reu | restricted existential uniqueness |
df-reu 3353 | β!π₯ β π΄π | Yes |
nfreud 3403, reurex 3356 |
rex | restricted existential quantification |
df-rex 3073 | βπ₯ β π΄π | Yes |
rexnal 3102, rexrnmpo 7488 |
rmo | restricted "at most one" |
df-rmo 3352 | β*π₯ β π΄π | Yes |
nfrmod 3402, nrexrmo 3373 |
rn | range | df-rn 5642 | ran π΄ |
Yes | elrng 5844, rncnvcnv 5886 |
rng | (unital) ring | df-ring 19890 |
Ring | Yes |
ringidval 19844, isring 19892, ringgrp 19893 |
rot | rotation | |
| No | 3anrot 1101, 3orrot 1093 |
s | eliminates need for syllogism (suffix) |
| | No | ancoms 460 |
sb | (proper) substitution (of a set) |
df-sb 2069 | [π¦ / π₯]π | Yes |
spsbe 2086, sbimi 2078 |
sbc | (proper) substitution of a class |
df-sbc 3739 | [π΄ / π₯]π | Yes |
sbc2or 3747, sbcth 3753 |
sca | scalar | df-sca 17084 |
(Scalarβπ») | Yes |
resssca 17159, mgpsca 19833 |
simp | simple, simplification | |
| No | simpl 484, simp3r3 1284 |
sn | singleton | df-sn 4586 |
{π΄} | Yes | eldifsn 4746 |
sp | specialization | |
| No | spsbe 2086, spei 2394 |
ss | subset | df-ss 3926 |
π΄ β π΅ | Yes | difss 4090 |
struct | structure | df-struct 16954 |
Struct | Yes | brstruct 16955, structfn 16963 |
sub | subtract | df-sub 11321 |
(π΄ β π΅) | Yes |
subval 11326, subaddi 11422 |
sup | supremum | df-sup 9312 |
sup(π΄, π΅, < ) | Yes |
fisupcl 9339, supmo 9322 |
supp | support (of a function) | df-supp 8061 |
(πΉ supp π) | Yes |
ressuppfi 9265, mptsuppd 8086 |
swap | swap (two parts within a theorem) |
| | No | rabswap 3415, 2reuswap 3703 |
syl | syllogism | syl 17 |
| No | 3syl 18 |
sym | symmetric | |
| No | df-symdif 4201, cnvsym 6063 |
symg | symmetric group | df-symg 19081 |
(SymGrpβπ΄) | Yes |
symghash 19091, pgrpsubgsymg 19123 |
t |
times (see "mul"), for all-constant theorems |
df-mul 10997 |
(3 Β· 2) = 6 | Yes |
3t2e6 12253 |
th, t |
theorem |
|
|
No |
nfth 1804, sbcth 3753, weth 10365, ancomst 466 |
tp | triple | df-tp 4590 |
{π΄, π΅, πΆ} | Yes |
eltpi 4647, tpeq1 4702 |
tr | transitive | |
| No | bitrd 279, biantr 805 |
tru, t |
true, truth |
df-tru 1545 |
β€ |
Yes |
bitru 1551, truanfal 1576, biimt 361 |
un | union | df-un 3914 |
(π΄ βͺ π΅) | Yes |
uneqri 4110, uncom 4112 |
unit | unit (in a ring) |
df-unit 19994 | (Unitβπ
) | Yes |
isunit 20009, nzrunit 20660 |
v |
setvar (especially for specializations of
theorems when a class is replaced by a setvar variable) |
|
x |
Yes |
cv 1541, vex 3448, velpw 4564, vtoclf 3515 |
v |
disjoint variable condition used in place of nonfreeness
hypothesis (suffix) |
|
|
No |
spimv 2390 |
vtx |
vertex |
df-vtx 27735 |
(VtxβπΊ) |
Yes |
vtxval0 27776, opvtxov 27742 |
vv |
two disjoint variable conditions used in place of nonfreeness
hypotheses (suffix) |
|
|
No |
19.23vv 1947 |
w | weak (version of a theorem) (suffix) | |
| No | ax11w 2127, spnfw 1984 |
wrd | word |
df-word 14331 | Word π | Yes |
iswrdb 14336, wrdfn 14344, ffz0iswrd 14357 |
xp | cross product (Cartesian product) |
df-xp 5637 | (π΄ Γ π΅) | Yes |
elxp 5654, opelxpi 5668, xpundi 5697 |
xr | eXtended reals | df-xr 11127 |
β* | Yes | ressxr 11133, rexr 11135, 0xr 11136 |
z | integers (from German "Zahlen") |
df-z 12434 | β€ | Yes |
elz 12435, zcn 12438 |
zn | ring of integers mod π | df-zn 20830 |
(β€/nβ€βπ) | Yes |
znval 20861, zncrng 20874, znhash 20888 |
zring | ring of integers | df-zring 20793 |
β€ring | Yes | zringbas 20798, zringcrng 20794
|
0, z |
slashed zero (empty set) | df-nul 4282 |
β
| Yes |
n0i 4292, vn0 4297; snnz 4736, prnz 4737 |
(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision.
(Revised by the Metamath team, 22-Sep-2022.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ π β β’ π |
|
Theorem | conventions-comments 29132 |
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 29130 and links therein.
- Input format.
The input format is ASCII. Tab characters are not allowed. If
non-ASCII characters have to be displayed in comments, use embedded
mathematical symbols when they have been defined (e.g., "` -> `" for
" β") or HTML entities (e.g., "é" for "Γ©").
Default indentation is by two spaces. Lines are hard-wrapped to be at
most 79-character 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". We use the serial comma (Oxford comma) in enumerations. We
use commas after "i.e." and "e.g.".
We avoid beginning a sentence with a symbol (for instance, by writing
"The function F is ..." instead of "F is...").
Since comments may contain many space-separated 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 79-character 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 hard-wrapped to be 79-character 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 first-order 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 space-separated
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 20-Sep-2022, 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, dd-Mmm-yyyy.)" (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, dd-Mmm-yyyy.)" tag.
- Usage of parentheticals.
Usually, the comment of a theorem should contain at most one of the
"Revised by" and "Proof shortened by" parentheticals, see Metamath Book,
pp. 142-143 (there must always be a "Contributed by" parenthetical for
every theorem). Exceptions for "Proof shortened by" parentheticals
are essential additional shortenings by a different person. If a proof
is shortened by the same person, the date within the "Proof shortened
by" parenthetical should be updated only. This also holds for "Revised
by" parentheticals, except that also more than one of such
parentheticals for the same person are acceptable (if there are good
reasons for this). A revision tag is optionally preceded by a short
description of the revision. Since this is somewhat subjective,
judgment and intellectual honesty should be applied, with collegial
settlement in case of dispute.
- Explaining new labels.
A comment should explain the first use of an abbreviation within a
label. This is often in a definition (e.g., Definition df-an 398
introduces the abbreviation "an" for conjunction ("and")), but not
always (e.g., Theorem alim 1813 introduces the abbreviation "al" for
the universal quantifier ("for all")). See conventions-labels 29131 for a
table of abbreviations.
(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision.
(Revised by the Metamath team, 22-Sep-2022.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ π β β’ π |
|
18.1.2 Natural deduction
|
|
Theorem | natded 29133 |
Here are typical natural deduction (ND) rules in the style of Gentzen
and JaΕkowski, 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 |
Ξβ’ π => Ξβ’ π |
idi 1 |
nothing | Reiteration is always redundant in Metamath.
Definition "new rule" in [Pfenning] p. 18,
definition IT in [Clemente] p. 10. |
β§I |
Ξβ’ π & Ξβ’ π => Ξβ’ π β§ π |
jca 513 |
jca 513, pm3.2i 472 |
Definition β§I in [Pfenning] p. 18,
definition Iβ§m,n in [Clemente] p. 10, and
definition β§I in [Indrzejczak] p. 34
(representing both Gentzen's system NK and JaΕkowski) |
β§EL |
Ξβ’ π β§ π => Ξβ’ π |
simpld 496 |
simpld 496, adantr 482 |
Definition β§EL in [Pfenning] p. 18,
definition Eβ§(1) in [Clemente] p. 11, and
definition β§E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and JaΕkowski) |
β§ER |
Ξβ’ π β§ π => Ξβ’ π |
simprd 497 |
simpr 486, adantl 483 |
Definition β§ER in [Pfenning] p. 18,
definition Eβ§(2) in [Clemente] p. 11, and
definition β§E in [Indrzejczak] p. 34
(representing both Gentzen's system NK and JaΕkowski) |
βI |
Ξ, πβ’ π => Ξβ’ π β π |
ex 414 | ex 414 |
Definition βI in [Pfenning] p. 18,
definition I=>m,n in [Clemente] p. 11, and
definition βI in [Indrzejczak] p. 33. |
βE |
Ξβ’ π β π & Ξβ’ π => Ξβ’ π |
mpd 15 | ax-mp 5, mpd 15, mpdan 686, imp 408 |
Definition βE in [Pfenning] p. 18,
definition E=>m,n in [Clemente] p. 11, and
definition βE in [Indrzejczak] p. 33. |
β¨IL | Ξβ’ π =>
Ξβ’ π β¨ π |
olcd 873 |
olc 867, olci 865, olcd 873 |
Definition β¨I in [Pfenning] p. 18,
definition Iβ¨n(1) in [Clemente] p. 12 |
β¨IR | Ξβ’ π =>
Ξβ’ π β¨ π |
orcd 872 |
orc 866, orci 864, orcd 872 |
Definition β¨IR in [Pfenning] p. 18,
definition Iβ¨n(2) in [Clemente] p. 12. |
β¨E | Ξβ’ π β¨ π & Ξ, πβ’ π &
Ξ, πβ’ π => Ξβ’ π |
mpjaodan 958 |
mpjaodan 958, jaodan 957, jaod 858 |
Definition β¨E in [Pfenning] p. 18,
definition Eβ¨m,n,p in [Clemente] p. 12. |
Β¬I | Ξ, πβ’ β₯ => Ξβ’ Β¬ π |
inegd 1562 | pm2.01d 189 |
|
Β¬I | Ξ, πβ’ π & Ξβ’ Β¬ π =>
Ξβ’ Β¬ π |
mtand 815 | mtand 815 |
definition I¬m,n,p in [Clemente] p. 13. |
Β¬I | Ξ, πβ’ π & Ξ, πβ’ Β¬ π =>
Ξβ’ Β¬ π |
pm2.65da 816 | pm2.65da 816 |
Contradiction. |
Β¬I |
Ξ, πβ’ Β¬ π => Ξβ’ Β¬ π |
pm2.01da 798 | pm2.01d 189, pm2.65da 816, pm2.65d 195 |
For an alternative falsum-free natural deduction ruleset |
Β¬E |
Ξβ’ π & Ξβ’ Β¬ π => Ξβ’ β₯ |
pm2.21fal 1564 |
pm2.21dd 194 | |
Β¬E |
Ξ, Β¬ πβ’ β₯ => Ξβ’ π |
|
pm2.21dd 194 |
definition βE in [Indrzejczak] p. 33. |
Β¬E |
Ξβ’ π & Ξβ’ Β¬ π => Ξβ’ π |
pm2.21dd 194 | pm2.21dd 194, pm2.21d 121, pm2.21 123 |
For an alternative falsum-free natural deduction ruleset.
Definition Β¬E in [Pfenning] p. 18. |
β€I | Ξβ’ β€ |
trud 1552 | tru 1546, trud 1552, mptru 1549 |
Definition β€I in [Pfenning] p. 18. |
β₯E | Ξ, β₯β’ π |
falimd 1560 | falim 1559 |
Definition β₯E in [Pfenning] p. 18. |
βI |
Ξβ’ [π / π₯]π => Ξβ’ βπ₯π |
alrimiv 1931 | alrimiv 1931, ralrimiva 3142 |
Definition βIa in [Pfenning] p. 18,
definition Iβn in [Clemente] p. 32. |
βE |
Ξβ’ βπ₯π => Ξβ’ [π‘ / π₯]π |
spsbcd 3752 | spcv 3563, rspcv 3576 |
Definition βE in [Pfenning] p. 18,
definition Eβn,t in [Clemente] p. 32. |
βI |
Ξβ’ [π‘ / π₯]π => Ξβ’ βπ₯π |
spesbcd 3838 | spcev 3564, rspcev 3580 |
Definition βI in [Pfenning] p. 18,
definition Iβn,t in [Clemente] p. 32. |
βE |
Ξβ’ βπ₯π & Ξ, [π / π₯]πβ’ π =>
Ξβ’ π |
exlimddv 1939 | exlimddv 1939, exlimdd 2214,
exlimdv 1937, rexlimdva 3151 |
Definition βEa,u in [Pfenning] p. 18,
definition Eβm,n,p,a in [Clemente] p. 32. |
β₯C |
Ξ, Β¬ πβ’ β₯ => Ξβ’ π |
efald 1563 | efald 1563 |
Proof by contradiction (classical logic),
definition β₯C in [Pfenning] p. 17. |
β₯C |
Ξ, Β¬ πβ’ π => Ξβ’ π |
pm2.18da 799 | pm2.18da 799, pm2.18d 127, pm2.18 128 |
For an alternative falsum-free natural deduction ruleset |
Β¬ Β¬C |
Ξβ’ Β¬ Β¬ π => Ξβ’ π |
notnotrd 133 | notnotrd 133, notnotr 130 |
Double negation rule (classical logic),
definition NNC in [Pfenning] p. 17,
definition E¬n in [Clemente] p. 14. |
EM | Ξβ’ π β¨ Β¬ π |
exmidd 895 | exmid 894 |
Excluded middle (classical logic),
definition XM in [Pfenning] p. 17,
proof 5.11 in [Clemente] p. 14. |
=I | Ξβ’ π΄ = π΄ |
eqidd 2739 | eqid 2738, eqidd 2739 |
Introduce equality,
definition =I in [Pfenning] p. 127. |
=E | Ξβ’ π΄ = π΅ & Ξ[π΄ / π₯]π =>
Ξβ’ [π΅ / π₯]π |
sbceq1dd 3744 | sbceq1d 3743, 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 Ξ
represents the set of (current) hypotheses. We use wff variable names
beginning with π to provide a closer representation
of the Metamath
equivalents (which typically use the antedent π to represent the
context Ξ).
Most of this information was developed by Mario Carneiro and posted on
3-Feb-2017. 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 ex-natded5.2 29134, ex-natded5.3 29137,
ex-natded5.5 29140, ex-natded5.7 29141, ex-natded5.8 29143, ex-natded5.13 29145,
ex-natded9.20 29147, and ex-natded9.26 29149.
(Contributed by DAW, 4-Feb-2017.) (New usage is discouraged.)
|
β’ π β β’ π |
|
18.1.3 Natural deduction examples
These are examples of how natural deduction rules can be applied in Metamath
(both as line-for-line translations of ND rules, and as a way to apply
deduction forms without being limited to applying ND rules). For more
information, see natded 29133 and mmnatded.html 29133. Since these examples should
not be used within proofs of other theorems, especially in mathboxes, they
are marked with "(New usage is discouraged.)".
|
|
Theorem | ex-natded5.2 29134 |
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 | ((π β§ π) β π) |
(π β ((π β§ π) β π)) |
Given |
$e. |
2 | 2 | (π β π) |
(π β (π β π)) |
Given |
$e. |
3 | 1 | π |
(π β π) |
Given |
$e. |
4 | 3 | π |
(π β π) |
βE 2,3 |
mpd 15, the MPE equivalent of βE, 1,2 |
5 | 4 | (π β§ π) |
(π β (π β§ π)) |
β§I 4,3 |
jca 513, the MPE equivalent of β§I, 3,1 |
6 | 6 | π |
(π β π) |
βE 1,5 |
mpd 15, the MPE equivalent of βE, 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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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 ex-natded5.2-2 29135.
A proof without context is shown in ex-natded5.2i 29136.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β ((π β§ π) β π)) & β’ (π β (π β π)) & β’ (π β π) β β’ (π β π) |
|
Theorem | ex-natded5.2-2 29135 |
A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with
ex-natded5.2 29134 and ex-natded5.2i 29136. (Contributed by Mario Carneiro,
9-Feb-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
β’ (π β ((π β§ π) β π)) & β’ (π β (π β π)) & β’ (π β π) β β’ (π β π) |
|
Theorem | ex-natded5.2i 29136 |
The same as ex-natded5.2 29134 and ex-natded5.2-2 29135 but with no context.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ ((π β§ π) β π)
& β’ (π β π)
& β’ π β β’ π |
|
Theorem | ex-natded5.3 29137 |
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 ex-natded5.3-2 29138.
A proof without context is shown in ex-natded5.3i 29139.
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 | (π β π) |
(π β (π β π)) |
Given |
$e; adantr 482 to move it into the ND hypothesis |
2 | 5;6 | (π β π) |
(π β (π β π)) |
Given |
$e; adantr 482 to move it into the ND hypothesis |
3 | 1 | ...| π |
((π β§ π) β π) |
ND hypothesis assumption |
simpr 486, to access the new assumption |
4 | 4 | ... π |
((π β§ π) β π) |
βE 1,3 |
mpd 15, the MPE equivalent of βE, 1.3.
adantr 482 was used to transform its dependency
(we could also use imp 408 to get this directly from 1)
|
5 | 7 | ... π |
((π β§ π) β π) |
βE 2,4 |
mpd 15, the MPE equivalent of βE, 4,6.
adantr 482 was used to transform its dependency |
6 | 8 | ... (π β§ π) |
((π β§ π) β (π β§ π)) |
β§I 4,5 |
jca 513, the MPE equivalent of β§I, 4,7 |
7 | 9 | (π β (π β§ π)) |
(π β (π β (π β§ π))) |
βI 3,6 |
ex 414, the MPE equivalent of βI, 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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β π)) & β’ (π β (π β π)) β β’ (π β (π β (π β§ π))) |
|
Theorem | ex-natded5.3-2 29138 |
A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with
ex-natded5.3 29137 and ex-natded5.3i 29139. (Contributed by Mario Carneiro,
9-Feb-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
β’ (π β (π β π)) & β’ (π β (π β π)) β β’ (π β (π β (π β§ π))) |
|
Theorem | ex-natded5.3i 29139 |
The same as ex-natded5.3 29137 and ex-natded5.3-2 29138 but with no context.
Identical to jccir 523, which should be used instead. (Contributed
by
Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
β’ (π β π)
& β’ (π β π) β β’ (π β (π β§ π)) |
|
Theorem | ex-natded5.5 29140 |
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 |
(π β π) |
(π β (π β π)) |
Given |
$e; adantr 482 to move it into the ND hypothesis |
2 | 5 | Β¬ π |
(π β Β¬ π) | Given |
$e; we'll use adantr 482 to move it into the ND hypothesis |
3 | 1 |
...| π | ((π β§ π) β π) |
ND hypothesis assumption |
simpr 486 |
4 | 4 | ... π |
((π β§ π) β π) |
βE 1,3 |
mpd 15 1,3 |
5 | 6 | ... Β¬ π |
((π β§ π) β Β¬ π) |
IT 2 |
adantr 482 5 |
6 | 7 | Β¬ π |
(π β Β¬ π) |
β§I 3,4,5 |
pm2.65da 816 4,6 |
The original used Latin letters; we have replaced them with
Greek letters to follow Metamath naming conventions and so that
it is easier to follow the Metamath translation.
The Metamath line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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 482; simpr 486 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 197;
a proof without context is shown in mto 196.
(Contributed by David A. Wheeler, 19-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β π)) & β’ (π β Β¬ π) β β’ (π β Β¬ π) |
|
Theorem | ex-natded5.7 29141 |
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 ex-natded5.7-2 29142.
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 |
(π β¨ (π β§ π)) |
(π β (π β¨ (π β§ π))) |
Given |
$e. No need for adantr 482 because we do not move this
into an ND hypothesis |
2 | 1 | ...| π |
((π β§ π) β π) |
ND hypothesis assumption (new scope) |
simpr 486 |
3 | 2 | ... (π β¨ π) |
((π β§ π) β (π β¨ π)) |
β¨IL 2 |
orcd 872, the MPE equivalent of β¨IL, 1 |
4 | 3 | ...| (π β§ π) |
((π β§ (π β§ π)) β (π β§ π)) |
ND hypothesis assumption (new scope) |
simpr 486 |
5 | 4 | ... π |
((π β§ (π β§ π)) β π) |
β§EL 4 |
simpld 496, the MPE equivalent of β§EL, 3 |
6 | 6 | ... (π β¨ π) |
((π β§ (π β§ π)) β (π β¨ π)) |
β¨IR 5 |
olcd 873, the MPE equivalent of β¨IR, 4 |
7 | 7 | (π β¨ π) |
(π β (π β¨ π)) |
β¨E 1,3,6 |
mpjaodan 958, the MPE equivalent of β¨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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π and uses the Metamath equivalents
of the natural deduction rules.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β¨ (π β§ π))) β β’ (π β (π β¨ π)) |
|
Theorem | ex-natded5.7-2 29142 |
A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with
ex-natded5.7 29141. (Contributed by Mario Carneiro,
9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β¨ (π β§ π))) β β’ (π β (π β¨ π)) |
|
Theorem | ex-natded5.8 29143 |
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 |
((π β§ π) β Β¬ π) |
(π β ((π β§ π) β Β¬ π)) |
Given |
$e; adantr 482 to move it into the ND hypothesis |
2 | 3;4 | (π β π) |
(π β (π β π)) | Given |
$e; adantr 482 to move it into the ND hypothesis |
3 | 7;8 |
π | (π β π) |
Given |
$e; adantr 482 to move it into the ND hypothesis |
4 | 1;2 | π | (π β π) |
Given |
$e. adantr 482 to move it into the ND hypothesis |
5 | 6 | ...| π |
((π β§ π) β π) |
ND Hypothesis/Assumption |
simpr 486. New ND hypothesis scope, each reference outside
the scope must change antecedent π to (π β§ π). |
6 | 9 | ... (π β§ π) |
((π β§ π) β (π β§ π)) |
β§I 5,3 |
jca 513 (β§I), 6,8 (adantr 482 to bring in scope) |
7 | 5 | ... Β¬ π |
((π β§ π) β Β¬ π) |
βE 1,6 |
mpd 15 (βE), 2,4 |
8 | 12 | ... π |
((π β§ π) β π) |
βE 2,4 |
mpd 15 (βE), 9,11;
note the contradiction with ND line 7 (MPE line 5) |
9 | 13 | Β¬ π |
(π β Β¬ π) |
Β¬I 5,7,8 |
pm2.65da 816 (Β¬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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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 482; simpr 486 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 ex-natded5.8-2 29144.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β ((π β§ π) β Β¬ π)) & β’ (π β (π β π)) & β’ (π β π)
& β’ (π β π) β β’ (π β Β¬ π) |
|
Theorem | ex-natded5.8-2 29144 |
A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer
line-by-line translation, see ex-natded5.8 29143. (Contributed by Mario
Carneiro, 9-Feb-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
β’ (π β ((π β§ π) β Β¬ π)) & β’ (π β (π β π)) & β’ (π β π)
& β’ (π β π) β β’ (π β Β¬ π) |
|
Theorem | ex-natded5.13 29145 |
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 ex-natded5.13-2 29146.
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 | (π β¨ π) |
(π β (π β¨ π)) |
Given |
$e. |
2;3 | 2 | (π β π) |
(π β (π β π)) | Given |
$e. adantr 482 to move it into the ND hypothesis |
3 | 9 | (Β¬ π β Β¬ π) |
(π β (Β¬ π β Β¬ π)) |
Given |
$e. ad2antrr 725 to move it into the ND sub-hypothesis |
4 | 1 | ...| π |
((π β§ π) β π) |
ND hypothesis assumption |
simpr 486 |
5 | 4 | ... π |
((π β§ π) β π) |
βE 2,4 |
mpd 15 1,3 |
6 | 5 | ... (π β¨ π) |
((π β§ π) β (π β¨ π)) |
β¨I 5 |
orcd 872 4 |
7 | 6 | ...| π |
((π β§ π) β π) |
ND hypothesis assumption |
simpr 486 |
8 | 8 | ... ...| Β¬ π |
(((π β§ π) β§ Β¬ π) β Β¬ π) |
(sub) ND hypothesis assumption |
simpr 486 |
9 | 11 | ... ... Β¬ π |
(((π β§ π) β§ Β¬ π) β Β¬ π) |
βE 3,8 |
mpd 15 8,10 |
10 | 7 | ... ... π |
(((π β§ π) β§ Β¬ π) β π) |
IT 7 |
adantr 482 6 |
11 | 12 | ... Β¬ Β¬ π |
((π β§ π) β Β¬ Β¬ π) |
Β¬I 8,9,10 |
pm2.65da 816 7,11 |
12 | 13 | ... π |
((π β§ π) β π) |
Β¬E 11 |
notnotrd 133 12 |
13 | 14 | ... (π β¨ π) |
((π β§ π) β (π β¨ π)) |
β¨I 12 |
olcd 873 13 |
14 | 16 | (π β¨ π) |
(π β (π β¨ π)) |
β¨E 1,6,13 |
mpjaodan 958 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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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 482; simpr 486 is useful when you want to
depend directly on the new assumption).
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β¨ π)) & β’ (π β (π β π)) & β’ (π β (Β¬ π β Β¬ π)) β β’ (π β (π β¨ π)) |
|
Theorem | ex-natded5.13-2 29146 |
A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare
with ex-natded5.13 29145. (Contributed by Mario Carneiro,
9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β¨ π)) & β’ (π β (π β π)) & β’ (π β (Β¬ π β Β¬ π)) β β’ (π β (π β¨ π)) |
|
Theorem | ex-natded9.20 29147 |
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 |
(π β§ (π β¨ π)) |
(π β (π β§ (π β¨ π))) |
Given |
$e |
2 | 2 | π |
(π β π) |
β§EL 1 |
simpld 496 1 |
3 | 11 |
(π β¨ π) |
(π β (π β¨ π)) |
β§ER 1 |
simprd 497 1 |
4 | 4 |
...| π |
((π β§ π) β π) |
ND hypothesis assumption |
simpr 486 |
5 | 5 |
... (π β§ π) |
((π β§ π) β (π β§ π)) |
β§I 2,4 |
jca 513 3,4 |
6 | 6 |
... ((π β§ π) β¨ (π β§ π)) |
((π β§ π) β ((π β§ π) β¨ (π β§ π))) |
β¨IR 5 |
orcd 872 5 |
7 | 8 |
...| π |
((π β§ π) β π) |
ND hypothesis assumption |
simpr 486 |
8 | 9 |
... (π β§ π) |
((π β§ π) β (π β§ π)) |
β§I 2,7 |
jca 513 7,8 |
9 | 10 |
... ((π β§ π) β¨ (π β§ π)) |
((π β§ π) β ((π β§ π) β¨ (π β§ π))) |
β¨IL 8 |
olcd 873 9 |
10 | 12 |
((π β§ π) β¨ (π β§ π)) |
(π β ((π β§ π) β¨ (π β§ π))) |
β¨E 3,6,9 |
mpjaodan 958 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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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 482; simpr 486 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 ex-natded9.20-2 29148.
(Contributed by David A. Wheeler, 19-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β§ (π β¨ π))) β β’ (π β ((π β§ π) β¨ (π β§ π))) |
|
Theorem | ex-natded9.20-2 29148 |
A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare
with ex-natded9.20 29147. (Contributed by David A. Wheeler,
19-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β (π β§ (π β¨ π))) β β’ (π β ((π β§ π) β¨ (π β§ π))) |
|
Theorem | ex-natded9.26 29149* |
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 π₯ 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 | βπ₯βπ¦π(π₯, π¦) |
(π β βπ₯βπ¦π) |
Given |
$e. |
2 | 6 | ...| βπ¦π(π₯, π¦) |
((π β§ βπ¦π) β βπ¦π) |
ND hypothesis assumption |
simpr 486. Later statements will have this scope. |
3 | 7;5,4 | ... π(π₯, π¦) |
((π β§ βπ¦π) β π) |
βE 2,y |
spsbcd 3752 (βE), 5,6. To use it we need a1i 11 and vex 3448.
This could be immediately done with 19.21bi 2183, but we want to show
the general approach for substitution.
|
4 | 12;8,9,10,11 | ... βπ₯π(π₯, π¦) |
((π β§ βπ¦π) β βπ₯π) |
βI 3,a |
spesbcd 3838 (βI), 11.
To use it we need sylibr 233, which in turn requires sylib 217 and
two uses of sbcid 3755.
This could be more immediately done using 19.8a 2175, but we want to show
the general approach for substitution.
|
5 | 13;1,2 | βπ₯π(π₯, π¦) |
(π β βπ₯π) | βE 1,2,4,a |
exlimdd 2214 (βE), 1,2,3,12.
We'll need supporting
assertions that the variable is free (not bound),
as provided in nfv 1918 and nfe1 2148 (MPE# 1,2) |
6 | 14 | βπ¦βπ₯π(π₯, π¦) |
(π β βπ¦βπ₯π) |
βI 5 |
alrimiv 1931 (βI), 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 line-for-line translation of this
natural deduction approach precedes every line with an antecedent
including π 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, π(π₯, π¦) 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 ex-natded9.26-2 29150.
(Contributed by Mario Carneiro, 9-Feb-2017.)
(Revised by David A. Wheeler, 18-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β βπ₯βπ¦π) β β’ (π β βπ¦βπ₯π) |
|
Theorem | ex-natded9.26-2 29150* |
A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare
with ex-natded9.26 29149. (Contributed by Mario Carneiro,
9-Feb-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
β’ (π β βπ₯βπ¦π) β β’ (π β βπ¦βπ₯π) |
|
18.1.4 Definitional examples
|
|
Theorem | ex-or 29151 |
Example for df-or 847. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9-May-2015.)
|
β’ (2 = 3 β¨ 4 = 4) |
|
Theorem | ex-an 29152 |
Example for df-an 398. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 9-May-2015.)
|
β’ (2 = 2 β§ 3 = 3) |
|
Theorem | ex-dif 29153 |
Example for df-dif 3912. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6-May-2015.)
|
β’ ({1, 3} β {1, 8}) =
{3} |
|
Theorem | ex-un 29154 |
Example for df-un 3914. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6-May-2015.)
|
β’ ({1, 3} βͺ {1, 8}) = {1, 3,
8} |
|
Theorem | ex-in 29155 |
Example for df-in 3916. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6-May-2015.)
|
β’ ({1, 3} β© {1, 8}) = {1} |
|
Theorem | ex-uni 29156 |
Example for df-uni 4865. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 2-Jul-2016.)
|
β’ βͺ {{1, 3}, {1,
8}} = {1, 3, 8} |
|
Theorem | ex-ss 29157 |
Example for df-ss 3926. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6-May-2015.)
|
β’ {1, 2} β {1, 2, 3} |
|
Theorem | ex-pss 29158 |
Example for df-pss 3928. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6-May-2015.)
|
β’ {1, 2} β {1, 2, 3} |
|
Theorem | ex-pw 29159 |
Example for df-pw 4561. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 2-Jul-2016.)
|
β’ (π΄ = {3, 5, 7} β π« π΄ = (({β
} βͺ {{3}, {5},
{7}}) βͺ ({{3, 5}, {3, 7}, {5, 7}} βͺ {{3, 5, 7}}))) |
|
Theorem | ex-pr 29160 |
Example for df-pr 4588. (Contributed by Mario Carneiro,
7-May-2015.)
|
β’ (π΄ β {1, -1} β (π΄β2) = 1) |
|
Theorem | ex-br 29161 |
Example for df-br 5105. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 6-May-2015.)
|
β’ (π
= {β¨2, 6β©, β¨3, 9β©}
β 3π
9) |
|
Theorem | ex-opab 29162* |
Example for df-opab 5167. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18-Jun-2015.)
|
β’ (π
= {β¨π₯, π¦β© β£ (π₯ β β β§ π¦ β β β§ (π₯ + 1) = π¦)} β 3π
4) |
|
Theorem | ex-eprel 29163 |
Example for df-eprel 5535. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18-Jun-2015.)
|
β’ 5 E {1, 5} |
|
Theorem | ex-id 29164 |
Example for df-id 5529. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18-Jun-2015.)
|
β’ (5 I 5 β§ Β¬ 4 I 5) |
|
Theorem | ex-po 29165 |
Example for df-po 5543. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18-Jun-2015.)
|
β’ ( < Po β β§ Β¬ β€ Po
β) |
|
Theorem | ex-xp 29166 |
Example for df-xp 5637. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7-May-2015.)
|
β’ ({1, 5} Γ {2, 7}) = ({β¨1,
2β©, β¨1, 7β©} βͺ {β¨5, 2β©, β¨5,
7β©}) |
|
Theorem | ex-cnv 29167 |
Example for df-cnv 5639. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 6-May-2015.)
|
β’ β‘{β¨2, 6β©, β¨3, 9β©} =
{β¨6, 2β©, β¨9, 3β©} |
|
Theorem | ex-co 29168 |
Example for df-co 5640. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7-May-2015.)
|
β’ ((exp β cos)β0) =
e |
|
Theorem | ex-dm 29169 |
Example for df-dm 5641. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7-May-2015.)
|
β’ (πΉ = {β¨2, 6β©, β¨3, 9β©}
β dom πΉ = {2,
3}) |
|
Theorem | ex-rn 29170 |
Example for df-rn 5642. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7-May-2015.)
|
β’ (πΉ = {β¨2, 6β©, β¨3, 9β©}
β ran πΉ = {6,
9}) |
|
Theorem | ex-res 29171 |
Example for df-res 5643. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7-May-2015.)
|
β’ ((πΉ = {β¨2, 6β©, β¨3, 9β©}
β§ π΅ = {1, 2}) β
(πΉ βΎ π΅) = {β¨2,
6β©}) |
|
Theorem | ex-ima 29172 |
Example for df-ima 5644. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 7-May-2015.)
|
β’ ((πΉ = {β¨2, 6β©, β¨3, 9β©}
β§ π΅ = {1, 2}) β
(πΉ β π΅) = {6}) |
|
Theorem | ex-fv 29173 |
Example for df-fv 6500. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 7-May-2015.)
|
β’ (πΉ = {β¨2, 6β©, β¨3, 9β©}
β (πΉβ3) =
9) |
|
Theorem | ex-1st 29174 |
Example for df-1st 7912. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18-Jun-2015.)
|
β’ (1st ββ¨3, 4β©) =
3 |
|
Theorem | ex-2nd 29175 |
Example for df-2nd 7913. Example by David A. Wheeler. (Contributed
by
Mario Carneiro, 18-Jun-2015.)
|
β’ (2nd ββ¨3, 4β©) =
4 |
|
Theorem | 1kp2ke3k 29176 |
Example for df-dec 12552, 1000 + 2000 = 3000.
This proof disproves (by counterexample) the assertion of Hao Wang, who
stated, "There is a theorem in the primitive notation of set theory
that
corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula
would be forbiddingly long... even if (one) knows the definitions and is
asked to simplify the long formula according to them, chances are he will
make errors and arrive at some incorrect result." (Hao Wang,
"Theory and
practice in mathematics" , In Thomas Tymoczko, editor, New
Directions in
the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc.,
Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)
This is noted in Metamath: A Computer Language for Pure
Mathematics by
Norman Megill (2007) section 1.1.3. Megill then states, "A number of
writers have conveyed the impression that the kind of absolute rigor
provided by Metamath is an impossible dream, suggesting that a complete,
formal verification of a typical theorem would take millions of steps in
untold volumes of books... These writers assume, however, that in order
to achieve the kind of complete formal verification they desire one must
break down a proof into individual primitive steps that make direct
reference to the axioms. This is not necessary. There is no reason not
to make use of previously proved theorems rather than proving them over
and over... A hierarchy of theorems and definitions permits an
exponential growth in the formula sizes and primitive proof steps to be
described with only a linear growth in the number of symbols used. Of
course, this is how ordinary informal mathematics is normally done anyway,
but with Metamath it can be done with absolute rigor and precision."
The proof here starts with (2 + 1) = 3, commutes
it, and repeatedly
multiplies both sides by ten. This is certainly longer than traditional
mathematical proofs, e.g., there are a number of steps explicitly shown
here to show that we're allowed to do operations such as multiplication.
However, while longer, the proof is clearly a manageable size - even
though every step is rigorously derived all the way back to the primitive
notions of set theory and logic. And while there's a risk of making
errors, the many independent verifiers make it much less likely that an
incorrect result will be accepted.
This proof heavily relies on the decimal constructor df-dec 12552 developed by
Mario Carneiro in 2015. The underlying Metamath language has an
intentionally very small set of primitives; it doesn't even have a
built-in construct for numbers. Instead, the digits are defined using
these primitives, and the decimal constructor is used to make it easy to
express larger numbers as combinations of digits.
(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario
Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)
|
β’ (;;;1000 + ;;;2000) = ;;;3000 |
|
Theorem | ex-fl 29177 |
Example for df-fl 13626. Example by David A. Wheeler. (Contributed
by Mario
Carneiro, 18-Jun-2015.)
|
β’ ((ββ(3 / 2)) = 1 β§
(ββ-(3 / 2)) = -2) |
|
Theorem | ex-ceil 29178 |
Example for df-ceil 13627. (Contributed by AV, 4-Sep-2021.)
|
β’ ((ββ(3 / 2)) = 2 β§
(ββ-(3 / 2)) = -1) |
|
Theorem | ex-mod 29179 |
Example for df-mod 13704. (Contributed by AV, 3-Sep-2021.)
|
β’ ((5 mod 3) = 2 β§ (-7 mod 2) =
1) |
|
Theorem | ex-exp 29180 |
Example for df-exp 13897. (Contributed by AV, 4-Sep-2021.)
|
β’ ((5β2) = ;25 β§ (-3β-2) = (1 / 9)) |
|
Theorem | ex-fac 29181 |
Example for df-fac 14102. (Contributed by AV, 4-Sep-2021.)
|
β’ (!β5) = ;;120 |
|
Theorem | ex-bc 29182 |
Example for df-bc 14131. (Contributed by AV, 4-Sep-2021.)
|
β’ (5C3) = ;10 |
|
Theorem | ex-hash 29183 |
Example for df-hash 14159. (Contributed by AV, 4-Sep-2021.)
|
β’ (β―β{0, 1, 2}) =
3 |
|
Theorem | ex-sqrt 29184 |
Example for df-sqrt 15054. (Contributed by AV, 4-Sep-2021.)
|
β’ (ββ;25) = 5 |
|
Theorem | ex-abs 29185 |
Example for df-abs 15055. (Contributed by AV, 4-Sep-2021.)
|
β’ (absβ-2) = 2 |
|
Theorem | ex-dvds 29186 |
Example for df-dvds 16072: 3 divides into 6. (Contributed by David A.
Wheeler, 19-May-2015.)
|
β’ 3 β₯ 6 |
|
Theorem | ex-gcd 29187 |
Example for df-gcd 16310. (Contributed by AV, 5-Sep-2021.)
|
β’ (-6 gcd 9) = 3 |
|
Theorem | ex-lcm 29188 |
Example for df-lcm 16401. (Contributed by AV, 5-Sep-2021.)
|
β’ (6 lcm 9) = ;18 |
|
Theorem | ex-prmo 29189 |
Example for df-prmo 16839: (#pβ10) =
2 Β· 3 Β· 5 Β· 7.
(Contributed by AV, 6-Sep-2021.)
|
β’ (#pβ;10) = ;;210 |
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18.1.5 Other examples
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Theorem | aevdemo 29190* |
Proof illustrating the comment of aev2 2062. (Contributed by BJ,
30-Mar-2021.) (Proof modification is discouraged.)
(New usage is discouraged.)
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β’ (βπ₯ π₯ = π¦ β ((βπβπ π = π β¨ βπ π = π) β§ ββ(π = π β π = π))) |
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Theorem | ex-ind-dvds 29191 |
Example of a proof by induction (divisibility result). (Contributed by
Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
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β’ (π β β0 β 3
β₯ ((4βπ) +
2)) |
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Theorem | ex-fpar 29192 |
Formalized example provided in the comment for fpar 8037.
(Contributed by
AV, 3-Jan-2024.)
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β’ π» = ((β‘(1st βΎ (V Γ V))
β (πΉ β
(1st βΎ (V Γ V)))) β© (β‘(2nd βΎ (V Γ V))
β (πΊ β
(2nd βΎ (V Γ V))))) & β’ π΄ = (0[,)+β) & β’ π΅ = β & β’ πΉ = (β βΎ π΄) & β’ πΊ = (sin βΎ π΅)
β β’ ((π β π΄ β§ π β π΅) β (π( + β π»)π) = ((ββπ) + (sinβπ))) |
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18.2 Humor
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18.2.1 April Fool's theorem
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Theorem | avril1 29193 |
Poisson d'Avril's Theorem. This theorem is noted for its
Selbstdokumentieren property, which means, literally,
"self-documenting" 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 π₯ 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, 1-Apr-2005.) (Proof modification is discouraged.)
(New usage is discouraged.)
A reply to skeptics can be found at mmnotes.txt, under the
1-Apr-2006 entry.
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β’ Β¬ (π΄π« β(iβ1) β§ πΉβ
(0 Β·
1)) |
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Theorem | 2bornot2b 29194 |
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 modern-day 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, 1-Apr-2006.) (Proof modification is discouraged.)
(New usage is discouraged.)
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β’ (2 Β· π΅ β¨ Β¬ 2 Β· π΅) |
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Theorem | helloworld 29195 |
The classic "Hello world" benchmark has been translated into 314
computer
programming languages - see http://helloworldcollection.de.
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 to put it to rest
with a remarkably short proof only four lines long. (Contributed by Prof.
Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.)
(New usage is discouraged.)
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β’ Β¬ (β β (πΏπΏ0) β§ πβ
(R1π)) |
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Theorem | 1p1e2apr1 29196 |
One plus one equals two. Using proof-shortening 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
360-page 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, 1-Apr-2008.)
(Proof modification is discouraged.) (New usage is discouraged.)
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β’ (1 + 1) = 2 |
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Theorem | eqid1 29197 |
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, 1-Apr-2009.)
(Proof modification is discouraged.) (New usage is discouraged.)
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β’ π΄ = π΄ |
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Theorem | 1div0apr 29198 |
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, 1-Apr-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
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β’ (1 / 0) = β
|
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Theorem | topnfbey 29199 |
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, 1-Apr-2020.) (Revised by Thierry Arnoux, 2-Aug-2020.)
(Proof modification is discouraged.) (New usage is discouraged.)
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β’ (π΅ β (0...+β) β +β <
π΅) |
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Theorem | 9p10ne21 29200 |
9 + 10 is not equal to 21. This disproves a popular meme which asserts
that 9 + 10 does equal 21. See
https://www.quora.com/Can-someone-try-to-prove-to-me-that-9+10-21
for
attempts to prove that 9 + 10 = 21, and see https://tinyurl.com/9p10e21
for the history of the 9 + 10 = 21 meme. (Contributed by BTernaryTau,
25-Aug-2023.)
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β’ (9 + ;10) β ;21 |