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Theorem List for Metamath Proof Explorer - 29601-29700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem2clwwlk2 29601* The set (𝑋𝐢2) of double loops of length 2 on a vertex 𝑋 is equal to the set of closed walks with length 2 on 𝑋. Considered as "double loops", the first of the two closed walks/loops is degenerated, i.e., has length 0. (Contributed by AV, 18-Feb-2022.) (Revised by AV, 20-Apr-2022.)
𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    β‡’   (𝑋 ∈ 𝑉 β†’ (𝑋𝐢2) = (𝑋(ClWWalksNOnβ€˜πΊ)2))
 
Theorem2clwwlkel 29602* Characterization of an element of the value of operation 𝐢, i.e., of a word being a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)
𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ (π‘Š ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋)))
 
Theorem2clwwlk2clwwlk 29603* An element of the value of operation 𝐢, i.e., a word being a double loop of length 𝑁 on vertex 𝑋, is composed of two closed walks. (Contributed by AV, 28-Apr-2022.) (Proof shortened by AV, 3-Nov-2022.)
𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ βˆƒπ‘Ž ∈ (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))βˆƒπ‘ ∈ (𝑋(ClWWalksNOnβ€˜πΊ)2)π‘Š = (π‘Ž ++ 𝑏)))
 
Theoremnumclwwlk1lem2foalem 29604 Lemma for numclwwlk1lem2foa 29607. (Contributed by AV, 29-May-2021.) (Revised by AV, 1-Nov-2022.)
(((π‘Š ∈ Word 𝑉 ∧ (β™―β€˜π‘Š) = (𝑁 βˆ’ 2)) ∧ (𝑋 ∈ 𝑉 ∧ π‘Œ ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) prefix (𝑁 βˆ’ 2)) = π‘Š ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 1)) = π‘Œ ∧ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©)β€˜(𝑁 βˆ’ 2)) = 𝑋))
 
Theoremextwwlkfab 29605* The set (𝑋𝐢𝑁) of double loops of length 𝑁 on vertex 𝑋 can be constructed from the set 𝐹 of closed walks on 𝑋 with length smaller by 2 than the fixed length by appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). 3 ≀ 𝑁 is required since for 𝑁 = 2: 𝐹 = (𝑋(ClWWalksNOnβ€˜πΊ)0) = βˆ… (see clwwlk0on0 29345 stating that a closed walk of length 0 is not represented as word), which would result in an empty set on the right hand side, but (𝑋𝐢𝑁) needs not be empty, see 2clwwlk2 29601. (Contributed by Alexander van der Vekens, 18-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑀 prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘€β€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋)})
 
Theoremextwwlkfabel 29606* Characterization of an element of the set (𝑋𝐢𝑁), i.e., a double loop of length 𝑁 on vertex 𝑋 with a construction from the set 𝐹 of closed walks on 𝑋 with length smaller by 2 than the fixed length by appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). (Contributed by AV, 22-Feb-2022.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (π‘Š ∈ (𝑋𝐢𝑁) ↔ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ∧ ((π‘Š prefix (𝑁 βˆ’ 2)) ∈ 𝐹 ∧ (π‘Šβ€˜(𝑁 βˆ’ 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (π‘Šβ€˜(𝑁 βˆ’ 2)) = 𝑋))))
 
Theoremnumclwwlk1lem2foa 29607* Going forth and back from the end of a (closed) walk: π‘Š represents the closed walk p0, ..., p(n-2), p0 = p(n-2). With 𝑋 = p(n-2) = p0 and π‘Œ = p(n-1), ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) represents the closed walk p0, ..., p(n-2), p(n-1), pn = p0 which is a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 2-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ ((π‘Š ∈ 𝐹 ∧ π‘Œ ∈ (𝐺 NeighbVtx 𝑋)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©) ++ βŸ¨β€œπ‘Œβ€βŸ©) ∈ (𝑋𝐢𝑁)))
 
Theoremnumclwwlk1lem2f 29608* 𝑇 is a function, mapping a double loop of length 𝑁 on vertex 𝑋 to the ordered pair of the first loop and the successor of 𝑋 in the second loop, which must be a neighbor of 𝑋. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 23-Feb-2022.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    &   π‘‡ = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)⟢(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
 
Theoremnumclwwlk1lem2fv 29609* Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    &   π‘‡ = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)    β‡’   (π‘Š ∈ (𝑋𝐢𝑁) β†’ (π‘‡β€˜π‘Š) = ⟨(π‘Š prefix (𝑁 βˆ’ 2)), (π‘Šβ€˜(𝑁 βˆ’ 1))⟩)
 
Theoremnumclwwlk1lem2f1 29610* 𝑇 is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 23-Feb-2022.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    &   π‘‡ = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–1-1β†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
 
Theoremnumclwwlk1lem2fo 29611* 𝑇 is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 13-Feb-2022.) (Revised by AV, 31-Oct-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    &   π‘‡ = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–ontoβ†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
 
Theoremnumclwwlk1lem2f1o 29612* 𝑇 is a 1-1 onto function. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    &   π‘‡ = (𝑒 ∈ (𝑋𝐢𝑁) ↦ ⟨(𝑒 prefix (𝑁 βˆ’ 2)), (π‘’β€˜(𝑁 βˆ’ 1))⟩)    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ 𝑇:(𝑋𝐢𝑁)–1-1-ontoβ†’(𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
 
Theoremnumclwwlk1lem2 29613* The set of double loops of length 𝑁 on vertex 𝑋 and the set of closed walks of length less by 2 on 𝑋 combined with the neighbors of 𝑋 are equinumerous. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Jul-2022.) (Proof shortened by AV, 3-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    β‡’   ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3)) β†’ (𝑋𝐢𝑁) β‰ˆ (𝐹 Γ— (𝐺 NeighbVtx 𝑋)))
 
Theoremnumclwwlk1 29614* 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 𝐺 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, but not for n=2, since 𝐹 = βˆ…, but (𝑋𝐢2), the set of closed walks with length 2 on 𝑋, see 2clwwlk2 29601, needs not be βˆ… in this case. This is because of the special definition of 𝐹 and the usage of words to represent (closed) walks, and does not contradict Huneke's statement, which would read "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty closed walks on v, see numclwlk1lem1 29622. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 29624. This case, however, is not required to prove the friendship theorem. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.) (Proof shortened by AV, 31-Jul-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   πΉ = (𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2))    β‡’   (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐢𝑁)) = (𝐾 Β· (β™―β€˜πΉ)))
 
Theoremclwwlknonclwlknonf1o 29615* 𝐹 is a bijection between the two representations of closed walks of a fixed positive length on a fixed vertex. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (Revised by AV, 1-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}    &   πΉ = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))    β‡’   ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
 
Theoremclwwlknonclwlknonen 29616* The sets of the two representations of closed walks of a fixed positive length on a fixed vertex are equinumerous. (Contributed by AV, 27-May-2022.) (Proof shortened by AV, 3-Nov-2022.)
((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtxβ€˜πΊ) ∧ 𝑁 ∈ β„•) β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β‰ˆ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
 
Theoremdlwwlknondlwlknonf1olem1 29617 Lemma 1 for dlwwlknondlwlknonf1o 29618. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)
(((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
 
Theoremdlwwlknondlwlknonf1o 29618* 𝐹 is a bijection between the two representations of double loops of a fixed positive length on a fixed vertex. (Contributed by AV, 30-May-2022.) (Revised by AV, 1-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}    &   π· = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}    &   πΉ = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))    β‡’   ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝐹:π‘Šβ€“1-1-onto→𝐷)
 
Theoremdlwwlknondlwlknonen 29619* The sets of the two representations of double loops of a fixed length on a fixed vertex are equinumerous. (Contributed by AV, 30-May-2022.) (Proof shortened by AV, 3-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}    &   π· = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}    β‡’   ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ π‘Š β‰ˆ 𝐷)
 
Theoremwlkl0 29620* There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (𝑋 ∈ 𝑉 β†’ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 0 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {βŸ¨βˆ…, {⟨0, π‘‹βŸ©}⟩})
 
Theoremclwlknon2num 29621* There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 29358, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (β™―β€˜{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 2 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) = 𝐾)
 
Theoremnumclwlk1lem1 29622* Lemma 1 for numclwlk1 29624 (Statement 9 in [Huneke] p. 2 for n=2): "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)". (Contributed by AV, 23-May-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}    &   πΉ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}    β‡’   (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
 
Theoremnumclwlk1lem2 29623* Lemma 2 for numclwlk1 29624 (Statement 9 in [Huneke] p. 2 for n>2). This theorem corresponds to numclwwlk1 29614, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}    &   πΉ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}    β‡’   (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
 
Theoremnumclwlk1 29624* 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 𝐺 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.)
𝑉 = (Vtxβ€˜πΊ)    &   πΆ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}    &   πΉ = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = (𝑁 βˆ’ 2) ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}    β‡’   (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ (β™―β€˜πΆ) = (𝐾 Β· (β™―β€˜πΉ)))
 
Theoremnumclwwlkovh0 29625* Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 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.)
𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻𝑁) = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  𝑋})
 
Theoremnumclwwlkovh 29626* Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 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.)
𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋𝐻𝑁) = {𝑀 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (π‘€β€˜(𝑁 βˆ’ 2)) β‰  (π‘€β€˜0))})
 
Theoremnumclwwlkovq 29627* Value of operation 𝑄, mapping a vertex 𝑣 and a positive integer 𝑛 to the not closed walks v(0) ... v(n) of length 𝑛 from a fixed vertex 𝑣 = v(0). "Not closed" means v(n) =/= v(0). Remark: 𝑛 ∈ β„•0 would not be useful: numclwwlkqhash 29628 would not hold, because (𝐾↑0) = 1! (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑋𝑄𝑁) = {𝑀 ∈ (𝑁 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑋 ∧ (lastSβ€˜π‘€) β‰  𝑋)})
 
Theoremnumclwwlkqhash 29628* In a 𝐾-regular graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set of closed walks of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 7-Jul-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(𝑋𝑄𝑁)) = ((𝐾↑𝑁) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))))
 
Theoremnumclwwlk2lem1 29629* In a friendship graph, for each walk of length 𝑛 starting at a fixed vertex 𝑣 and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation 𝐻. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation 𝐻, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem generally holds only for friendship graphs, because these guarantee that for the first and last vertex there is a (unique) third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 1-May-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝑄𝑁) β†’ βˆƒ!𝑣 ∈ 𝑉 (π‘Š ++ βŸ¨β€œπ‘£β€βŸ©) ∈ (𝑋𝐻(𝑁 + 2))))
 
Theoremnumclwlk2lem2f 29630* 𝑅 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 𝑋 = 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.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    &   π‘… = (π‘₯ ∈ (𝑋𝐻(𝑁 + 2)) ↦ (π‘₯ prefix (𝑁 + 1)))    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑅:(𝑋𝐻(𝑁 + 2))⟢(𝑋𝑄𝑁))
 
Theoremnumclwlk2lem2fv 29631* Value of the function 𝑅. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    &   π‘… = (π‘₯ ∈ (𝑋𝐻(𝑁 + 2)) ↦ (π‘₯ prefix (𝑁 + 1)))    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (π‘Š ∈ (𝑋𝐻(𝑁 + 2)) β†’ (π‘…β€˜π‘Š) = (π‘Š prefix (𝑁 + 1))))
 
Theoremnumclwlk2lem2f1o 29632* 𝑅 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.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    &   π‘… = (π‘₯ ∈ (𝑋𝐻(𝑁 + 2)) ↦ (π‘₯ prefix (𝑁 + 1)))    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-ontoβ†’(𝑋𝑄𝑁))
 
Theoremnumclwwlk2lem3 29633* In a friendship graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex equals the size of the set of all closed walks of length (𝑁 + 2) starting at this vertex 𝑋 and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Proof shortened by AV, 3-Nov-2022.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜(𝑋𝑄𝑁)) = (β™―β€˜(𝑋𝐻(𝑁 + 2))))
 
Theoremnumclwwlk2 29634* 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 29231, 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.)
𝑉 = (Vtxβ€˜πΊ)    &   π‘„ = (𝑣 ∈ 𝑉, 𝑛 ∈ β„• ↦ {𝑀 ∈ (𝑛 WWalksN 𝐺) ∣ ((π‘€β€˜0) = 𝑣 ∧ (lastSβ€˜π‘€) β‰  𝑣)})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 βˆ’ 2)) βˆ’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))))
 
Theoremnumclwwlk3lem1 29635 Lemma 2 for numclwwlk3 29638. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Proof shortened by AV, 23-Jan-2022.)
((𝐾 ∈ β„‚ ∧ π‘Œ ∈ β„‚ ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (((𝐾↑(𝑁 βˆ’ 2)) βˆ’ π‘Œ) + (𝐾 Β· π‘Œ)) = (((𝐾 βˆ’ 1) Β· π‘Œ) + (𝐾↑(𝑁 βˆ’ 2))))
 
Theoremnumclwwlk3lem2lem 29636* Lemma for numclwwlk3lem2 29637: The set of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the union of the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by AV, 1-May-2022.)
𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = ((𝑋𝐻𝑁) βˆͺ (𝑋𝐢𝑁)))
 
Theoremnumclwwlk3lem2 29637* Lemma 1 for numclwwlk3 29638: The number of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the sum of the number of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 1-Jun-2021.) (Revised by AV, 1-May-2022.)
𝐢 = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) = 𝑣})    &   π» = (𝑣 ∈ 𝑉, 𝑛 ∈ (β„€β‰₯β€˜2) ↦ {𝑀 ∈ (𝑣(ClWWalksNOnβ€˜πΊ)𝑛) ∣ (π‘€β€˜(𝑛 βˆ’ 2)) β‰  𝑣})    β‡’   (((𝐺 ∈ FinUSGraph ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)) = ((β™―β€˜(𝑋𝐻𝑁)) + (β™―β€˜(𝑋𝐢𝑁))))
 
Theoremnumclwwlk3 29638 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 29614) and with v(n-2) =/= v(n) (see numclwwlk2 29634): 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜3))) β†’ (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑁)) = (((𝐾 βˆ’ 1) Β· (β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)(𝑁 βˆ’ 2)))) + (𝐾↑(𝑁 βˆ’ 2))))
 
Theoremnumclwwlk4 29639* 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ β„•) β†’ (β™―β€˜(𝑁 ClWWalksN 𝐺)) = Ξ£π‘₯ ∈ 𝑉 (β™―β€˜(π‘₯(ClWWalksNOnβ€˜πΊ)𝑁)))
 
Theoremnumclwwlk5lem 29640 Lemma for numclwwlk5 29641. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.) (Revised by AV, 7-Mar-2022.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ β„•0) β†’ (2 βˆ₯ (𝐾 βˆ’ 1) β†’ ((β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)2)) mod 2) = 1))
 
Theoremnumclwwlk5 29641 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ β„™ ∧ 𝑃 βˆ₯ (𝐾 βˆ’ 1))) β†’ ((β™―β€˜(𝑋(ClWWalksNOnβ€˜πΊ)𝑃)) mod 𝑃) = 1)
 
Theoremnumclwwlk7lem 29642 Lemma for numclwwlk7 29644, frgrreggt1 29646 and frgrreg 29647: If a finite, nonempty friendship graph is 𝐾-regular, the 𝐾 is a nonnegative integer. (Contributed by AV, 3-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 β‰  βˆ… ∧ 𝑉 ∈ Fin)) β†’ 𝐾 ∈ β„•0)
 
Theoremnumclwwlk6 29643 For a prime divisor 𝑃 of 𝐾 βˆ’ 1, the total number of closed walks of length 𝑃 in a 𝐾-regular friendship graph is equal modulo 𝑃 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ β„™ ∧ 𝑃 βˆ₯ (𝐾 βˆ’ 1))) β†’ ((β™―β€˜(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((β™―β€˜π‘‰) mod 𝑃))
 
Theoremnumclwwlk7 29644 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 29593 or frrusgrord 29594, 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 29518, as well as k-regular (for any k), see 0vtxrgr 28833, but has no closed walk, see 0clwlk0 29385, this theorem would be false for a null graph: ((β™―β€˜(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0 β‰  1, so this case must be excluded (by assuming 𝑉 β‰  βˆ…). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 3-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 β‰  βˆ… ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ β„™ ∧ 𝑃 βˆ₯ (𝐾 βˆ’ 1))) β†’ ((β™―β€˜(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1)
 
Theoremnumclwwlk8 29645 The size of the set of closed walks of length 𝑃, 𝑃 prime, is divisible by 𝑃. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p", see also clwlksndivn 29339. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) (Proof shortened by AV, 2-Mar-2022.)
((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ β„™) β†’ ((β™―β€˜(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)
 
Theoremfrgrreggt1 29646 If a finite nonempty friendship graph is 𝐾-regular with 𝐾 > 1, then 𝐾 must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) β†’ 𝐾 = 2))
 
Theoremfrgrreg 29647 If a finite nonempty friendship graph is 𝐾-regular, then 𝐾 must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) β†’ ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) β†’ (𝐾 = 0 ∨ 𝐾 = 2)))
 
Theoremfrgrregord013 29648 If a finite friendship graph is 𝐾-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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 0 ∨ (β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
 
Theoremfrgrregord13 29649 If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 29648. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 β‰  βˆ…) ∧ 𝐺 RegUSGraph 𝐾) β†’ ((β™―β€˜π‘‰) = 1 ∨ (β™―β€˜π‘‰) = 3))
 
Theoremfrgrogt3nreg 29650* 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (β™―β€˜π‘‰)) β†’ βˆ€π‘˜ ∈ β„•0 Β¬ 𝐺 RegUSGraph π‘˜)
 
Theoremfriendshipgt3 29651* 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (β™―β€˜π‘‰)) β†’ βˆƒπ‘£ ∈ 𝑉 βˆ€π‘€ ∈ (𝑉 βˆ– {𝑣}){𝑣, 𝑀} ∈ (Edgβ€˜πΊ))
 
Theoremfriendship 29652* 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.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ FriendGraph ∧ 𝑉 β‰  βˆ… ∧ 𝑉 ∈ Fin) β†’ βˆƒπ‘£ ∈ 𝑉 βˆ€π‘€ ∈ (𝑉 βˆ– {𝑣}){𝑣, 𝑀} ∈ (Edgβ€˜πΊ))
 
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].
 
Theoremconventions 29653

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‘˜ ∈ 𝐴𝐡 (df-sum 15633) which denotes that index variable π‘˜ ranges over 𝐴 when evaluating 𝐡. Thus, Ξ£π‘˜ ∈ β„•(1 / (2β†‘π‘˜)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 15828). The notation is usually explained in more detail when first introduced.

  • Axiomatic assertions ($a). All axiomatic assertions ($a statements) starting with " " have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats $a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound as discussed below, except for four of them (df-bi 206, df-clab 2711, df-cleq 2725, df-clel 2811) 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 2811. 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 11168, proven by the preceding theorem ax1cn 11144. 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 " πœ‘ & (πœ‘ β†’ πœ“) β‡’ πœ“". 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 11168 (not ax1cn 11144) and ax-1ne0 11179 (not ax1ne0 11155), as these are proven axioms for complex arithmetic. Thus, both ax1cn 11144 and ax1ne0 11155 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".

  • New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use < and ≀ for inequality expressions, and use ((sinβ€˜(i Β· 𝐴)) / i) instead of (sinhβ€˜π΄) 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 10111, we prefer proofs that do not use it, or use weaker versions like countable choice ax-cc 10430 or dependent choice ax-dc 10441. An example is our proof of the Schroeder-Bernstein Theorem sbth 9093, 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 2711, df-cleq 2725, and df-clel 2811, 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œ‘ = phi, πœ“ = psi, πœ’ = chi, πœƒ = theta, 𝜏 = tau, πœ‚ = eta, 𝜁 = zeta, and 𝜎 = sigma. Individual setvar variables are represented with lowercase Latin letters and are generally used in this order: π‘₯, 𝑦, 𝑧, 𝑀, 𝑣, 𝑒, and 𝑑. In addition, the surreal number section uses subscripted lowercase Latin letters such as π‘₯𝑂, π‘₯𝐿, and π‘₯𝑅. These match the conventional literature on surreal numbers. These variables should not be used outside of that section. Variables that represent classes are often represented by uppercase Latin letters: 𝐴, 𝐡, 𝐢, 𝐷, 𝐸, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g., 0 for a zero element (e.g., fsuppcor 9399) and connective symbols such as + for some group addition operation (e.g., grprinvd ). Class variables are selected in alphabetical order starting from 𝐴 if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.

  • Turnstile. "", meaning "It is provable that", is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff Β¬ πœ‘".

  • Biconditional (↔). There are basically two ways to maximize the effectiveness of biconditionals (↔): 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:

    • βˆ€: universal quantifier (wal 1540);
    • βˆƒ: existential quantifier (df-ex 1783);
    • βˆƒ*: at-most-one quantifier (df-mo 2535);
    • βˆƒ!: unique existential quantifier (df-eu 2564).

    The phrase "uniqueness quantifier" is avoided since it is ambiguous: it can be understood as claiming either uniqueness (βˆƒ*) or unique existence (βˆƒ!).

  • Substitution. The expression "[𝑦 / π‘₯]πœ‘" should be read "the formula that results from the proper substitution of 𝑦 for π‘₯ in the formula πœ‘". See df-sb 2069 and the related df-sbc 3779 and df-csb 3895.

  • Is-a-set. " 𝐴 ∈ V" should be read "Class 𝐴 is a set (i.e., exists)." This is a convention based on Definition 2.9 of [Quine] p. 19. See df-v 3477 and isset 3488. However, instead of using 𝐼 ∈ V in the antecedent of a theorem for some variable 𝐼, we now prefer to use 𝐼 ∈ 𝑉 (or another variable if 𝑉 is not available) to make it more general. That way we can often avoid extra uses of elex 3493 and syl 17 in the common case where 𝐼 is already a member of something. For hypotheses ($e statement) of theorems (mostly in inference form), however, 𝐴 ∈ V is used rather than 𝐴 ∈ 𝑉 (e.g., difexi 5329). This is because 𝐴 ∈ V is almost always satisfied using an existence theorem stating " ... ∈ V", and a hard-coded V in the $e statement saves a couple of syntax building steps that substitute V into 𝑉. Notice that this does not hold for hypotheses of theorems in deduction form: Here still (πœ‘ β†’ 𝐴 ∈ 𝑉) should be used rather than (πœ‘ β†’ 𝐴 ∈ V).

  • Converse. The symbol " β—‘" denotes the converse of a relation, so " ◑𝑅" denotes the converse of the class 𝑅, which is typically a relation in that context (see df-cnv 5685). The converse of a relation 𝑅 is sometimes denoted by R-1 in textbooks, especially when 𝑅 is a function, but we avoid this notation since it is generally not a genuine inverse (see f1cocnv1 6864 and funcocnv2 6859 for cases where it is a left or right-inverse). This can be used to define a subset, e.g., df-tan 16015 notates "the set of values whose cosine is a nonzero complex number" as (β—‘cos β€œ (β„‚ βˆ– {0})).

  • Function application. The symbols "(πΉβ€˜π‘₯)" should be read "the value of (function) 𝐹 at π‘₯" and has the same meaning as the more familiar but ambiguous notation F(x). For example, (cosβ€˜0) = 1 (see cos0 16093). 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 6552. 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 7412). For example, the + in (2 + 2); see 2p2e4 12347. 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œ‘ β†’ πœ“), (πœ‘ ∨ πœ“), (πœ‘ ∧ πœ“), and (πœ‘ ↔ πœ“) (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 𝐴 ∈ 𝐡 (see wel 2108), equality 𝐴 = 𝐡 (see df-cleq 2725), subset 𝐴 βŠ† 𝐡 (see df-ss 3966), and less-than 𝐴 < 𝐡 (see df-lt 11123). For the general definition of a binary relation in the form 𝐴𝑅𝐡, see df-br 5150. For example, 0 < 1 (see 0lt1 11736) 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 11445.

  • 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 16008). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 16014). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 16070), its function application form labeled NAMEval (e.g., cosval 16066), and at least one simple value (e.g., cos0 16093). 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 47346 (which is based on the sequence builder seq, see df-seq 13967).

  • Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g., (!β€˜4) = 24 (df-fac 14234 and fac4 14241).

  • 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 11117), while "0g" is the group identity element (df-0g 17387), "0." is the poset zero (df-p0 18378), "0𝑝" is the zero polynomial (df-0p 25187), "0vec" is the zero vector in a normed subcomplex vector space (df-0v 29851), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 18380). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including "1", "+", "βˆ—", and "βˆ₯". 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 βˆ€, are often represented by that letter followed by a period. For example, "A." is used to represent βˆ€, "e." is used to represent ∈, and "E." is used to represent βˆƒ. Single letters are now always variable names, so constants that are often shown as single letters are now typically preceded with "_" in their ASCII representation, for example, "_i" is the ASCII representation for the imaginary unit i. A script font constant is often the letter preceded by "~" meaning "curly", such as "~P" to represent the power class 𝒫.

    Originally, all setvar and class variables used only single letters 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 7412, each operation value can be written as function value of an ordered pair), the convention for its primary usage should be used, e.g., (iEdgβ€˜πΊ) versus (𝑉iEdg𝐸) for the edges of a graph 𝐺 = βŸ¨π‘‰, 𝐸⟩.

  • LaTeX definitions. Each token has a "LaTeX definition" which is used by the Metamath program to output tex files. When writing LaTeX definitions, contributors should favor simplicity over perfection of the display, and should only use core LaTeX symbols or symbols from standard packages; if packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic, graphicx are needed, this should be discussed. A useful resource is The Comprehensive LaTeX Symbol List.

  • Number construction independence. There are many ways to model complex numbers. After deriving the complex number postulates we reintroduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. This also lets us be independent of the specific construction, which we believe is valuable. See mmcomplex.html 7412 for details. Thus, for example, we don't allow the use of βˆ… βˆ‰ 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 βˆ… ∈ 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 (𝐴 < 𝐡 β†’ 𝐡 β‰  𝐴) without any hypotheses, but we require that theorems using this result prove that 𝐴 and 𝐡 are real numbers, so that the statement we use is ltnei 11338. Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 11338:

    1. Having the hypotheses immediately shows the intended domain of applicability (is it ℝ, ℝ*, Ο‰, or something else?), without having to trace back to definitions.
    2. Having the hypotheses forces the intended use of the statement, which generally is desirable.
    3. Many out-of-domain values are dependent on contingent details of definitions, so hypothesis-free theorems would be non-portable and "brittle".
    4. 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.
    5. 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 12213) for the set of positive integers starting from 1, and β„•0 (df-n0 12473) 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 12678, e.g., 4001 (see 4001prm 17078 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 4185). 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 $e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually πœ‘), and every hypothesis ($e statement) is either:
      1. an implication with the same antecedent as the conclusion, or
      2. a definition. A definition can be for a class variable (this is a class variable followed by =, e.g., the definition of 𝐷 in lhop 25533) or a wff variable (this is a wff variable followed by ↔); class variable definitions are more common.
      In practice, a proof of a theorem in deduction form will also contain many steps that are implications where the antecedent is either that wff variable (usually πœ‘) or is a conjunction (πœ‘ ∩ ...) including that wff variable (πœ‘). E.g., a1d 25, unssd 4187. Although they are no real deductions, theorems without $e hypotheses, but in the form (πœ‘ 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 2734, 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 4186).

    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., 𝐴 ∈ V) is converted to a theorem (in closed form) by replacing the hypothesis with an antecedent of the form (𝐴 ∈ 𝑉 β†’, we sometimes suffix the closed form with "g" (for "more general") as in uniex 7731 versus uniexg 7730. 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 5989. Its inference form which was the original "brres", now is brresi 5991. 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 5991. 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 4587, which works in certain cases in set theory. We also sometimes use dedhb 3700. 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œ‘ 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 3700; a list of translations for common natural deduction rules is given in natded 29656.

  • 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(𝐹) in df-recs 8371, rec(𝐹, 𝐼) in df-rdg 8410, seqΟ‰(𝐹, 𝐼) in df-seqom 8448, and seq𝑀( + , 𝐹) in df-seq 13967. These have characteristic function 𝐹 and initial value 𝐼. (Ξ£g in df-gsum 17388 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 8371, but for the "average user" the most useful one is probably df-seq 13967- 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 17080.

  • 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œ‘) is meant to be "the π‘₯ such that πœ‘" (where πœ‘ 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 𝐴 ∈ (πΉβ€˜π΅) β†’ 𝐡 ∈ dom 𝐹, which is useful when 𝐹 is a family of sets. (by this we mean it's a set set even in a type theoretic interpretation.)

    The second approach uses "(New usage is discouraged.)" to prevent unintentional uses of certain properties. For example, you could define some construct 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 𝑦π‘₯ $. $d π‘¦πœ‘ $. df-iota $a (βˆƒ!π‘₯πœ‘ β†’ (β„©π‘₯πœ‘) = βˆͺ {π‘₯ ∣ πœ‘}) $. $}. 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‘₯ = 𝑦 β†’ P(x) = P(y) ) fails to hold if definitions don't unfold without some assumptions. (That is, iotabidv 6528 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 6905 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 9083 is the first lemma for sbth 9093. 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 9083 is the first lemma for sbth 9093.

  • 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 16609"). 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 16609 entries) have an anchor in mmtheorems.html 16609 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 28246. 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 28246. 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 28246.

  • 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:

  • β„²π‘₯πœ‘ is read " π‘₯ is not free in (wff) πœ‘"; see df-nf 1787 (whose description has some important technical details). Similarly, β„²π‘₯𝐴 is read π‘₯ is not free in (class) 𝐴, see df-nfc 2886.
  • "$d π‘₯𝑦 $." should be read "Assume π‘₯ and 𝑦 are distinct variables."
  • "$d πœ‘π‘₯ $." should be read "Assume π‘₯ does not occur in Ο•." Sometimes a theorem is proved using β„²π‘₯πœ‘ (df-nf 1787) in place of "$d πœ‘π‘₯ $." 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 $e version, see the proof of euf 2571 from eu6 2569.
  • "$d 𝐴π‘₯ $." should be read "Assume π‘₯ is not a variable occurring in class 𝐴."
  • "$d 𝐴π‘₯ $. $d πœ“π‘₯ $. $e |- (π‘₯ = 𝐴 β†’ (πœ‘ ↔ πœ“)) $." is an idiom often used instead of explicit substitution, meaning "Assume ψ results from the proper substitution of 𝐴 for π‘₯ in Ο•." Therefore, we often use the term "implicit substitution" for such a hypothesis.
  • Class and wff variables should appear at the beginning of distinct variable conditions, and setvars should be in alphabetical order. E.g., "$d 𝑍π‘₯𝑦 $.", "$d πœ“π‘Žπ‘₯ $.". 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¬ βˆ€π‘₯π‘₯ = 𝑦 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 β„²π‘₯𝐴 or β„²π‘₯πœ‘; here it is. T[x, A] where $d π‘₯𝐴, and you wish to prove β„²π‘₯𝐴 β‡’ T[x, A]. You apply the theorem substituting 𝑦 for π‘₯ and 𝐴 for 𝐴, where 𝑦 is a new dummy variable, so that $d y A is satisfied. You obtain T[y, A], and apply chvar to obtain T[x, A] (or just use mpbir 230 if T[x, A] binds π‘₯). The side goal is (π‘₯ = 𝑦 β†’ ( T[y, A] ↔ 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 2711, df-cleq 2725, and df-clel 2811); 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 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):

  1. The expression must be a biconditional or an equality (i.e. its root-symbol must be ↔ or =). If the proposed definition passes this first rule, we then define its definiendum as its left hand side (LHS) and its definiens as its right hand side (RHS). We define the *defined symbol* as the 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).

  2. 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).

  3. No two variables occurring in the LHS may share a disjoint variable (DV) condition.

  4. All dummy variables are required to be disjoint from any other (dummy or not) variable occurring in this labeled expression.

  5. 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 ( 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 ↔ or =, and that setvar variables are simply variables with the setvar typecode.

  6. 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œ‘ β†’ βˆ€π‘₯πœ‘) for each setvar dummy variable π‘₯ where πœ‘ 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œ‘ β†’ βˆ€π‘₯πœ‘) 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 3476. Definition df-v 3477 V = {π‘₯ ∣ π‘₯ = π‘₯} is justified by the justification theorem vjust 3476 {π‘₯ ∣ π‘₯ = π‘₯} = {𝑦 ∣ 𝑦 = 𝑦}. Another example of a justification theorem is trujust 1544; Definition df-tru 1545 (⊀ ↔ (βˆ€π‘₯π‘₯ = π‘₯ β†’ βˆ€π‘₯π‘₯ = π‘₯)) is justified by trujust 1544 ((βˆ€π‘₯π‘₯ = π‘₯ β†’ βˆ€π‘₯π‘₯ = π‘₯) ↔ (βˆ€π‘¦π‘¦ = 𝑦 β†’ βˆ€π‘¦π‘¦ = 𝑦)).

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.)

πœ‘    β‡’   πœ‘
 
Theoremconventions-labels 29654

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 29653 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 3064"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g., for mdet0 22108: "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 2659 and stirling 44805.
  • 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 3252.
  • 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... 15827. 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 (𝐴 βˆ– 𝐡) is defined in df-dif 3952, and thus its syntax label fragment is "dif". Similarly, the subclass relation 𝐴 βŠ† 𝐡 has syntax label fragment "ss" because it is defined in df-ss 3966. Most theorem names follow from these fragments, for example, the theorem proving (𝐴 βˆ– 𝐡) βŠ† 𝐴 involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 4132. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4630), and the pair construct {𝐴, 𝐡} has fragment "pr" ( from df-pr 4632). 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 𝐴 ∈ 𝐡 does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with (𝐴 ∈ (𝐡 βˆ– {𝐢}) uses is-element-of ("el") of a class difference ("dif") of a singleton ("sn"), it is labeled eldifsn 4791. 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 11116) and "re" represents real numbers ℝ (Definition df-r 11120). The empty set βˆ… often uses fragment 0, even though it is defined in df-nul 4324. The syntax construct (𝐴 + 𝐡) usually uses the fragment "add" (which is consistent with df-add 11121), but "p" is used as the fragment for constant theorems. Equality (𝐴 = 𝐡) often uses "e" as the fragment. As a result, "two plus two equals four" is labeled 2p2e4 12347.
  • 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 16093 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 16014) we have value cosval 16066 and closure coscl 16070.
  • 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 29656 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 β„²π‘₯πœ‘ 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 2571 derived from eu6 2569. The "f" stands for "not free in" which is less restrictive than "does not occur in." The suffix "b" often means "biconditional" (↔, "iff" , "if and only if"), e.g., sspwb 5450. 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 3542. 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 β„²π‘₯πœ‘
    • 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.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
aand (suffix) No biimpa 478, rexlimiva 3148
ablAbelian group df-abl 19651 Abel Yes ablgrp 19653, zringabl 21021
absabsorption No ressabs 17194
absabsolute value (of a complex number) df-abs 15183 (absβ€˜π΄) Yes absval 15185, absneg 15224, abs1 15244
adadding No adantr 482, ad2antlr 726
addadd (see "p") df-add 11121 (𝐴 + 𝐡) Yes addcl 11192, addcom 11400, addass 11197
al"for all" βˆ€π‘₯πœ‘ No alim 1813, alex 1829
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 398 (πœ‘ ∧ πœ“) Yes anor 982, iman 403, imnan 401
antantecedent No adantr 482
assassociative No biass 386, orass 921, mulass 11198
asymasymmetric, antisymmetric No intasym 6117, asymref 6118, posasymb 18272
axaxiom No ax6dgen 2125, ax1cn 11144
bas, base base (set of an extensible structure) df-base 17145 (Baseβ€˜π‘†) Yes baseval 17146, ressbas 17179, cnfldbas 20948
b, bibiconditional ("iff", "if and only if") df-bi 206 (πœ‘ ↔ πœ“) Yes impbid 211, sspwb 5450
brbinary relation df-br 5150 𝐴𝑅𝐡 Yes brab1 5197, brun 5200
ccommutes, commuted (suffix) No biimpac 480
ccontraction (suffix) No sylc 65, syl2anc 585
cbvchange bound variable No cbvalivw 2011, cbvrex 3360
cdmcodomain No ffvelcdm 7084, focdmex 7942
clclosure No ifclda 4564, ovrcl 7450, zaddcl 12602
cncomplex numbers df-c 11116 β„‚ Yes nnsscn 12217, nncn 12220
cnfldfield of complex numbers df-cnfld 20945 β„‚fld Yes cnfldbas 20948, cnfldinv 20976
cntzcentralizer df-cntz 19181 (Cntzβ€˜π‘€) Yes cntzfval 19184, dprdfcntz 19885
cnvconverse df-cnv 5685 ◑𝐴 Yes opelcnvg 5881, f1ocnv 6846
cocomposition df-co 5686 (𝐴 ∘ 𝐡) Yes cnvco 5886, fmptco 7127
comcommutative No orcom 869, bicomi 223, eqcomi 2742
concontradiction, contraposition No condan 817, con2d 134
csbclass substitution df-csb 3895 ⦋𝐴 / π‘₯⦌𝐡 Yes csbid 3907, csbie2g 3937
cygcyclic group df-cyg 19746 CycGrp Yes iscyg 19747, zringcyg 21039
ddeduction form (suffix) No idd 24, impbid 211
df(alternate) definition (prefix) No dfrel2 6189, dffn2 6720
di, distrdistributive No andi 1007, imdi 391, ordi 1005, difindi 4282, ndmovdistr 7596
difclass difference df-dif 3952 (𝐴 βˆ– 𝐡) Yes difss 4132, difindi 4282
divdivision df-div 11872 (𝐴 / 𝐡) Yes divcl 11878, divval 11874, divmul 11875
dmdomain df-dm 5687 dom 𝐴 Yes dmmpt 6240, iswrddm0 14488
e, eq, equequals (equ for setvars, eq for classes) df-cleq 2725 𝐴 = 𝐡 Yes 2p2e4 12347, uneqri 4152, equtr 2025
edgedge df-edg 28308 (Edgβ€˜πΊ) Yes edgopval 28311, usgredgppr 28453
elelement of 𝐴 ∈ 𝐡 Yes eldif 3959, eldifsn 4791, elssuni 4942
enequinumerous df-en 𝐴 β‰ˆ 𝐡 Yes domen 8957, enfi 9190
eu"there exists exactly one" eu6 2569 βˆƒ!π‘₯πœ‘ Yes euex 2572, euabsn 4731
exexists (i.e. is a set) ∈ V No brrelex1 5730, 0ex 5308
ex, e"there exists (at least one)" df-ex 1783 βˆƒπ‘₯πœ‘ Yes exim 1837, alex 1829
expexport No expt 177, expcom 415
f"not free in" (suffix) No equs45f 2459, sbf 2263
ffunction df-f 6548 𝐹:𝐴⟢𝐡 Yes fssxp 6746, opelf 6753
falfalse df-fal 1555 βŠ₯ Yes bifal 1558, falantru 1577
fifinite intersection df-fi 9406 (fiβ€˜π΅) Yes fival 9407, inelfi 9413
fi, finfinite df-fin 8943 Fin Yes isfi 8972, snfi 9044, onfin 9230
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 36860) df-field 20360 Field Yes isfld 20368, fldidom 20923
fnfunction with domain df-fn 6547 𝐴 Fn 𝐡 Yes ffn 6718, fndm 6653
frgpfree group df-frgp 19578 (freeGrpβ€˜πΌ) Yes frgpval 19626, frgpadd 19631
fsuppfinitely supported function df-fsupp 9362 𝑅 finSupp 𝑍 Yes isfsupp 9365, fdmfisuppfi 9372, fsuppco 9397
funfunction df-fun 6546 Fun 𝐹 Yes funrel 6566, ffun 6721
fvfunction value df-fv 6552 (πΉβ€˜π΄) Yes fvres 6911, swrdfv 14598
fzfinite set of sequential integers df-fz 13485 (𝑀...𝑁) Yes fzval 13486, eluzfz 13496
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 13599, fz0tp 13602
fzohalf-open integer range df-fzo 13628 (𝑀..^𝑁) Yes elfzo 13634, elfzofz 13648
gmore general (suffix); eliminates "is a set" hypotheses No uniexg 7730
grgraph No uhgrf 28322, isumgr 28355, usgrres1 28572
grpgroup df-grp 18822 Grp Yes isgrp 18825, tgpgrp 23582
gsumgroup sum df-gsum 17388 (𝐺 Ξ£g 𝐹) Yes gsumval 18596, gsumwrev 19233
hashsize (of a set) df-hash 14291 (β™―β€˜π΄) Yes hashgval 14293, hashfz1 14306, hashcl 14316
hbhypothesis builder (prefix) No hbxfrbi 1828, hbald 2169, hbequid 37779
hm(monoid, group, ring) homomorphism No ismhm 18673, isghm 19092, isrhm 20257
iinference (suffix) No eleq1i 2825, tcsni 9738
iimplication (suffix) No brwdomi 9563, infeq5i 9631
ididentity No biid 261
iedgindexed edge df-iedg 28259 (iEdgβ€˜πΊ) Yes iedgval0 28300, edgiedgb 28314
idmidempotent No anidm 566, tpidm13 4761
im, impimplication (label often omitted) df-im 15048 (𝐴 β†’ 𝐡) Yes iman 403, imnan 401, impbidd 209
imaimage df-ima 5690 (𝐴 β€œ 𝐡) Yes resima 6016, imaundi 6150
impimport No biimpa 478, impcom 409
inintersection df-in 3956 (𝐴 ∩ 𝐡) Yes elin 3965, incom 4202
infinfimum df-inf 9438 inf(ℝ+, ℝ*, < ) Yes fiinfcl 9496, infiso 9503
is...is (something a) ...? No isring 20060
jjoining, disjoining No jc 161, jaoi 856
lleft No olcd 873, simpl 484
mapmapping operation or set exponentiation df-map 8822 (𝐴 ↑m 𝐡) Yes mapvalg 8830, elmapex 8842
matmatrix df-mat 21908 (𝑁 Mat 𝑅) Yes matval 21911, matring 21945
mdetdeterminant (of a square matrix) df-mdet 22087 (𝑁 maDet 𝑅) Yes mdetleib 22089, mdetrlin 22104
mgmmagma df-mgm 18561 Magma Yes mgmidmo 18579, mgmlrid 18586, ismgm 18562
mgpmultiplicative group df-mgp 19988 (mulGrpβ€˜π‘…) Yes mgpress 20002, ringmgp 20062
mndmonoid df-mnd 18626 Mnd Yes mndass 18634, mndodcong 19410
mo"there exists at most one" df-mo 2535 βˆƒ*π‘₯πœ‘ Yes eumo 2573, moim 2539
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mpomaps-to notation for an operation df-mpo 7414 (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢) Yes mpompt 7522, resmpo 7528
mptmodus ponendo tollens No mptnan 1771, mptxor 1772
mptmaps-to notation for a function df-mpt 5233 (π‘₯ ∈ 𝐴 ↦ 𝐡) Yes fconstmpt 5739, resmpt 6038
mpt2maps-to notation for an operation (deprecated). We are in the process of replacing mpt2 with mpo in labels. df-mpo 7414 (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ 𝐢) Yes mpompt 7522, resmpo 7528
mulmultiplication (see "t") df-mul 11122 (𝐴 Β· 𝐡) Yes mulcl 11194, divmul 11875, mulcom 11196, mulass 11198
n, notnot Β¬ πœ‘ Yes nan 829, notnotr 130
nenot equaldf-ne 𝐴 β‰  𝐡 Yes exmidne 2951, neeqtrd 3011
nelnot element ofdf-nel 𝐴 βˆ‰ 𝐡 Yes neli 3049, nnel 3057
ne0not equal to zero (see n0) β‰  0 No negne0d 11569, ine0 11649, gt0ne0 11679
nf "not free in" (prefix) No nfnd 1862
ngpnormed group df-ngp 24092 NrmGrp Yes isngp 24105, ngptps 24111
nmnorm (on a group or ring) df-nm 24091 (normβ€˜π‘Š) Yes nmval 24098, subgnm 24142
nnpositive integers df-nn 12213 β„• Yes nnsscn 12217, nncn 12220
nn0nonnegative integers df-n0 12473 β„•0 Yes nnnn0 12479, nn0cn 12482
n0not the empty set (see ne0) β‰  βˆ… No n0i 4334, vn0 4339, ssn0 4401
OLDold, obsolete (to be removed soon) No 19.43OLD 1887
onordinal number df-on 6369 𝐴 ∈ On Yes elon 6374, 1on 8478 onelon 6390
opordered pair df-op 4636 ⟨𝐴, 𝐡⟩ Yes dfopif 4871, opth 5477
oror df-or 847 (πœ‘ ∨ πœ“) Yes orcom 869, anor 982
otordered triple df-ot 4638 ⟨𝐴, 𝐡, 𝐢⟩ Yes euotd 5514, fnotovb 7459
ovoperation value df-ov 7412 (𝐴𝐹𝐡) Yes fnotovb 7459, fnovrn 7582
pplus (see "add"), for all-constant theorems df-add 11121 (3 + 2) = 5 Yes 3p2e5 12363
pfxprefix df-pfx 14621 (π‘Š prefix 𝐿) Yes pfxlen 14633, ccatpfx 14651
pmPrincipia Mathematica No pm2.27 42
pmpartial mapping (operation) df-pm 8823 (𝐴 ↑pm 𝐡) Yes elpmi 8840, pmsspw 8871
prpair df-pr 4632 {𝐴, 𝐡} Yes elpr 4652, prcom 4737, prid1g 4765, prnz 4782
prm, primeprime (number) df-prm 16609 β„™ Yes 1nprm 16616, dvdsprime 16624
pssproper subset df-pss 3968 𝐴 ⊊ 𝐡 Yes pssss 4096, sspsstri 4103
q rational numbers ("quotients") df-q 12933 β„š Yes elq 12934
rreversed (suffix) No pm4.71r 560, caovdir 7641
rright No orcd 872, simprl 770
rabrestricted class abstraction df-rab 3434 {π‘₯ ∈ 𝐴 ∣ πœ‘} Yes rabswap 3442, df-oprab 7413
ralrestricted universal quantification df-ral 3063 βˆ€π‘₯ ∈ π΄πœ‘ Yes ralnex 3073, ralrnmpo 7547
rclreverse closure No ndmfvrcl 6928, nnarcl 8616
rereal numbers df-r 11120 ℝ Yes recn 11200, 0re 11216
relrelation df-rel 5684 Rel 𝐴 Yes brrelex1 5730, relmpoopab 8080
resrestriction df-res 5689 (𝐴 β†Ύ 𝐡) Yes opelres 5988, f1ores 6848
reurestricted existential uniqueness df-reu 3378 βˆƒ!π‘₯ ∈ π΄πœ‘ Yes nfreud 3430, reurex 3381
rexrestricted existential quantification df-rex 3072 βˆƒπ‘₯ ∈ π΄πœ‘ Yes rexnal 3101, rexrnmpo 7548
rmorestricted "at most one" df-rmo 3377 βˆƒ*π‘₯ ∈ π΄πœ‘ Yes nfrmod 3429, nrexrmo 3398
rnrange df-rn 5688 ran 𝐴 Yes elrng 5892, rncnvcnv 5934
ring(unital) ring df-ring 20058 Ring Yes ringidval 20006, isring 20060, ringgrp 20061
rngnon-unital ring df-rng 46649 Rng Yes isrng 46650, rngabl 46651, rnglz 46664
rotrotation No 3anrot 1101, 3orrot 1093
seliminates need for syllogism (suffix) No ancoms 460
sb(proper) substitution (of a set) df-sb 2069 [𝑦 / π‘₯]πœ‘ Yes spsbe 2086, sbimi 2078
sbc(proper) substitution of a class df-sbc 3779 [𝐴 / π‘₯]πœ‘ Yes sbc2or 3787, sbcth 3793
scascalar df-sca 17213 (Scalarβ€˜π») Yes resssca 17288, mgpsca 19995
simpsimple, simplification No simpl 484, simp3r3 1284
snsingleton df-sn 4630 {𝐴} Yes eldifsn 4791
spspecialization No spsbe 2086, spei 2394
sssubset df-ss 3966 𝐴 βŠ† 𝐡 Yes difss 4132
structstructure df-struct 17080 Struct Yes brstruct 17081, structfn 17089
subsubtract df-sub 11446 (𝐴 βˆ’ 𝐡) Yes subval 11451, subaddi 11547
supsupremum df-sup 9437 sup(𝐴, 𝐡, < ) Yes fisupcl 9464, supmo 9447
suppsupport (of a function) df-supp 8147 (𝐹 supp 𝑍) Yes ressuppfi 9390, mptsuppd 8172
swapswap (two parts within a theorem) No rabswap 3442, 2reuswap 3743
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 4243, cnvsym 6114
symgsymmetric group df-symg 19235 (SymGrpβ€˜π΄) Yes symghash 19245, pgrpsubgsymg 19277
t times (see "mul"), for all-constant theorems df-mul 11122 (3 Β· 2) = 6 Yes 3t2e6 12378
th, t theorem No nfth 1804, sbcth 3793, weth 10490, ancomst 466
tptriple df-tp 4634 {𝐴, 𝐡, 𝐢} Yes eltpi 4692, tpeq1 4747
trtransitive No bitrd 279, biantr 805
tru, t true, truth df-tru 1545 ⊀ Yes bitru 1551, truanfal 1576, biimt 361
ununion df-un 3954 (𝐴 βˆͺ 𝐡) Yes uneqri 4152, uncom 4154
unitunit (in a ring) df-unit 20172 (Unitβ€˜π‘…) Yes isunit 20187, nzrunit 20301
v setvar (especially for specializations of theorems when a class is replaced by a setvar variable) x Yes cv 1541, vex 3479, velpw 4608, vtoclf 3548
v disjoint variable condition used in place of nonfreeness hypothesis (suffix) No spimv 2390
vtx vertex df-vtx 28258 (Vtxβ€˜πΊ) Yes vtxval0 28299, opvtxov 28265
vv two disjoint variable conditions used in place of nonfreeness hypotheses (suffix) No 19.23vv 1947
wweak (version of a theorem) (suffix) No ax11w 2127, spnfw 1984
wrdword df-word 14465 Word 𝑆 Yes iswrdb 14470, wrdfn 14478, ffz0iswrd 14491
xpcross product (Cartesian product) df-xp 5683 (𝐴 Γ— 𝐡) Yes elxp 5700, opelxpi 5714, xpundi 5745
xreXtended reals df-xr 11252 ℝ* Yes ressxr 11258, rexr 11260, 0xr 11261
z integers (from German "Zahlen") df-z 12559 β„€ Yes elz 12560, zcn 12563
zn ring of integers mod 𝑁 df-zn 21056 (β„€/nβ„€β€˜π‘) Yes znval 21087, zncrng 21100, znhash 21114
zringring of integers df-zring 21018 β„€ring Yes zringbas 21023, zringcrng 21019
0, z slashed zero (empty set) df-nul 4324 βˆ… Yes n0i 4334, vn0 4339; snnz 4781, prnz 4782

(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.)

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Theoremconventions-comments 29655

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 29653 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., "&eacute;" 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 29654 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.)

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18.1.2  Natural deduction
 
Theoremnatded 29656 Here are typical natural deduction (ND) rules in the style of Gentzen and JaΕ›kowski, 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.

NameNatural Deduction RuleTranslation RecommendationComments
IT Ξ“πœ“ => Ξ“πœ“ idi 1 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
∧I Ξ“πœ“ & Ξ“πœ’ => Ξ“πœ“ ∧ πœ’ jca 513 jca 513, pm3.2i 472 Definition ∧I in [Pfenning] p. 18, definition I∧m,n in [Clemente] p. 10, and definition ∧I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and JaΕ›kowski)
∧EL Ξ“πœ“ ∧ πœ’ => Ξ“πœ“ simpld 496 simpld 496, adantr 482 Definition ∧EL in [Pfenning] p. 18, definition E∧(1) in [Clemente] p. 11, and definition ∧E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and JaΕ›kowski)
∧ER Ξ“πœ“ ∧ πœ’ => Ξ“πœ’ simprd 497 simpr 486, adantl 483 Definition ∧ER in [Pfenning] p. 18, definition E∧(2) in [Clemente] p. 11, and definition ∧E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and JaΕ›kowski)
β†’I Ξ“, πœ“πœ’ => Ξ“πœ“ β†’ πœ’ ex 414 ex 414 Definition β†’I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition β†’I in [Indrzejczak] p. 33.
β†’E Ξ“πœ“ β†’ πœ’ & Ξ“πœ“ => Ξ“πœ’ mpd 15 ax-mp 5, mpd 15, mpdan 686, imp 408 Definition β†’E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition β†’E in [Indrzejczak] p. 33.
∨IL Ξ“πœ“ => Ξ“πœ“ ∨ πœ’ olcd 873 olc 867, olci 865, olcd 873 Definition ∨I in [Pfenning] p. 18, definition I∨n(1) in [Clemente] p. 12
∨IR Ξ“πœ’ => Ξ“πœ“ ∨ πœ’ orcd 872 orc 866, orci 864, orcd 872 Definition ∨IR in [Pfenning] p. 18, definition I∨n(2) in [Clemente] p. 12.
∨E Ξ“πœ“ ∨ πœ’ & Ξ“, πœ“πœƒ & Ξ“, πœ’πœƒ => Ξ“πœƒ mpjaodan 958 mpjaodan 958, jaodan 957, jaod 858 Definition ∨E in [Pfenning] p. 18, definition E∨m,n,p in [Clemente] p. 12.
Β¬I Ξ“, πœ“βŠ₯ => Γ¬ πœ“ inegd 1562 pm2.01d 189
Β¬I Ξ“, πœ“πœƒ & Γ¬ πœƒ => Γ¬ πœ“ mtand 815 mtand 815 definition IΒ¬m,n,p in [Clemente] p. 13.
Β¬I Ξ“, πœ“πœ’ & Ξ“, πœ“Β¬ πœ’ => Γ¬ πœ“ pm2.65da 816 pm2.65da 816 Contradiction.
Β¬I Ξ“, πœ“Β¬ πœ“ => Γ¬ πœ“ pm2.01da 798 pm2.01d 189, pm2.65da 816, pm2.65d 195 For an alternative falsum-free natural deduction ruleset
Β¬E Ξ“πœ“ & Γ¬ πœ“ => Ξ“βŠ₯ pm2.21fal 1564 pm2.21dd 194
Β¬E Ξ“, Β¬ πœ“βŠ₯ => Ξ“πœ“ pm2.21dd 194 definition β†’E in [Indrzejczak] p. 33.
Β¬E Ξ“πœ“ & Γ¬ πœ“ => Ξ“πœƒ 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.
⊀I Ξ“βŠ€ trud 1552 tru 1546, trud 1552, mptru 1549 Definition ⊀I in [Pfenning] p. 18.
βŠ₯E Ξ“, βŠ₯πœƒ falimd 1560 falim 1559 Definition βŠ₯E in [Pfenning] p. 18.
βˆ€I Ξ“[π‘Ž / π‘₯]πœ“ => Ξ“βˆ€π‘₯πœ“ alrimiv 1931 alrimiv 1931, ralrimiva 3147 Definition βˆ€Ia in [Pfenning] p. 18, definition Iβˆ€n in [Clemente] p. 32.
βˆ€E Ξ“βˆ€π‘₯πœ“ => Ξ“[𝑑 / π‘₯]πœ“ spsbcd 3792 spcv 3596, rspcv 3609 Definition βˆ€E in [Pfenning] p. 18, definition Eβˆ€n,t in [Clemente] p. 32.
βˆƒI Ξ“[𝑑 / π‘₯]πœ“ => Ξ“βˆƒπ‘₯πœ“ spesbcd 3878 spcev 3597, rspcev 3613 Definition βˆƒI in [Pfenning] p. 18, definition Iβˆƒn,t in [Clemente] p. 32.
βˆƒE Ξ“βˆƒπ‘₯πœ“ & Ξ“, [π‘Ž / π‘₯]πœ“πœƒ => Ξ“πœƒ exlimddv 1939 exlimddv 1939, exlimdd 2214, exlimdv 1937, rexlimdva 3156 Definition βˆƒEa,u in [Pfenning] p. 18, definition Eβˆƒm,n,p,a in [Clemente] p. 32.
βŠ₯C Ξ“, Β¬ πœ“βŠ₯ => Ξ“πœ“ efald 1563 efald 1563 Proof by contradiction (classical logic), definition βŠ₯C in [Pfenning] p. 17.
βŠ₯C Ξ“, Β¬ πœ“πœ“ => Ξ“πœ“ pm2.18da 799 pm2.18da 799, pm2.18d 127, pm2.18 128 For an alternative falsum-free natural deduction ruleset
Β¬ Β¬C Γ¬ Β¬ πœ“ => Ξ“πœ“ notnotrd 133 notnotrd 133, notnotr 130 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition EΒ¬n in [Clemente] p. 14.
EM Ξ“πœ“ ∨ Β¬ πœ“ exmidd 895 exmid 894 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2734 eqid 2733, eqidd 2734 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐡 & Ξ“[𝐴 / π‘₯]πœ“ => Ξ“[𝐡 / π‘₯]πœ“ sbceq1dd 3784 sbceq1d 3783, 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œ“ to provide a closer representation of the Metamath equivalents (which typically use the antedent πœ‘ 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 29657, ex-natded5.3 29660, ex-natded5.5 29663, ex-natded5.7 29664, ex-natded5.8 29666, ex-natded5.13 29668, ex-natded9.20 29670, and ex-natded9.26 29672.

(Contributed by DAW, 4-Feb-2017.) (New usage is discouraged.)

πœ‘    β‡’   πœ‘
 
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 29656 and mmnatded.html 29656. Since these examples should not be used within proofs of other theorems, especially in mathboxes, they are marked with "(New usage is discouraged.)".

 
Theoremex-natded5.2 29657 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 TranslationND Rationale MPE Rationale
15 ((πœ“ ∧ πœ’) β†’ πœƒ) (πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ πœƒ)) Given $e.
22 (πœ’ β†’ πœ“) (πœ‘ β†’ (πœ’ β†’ πœ“)) Given $e.
31 πœ’ (πœ‘ β†’ πœ’) Given $e.
43 πœ“ (πœ‘ β†’ πœ“) β†’E 2,3 mpd 15, the MPE equivalent of β†’E, 1,2
54 (πœ“ ∧ πœ’) (πœ‘ β†’ (πœ“ ∧ πœ’)) ∧I 4,3 jca 513, the MPE equivalent of ∧I, 3,1
66 πœƒ (πœ‘ β†’ πœƒ) β†’E 1,5 mpd 15, the MPE equivalent of β†’E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including πœ‘ 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 29658. A proof without context is shown in ex-natded5.2i 29659. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ πœƒ))    &   (πœ‘ β†’ (πœ’ β†’ πœ“))    &   (πœ‘ β†’ πœ’)    β‡’   (πœ‘ β†’ πœƒ)
 
Theoremex-natded5.2-2 29658 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 29657 and ex-natded5.2i 29659. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ πœƒ))    &   (πœ‘ β†’ (πœ’ β†’ πœ“))    &   (πœ‘ β†’ πœ’)    β‡’   (πœ‘ β†’ πœƒ)
 
Theoremex-natded5.2i 29659 The same as ex-natded5.2 29657 and ex-natded5.2-2 29658 but with no context. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((πœ“ ∧ πœ’) β†’ πœƒ)    &   (πœ’ β†’ πœ“)    &   πœ’    β‡’   πœƒ
 
Theoremex-natded5.3 29660 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 29661. A proof without context is shown in ex-natded5.3i 29662. 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 TranslationND Rationale MPE Rationale
12;3 (πœ“ β†’ πœ’) (πœ‘ β†’ (πœ“ β†’ πœ’)) Given $e; adantr 482 to move it into the ND hypothesis
25;6 (πœ’ β†’ πœƒ) (πœ‘ β†’ (πœ’ β†’ πœƒ)) Given $e; adantr 482 to move it into the ND hypothesis
31 ...| πœ“ ((πœ‘ ∧ πœ“) β†’ πœ“) ND hypothesis assumption simpr 486, to access the new assumption
44 ... πœ’ ((πœ‘ ∧ πœ“) β†’ πœ’) β†’E 1,3 mpd 15, the MPE equivalent of β†’E, 1.3. adantr 482 was used to transform its dependency (we could also use imp 408 to get this directly from 1)
57 ... πœƒ ((πœ‘ ∧ πœ“) β†’ πœƒ) β†’E 2,4 mpd 15, the MPE equivalent of β†’E, 4,6. adantr 482 was used to transform its dependency
68 ... (πœ’ ∧ πœƒ) ((πœ‘ ∧ πœ“) β†’ (πœ’ ∧ πœƒ)) ∧I 4,5 jca 513, the MPE equivalent of ∧I, 4,7
79 (πœ“ β†’ (πœ’ ∧ πœƒ)) (πœ‘ β†’ (πœ“ β†’ (πœ’ ∧ πœƒ))) β†’I 3,6 ex 414, the MPE equivalent of β†’I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including πœ‘ 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.)

(πœ‘ β†’ (πœ“ β†’ πœ’))    &   (πœ‘ β†’ (πœ’ β†’ πœƒ))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ ∧ πœƒ)))
 
Theoremex-natded5.3-2 29661 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 29660 and ex-natded5.3i 29662. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    &   (πœ‘ β†’ (πœ’ β†’ πœƒ))    β‡’   (πœ‘ β†’ (πœ“ β†’ (πœ’ ∧ πœƒ)))
 
Theoremex-natded5.3i 29662 The same as ex-natded5.3 29660 and ex-natded5.3-2 29661 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.)
(πœ“ β†’ πœ’)    &   (πœ’ β†’ πœƒ)    β‡’   (πœ“ β†’ (πœ’ ∧ πœƒ))
 
Theoremex-natded5.5 29663 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 TranslationND Rationale MPE Rationale
12;3 (πœ“ β†’ πœ’) (πœ‘ β†’ (πœ“ β†’ πœ’)) Given $e; adantr 482 to move it into the ND hypothesis
25 Β¬ πœ’ (πœ‘ β†’ Β¬ πœ’) Given $e; we'll use adantr 482 to move it into the ND hypothesis
31 ...| πœ“ ((πœ‘ ∧ πœ“) β†’ πœ“) ND hypothesis assumption simpr 486
44 ... πœ’ ((πœ‘ ∧ πœ“) β†’ πœ’) β†’E 1,3 mpd 15 1,3
56 ... Β¬ πœ’ ((πœ‘ ∧ πœ“) β†’ Β¬ πœ’) IT 2 adantr 482 5
67 Β¬ πœ“ (πœ‘ β†’ Β¬ πœ“) ∧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œ‘ 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.)

(πœ‘ β†’ (πœ“ β†’ πœ’))    &   (πœ‘ β†’ Β¬ πœ’)    β‡’   (πœ‘ β†’ Β¬ πœ“)
 
Theoremex-natded5.7 29664 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 29665. 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 TranslationND Rationale MPE Rationale
16 (πœ“ ∨ (πœ’ ∧ πœƒ)) (πœ‘ β†’ (πœ“ ∨ (πœ’ ∧ πœƒ))) Given $e. No need for adantr 482 because we do not move this into an ND hypothesis
21 ...| πœ“ ((πœ‘ ∧ πœ“) β†’ πœ“) ND hypothesis assumption (new scope) simpr 486
32 ... (πœ“ ∨ πœ’) ((πœ‘ ∧ πœ“) β†’ (πœ“ ∨ πœ’)) ∨IL 2 orcd 872, the MPE equivalent of ∨IL, 1
43 ...| (πœ’ ∧ πœƒ) ((πœ‘ ∧ (πœ’ ∧ πœƒ)) β†’ (πœ’ ∧ πœƒ)) ND hypothesis assumption (new scope) simpr 486
54 ... πœ’ ((πœ‘ ∧ (πœ’ ∧ πœƒ)) β†’ πœ’) ∧EL 4 simpld 496, the MPE equivalent of ∧EL, 3
66 ... (πœ“ ∨ πœ’) ((πœ‘ ∧ (πœ’ ∧ πœƒ)) β†’ (πœ“ ∨ πœ’)) ∨IR 5 olcd 873, the MPE equivalent of ∨IR, 4
77 (πœ“ ∨ πœ’) (πœ‘ β†’ (πœ“ ∨ πœ’)) ∨E 1,3,6 mpjaodan 958, the MPE equivalent of ∨E, 2,5,6

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including πœ‘ 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.)

(πœ‘ β†’ (πœ“ ∨ (πœ’ ∧ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ ∨ πœ’))
 
Theoremex-natded5.7-2 29665 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 29664. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ (πœ“ ∨ (πœ’ ∧ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ ∨ πœ’))
 
Theoremex-natded5.8 29666 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 TranslationND Rationale MPE Rationale
110;11 ((πœ“ ∧ πœ’) β†’ Β¬ πœƒ) (πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ Β¬ πœƒ)) Given $e; adantr 482 to move it into the ND hypothesis
23;4 (𝜏 β†’ πœƒ) (πœ‘ β†’ (𝜏 β†’ πœƒ)) Given $e; adantr 482 to move it into the ND hypothesis
37;8 πœ’ (πœ‘ β†’ πœ’) Given $e; adantr 482 to move it into the ND hypothesis
41;2 𝜏 (πœ‘ β†’ 𝜏) Given $e. adantr 482 to move it into the ND hypothesis
56 ...| πœ“ ((πœ‘ ∧ πœ“) β†’ πœ“) ND Hypothesis/Assumption simpr 486. New ND hypothesis scope, each reference outside the scope must change antecedent πœ‘ to (πœ‘ ∧ πœ“).
69 ... (πœ“ ∧ πœ’) ((πœ‘ ∧ πœ“) β†’ (πœ“ ∧ πœ’)) ∧I 5,3 jca 513 (∧I), 6,8 (adantr 482 to bring in scope)
75 ... Β¬ πœƒ ((πœ‘ ∧ πœ“) β†’ Β¬ πœƒ) β†’E 1,6 mpd 15 (β†’E), 2,4
812 ... πœƒ ((πœ‘ ∧ πœ“) β†’ πœƒ) β†’E 2,4 mpd 15 (β†’E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 Β¬ πœ“ (πœ‘ β†’ Β¬ πœ“) 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œ‘ 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 29667.

(Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ Β¬ πœƒ))    &   (πœ‘ β†’ (𝜏 β†’ πœƒ))    &   (πœ‘ β†’ πœ’)    &   (πœ‘ β†’ 𝜏)    β‡’   (πœ‘ β†’ Β¬ πœ“)
 
Theoremex-natded5.8-2 29667 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 29666. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ ((πœ“ ∧ πœ’) β†’ Β¬ πœƒ))    &   (πœ‘ β†’ (𝜏 β†’ πœƒ))    &   (πœ‘ β†’ πœ’)    &   (πœ‘ β†’ 𝜏)    β‡’   (πœ‘ β†’ Β¬ πœ“)
 
Theoremex-natded5.13 29668 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 29669. 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 TranslationND Rationale MPE Rationale
115 (πœ“ ∨ πœ’) (πœ‘ β†’ (πœ“ ∨ πœ’)) Given $e.
2;32 (πœ“ β†’ πœƒ) (πœ‘ β†’ (πœ“ β†’ πœƒ)) Given $e. adantr 482 to move it into the ND hypothesis
39 (Β¬ 𝜏 β†’ Β¬ πœ’) (πœ‘ β†’ (Β¬ 𝜏 β†’ Β¬ πœ’)) Given $e. ad2antrr 725 to move it into the ND sub-hypothesis
41 ...| πœ“ ((πœ‘ ∧ πœ“) β†’ πœ“) ND hypothesis assumption simpr 486
54 ... πœƒ ((πœ‘ ∧ πœ“) β†’ πœƒ) β†’E 2,4 mpd 15 1,3
65 ... (πœƒ ∨ 𝜏) ((πœ‘ ∧ πœ“) β†’ (πœƒ ∨ 𝜏)) ∨I 5 orcd 872 4
76 ...| πœ’ ((πœ‘ ∧ πœ’) β†’ πœ’) ND hypothesis assumption simpr 486
88 ... ...| Β¬ 𝜏 (((πœ‘ ∧ πœ’) ∧ Β¬ 𝜏) β†’ Β¬ 𝜏) (sub) ND hypothesis assumption simpr 486
911 ... ... Β¬ πœ’ (((πœ‘ ∧ πœ’) ∧ Β¬ 𝜏) β†’ Β¬ πœ’) β†’E 3,8 mpd 15 8,10
107 ... ... πœ’ (((πœ‘ ∧ πœ’) ∧ Β¬ 𝜏) β†’ πœ’) IT 7 adantr 482 6
1112 ... Β¬ Β¬ 𝜏 ((πœ‘ ∧ πœ’) β†’ Β¬ Β¬ 𝜏) Β¬I 8,9,10 pm2.65da 816 7,11
1213 ... 𝜏 ((πœ‘ ∧ πœ’) β†’ 𝜏) Β¬E 11 notnotrd 133 12
1314 ... (πœƒ ∨ 𝜏) ((πœ‘ ∧ πœ’) β†’ (πœƒ ∨ 𝜏)) ∨I 12 olcd 873 13
1416 (πœƒ ∨ 𝜏) (πœ‘ β†’ (πœƒ ∨ 𝜏)) ∨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œ‘ 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.)

(πœ‘ β†’ (πœ“ ∨ πœ’))    &   (πœ‘ β†’ (πœ“ β†’ πœƒ))    &   (πœ‘ β†’ (Β¬ 𝜏 β†’ Β¬ πœ’))    β‡’   (πœ‘ β†’ (πœƒ ∨ 𝜏))
 
Theoremex-natded5.13-2 29669 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 29668. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ (πœ“ ∨ πœ’))    &   (πœ‘ β†’ (πœ“ β†’ πœƒ))    &   (πœ‘ β†’ (Β¬ 𝜏 β†’ Β¬ πœ’))    β‡’   (πœ‘ β†’ (πœƒ ∨ 𝜏))
 
Theoremex-natded9.20 29670 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 TranslationND Rationale MPE Rationale
11 (πœ“ ∧ (πœ’ ∨ πœƒ)) (πœ‘ β†’ (πœ“ ∧ (πœ’ ∨ πœƒ))) Given $e
22 πœ“ (πœ‘ β†’ πœ“) ∧EL 1 simpld 496 1
311 (πœ’ ∨ πœƒ) (πœ‘ β†’ (πœ’ ∨ πœƒ)) ∧ER 1 simprd 497 1
44 ...| πœ’ ((πœ‘ ∧ πœ’) β†’ πœ’) ND hypothesis assumption simpr 486
55 ... (πœ“ ∧ πœ’) ((πœ‘ ∧ πœ’) β†’ (πœ“ ∧ πœ’)) ∧I 2,4 jca 513 3,4
66 ... ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ)) ((πœ‘ ∧ πœ’) β†’ ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ))) ∨IR 5 orcd 872 5
78 ...| πœƒ ((πœ‘ ∧ πœƒ) β†’ πœƒ) ND hypothesis assumption simpr 486
89 ... (πœ“ ∧ πœƒ) ((πœ‘ ∧ πœƒ) β†’ (πœ“ ∧ πœƒ)) ∧I 2,7 jca 513 7,8
910 ... ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ)) ((πœ‘ ∧ πœƒ) β†’ ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ))) ∨IL 8 olcd 873 9
1012 ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ)) (πœ‘ β†’ ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ))) ∨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œ‘ 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 29671. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(πœ‘ β†’ (πœ“ ∧ (πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ)))
 
Theoremex-natded9.20-2 29671 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 29670. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ (πœ“ ∧ (πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ ((πœ“ ∧ πœ’) ∨ (πœ“ ∧ πœƒ)))
 
Theoremex-natded9.26 29672* 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 TranslationND Rationale MPE Rationale
13 βˆƒπ‘₯βˆ€π‘¦πœ“(π‘₯, 𝑦) (πœ‘ β†’ βˆƒπ‘₯βˆ€π‘¦πœ“) Given $e.
26 ...| βˆ€π‘¦πœ“(π‘₯, 𝑦) ((πœ‘ ∧ βˆ€π‘¦πœ“) β†’ βˆ€π‘¦πœ“) ND hypothesis assumption simpr 486. Later statements will have this scope.
37;5,4 ... πœ“(π‘₯, 𝑦) ((πœ‘ ∧ βˆ€π‘¦πœ“) β†’ πœ“) βˆ€E 2,y spsbcd 3792 (βˆ€E), 5,6. To use it we need a1i 11 and vex 3479. This could be immediately done with 19.21bi 2183, but we want to show the general approach for substitution.
412;8,9,10,11 ... βˆƒπ‘₯πœ“(π‘₯, 𝑦) ((πœ‘ ∧ βˆ€π‘¦πœ“) β†’ βˆƒπ‘₯πœ“) βˆƒI 3,a spesbcd 3878 (βˆƒI), 11. To use it we need sylibr 233, which in turn requires sylib 217 and two uses of sbcid 3795. This could be more immediately done using 19.8a 2175, but we want to show the general approach for substitution.
513;1,2 βˆƒπ‘₯πœ“(π‘₯, 𝑦) (πœ‘ β†’ βˆƒπ‘₯πœ“) βˆƒE 1,2,4,a exlimdd 2214 (βˆƒE), 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)
614 βˆ€π‘¦βˆƒπ‘₯πœ“(π‘₯, 𝑦) (πœ‘ β†’ βˆ€π‘¦βˆƒπ‘₯πœ“) βˆ€I 5 alrimiv 1931 (βˆ€I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including πœ‘ 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œ“(π‘₯, 𝑦) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 29673.

(Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(πœ‘ β†’ βˆƒπ‘₯βˆ€π‘¦πœ“)    β‡’   (πœ‘ β†’ βˆ€π‘¦βˆƒπ‘₯πœ“)
 
Theoremex-natded9.26-2 29673* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 29672. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(πœ‘ β†’ βˆƒπ‘₯βˆ€π‘¦πœ“)    β‡’   (πœ‘ β†’ βˆ€π‘¦βˆƒπ‘₯πœ“)
 
18.1.4  Definitional examples
 
Theoremex-or 29674 Example for df-or 847. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 ∨ 4 = 4)
 
Theoremex-an 29675 Example for df-an 398. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 ∧ 3 = 3)
 
Theoremex-dif 29676 Example for df-dif 3952. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} βˆ– {1, 8}) = {3}
 
Theoremex-un 29677 Example for df-un 3954. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} βˆͺ {1, 8}) = {1, 3, 8}
 
Theoremex-in 29678 Example for df-in 3956. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∩ {1, 8}) = {1}
 
Theoremex-uni 29679 Example for df-uni 4910. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
βˆͺ {{1, 3}, {1, 8}} = {1, 3, 8}
 
Theoremex-ss 29680 Example for df-ss 3966. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} βŠ† {1, 2, 3}
 
Theoremex-pss 29681 Example for df-pss 3968. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊊ {1, 2, 3}
 
Theoremex-pw 29682 Example for df-pw 4605. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 = {3, 5, 7} β†’ 𝒫 𝐴 = (({βˆ…} βˆͺ {{3}, {5}, {7}}) βˆͺ ({{3, 5}, {3, 7}, {5, 7}} βˆͺ {{3, 5, 7}})))
 
Theoremex-pr 29683 Example for df-pr 4632. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐴 ∈ {1, -1} β†’ (𝐴↑2) = 1)
 
Theoremex-br 29684 Example for df-br 5150. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} β†’ 3𝑅9)
 
Theoremex-opab 29685* Example for df-opab 5212. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑅 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ β„‚ ∧ 𝑦 ∈ β„‚ ∧ (π‘₯ + 1) = 𝑦)} β†’ 3𝑅4)
 
Theoremex-eprel 29686 Example for df-eprel 5581. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
5 E {1, 5}
 
Theoremex-id 29687 Example for df-id 5575. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(5 I 5 ∧ ¬ 4 I 5)
 
Theoremex-po 29688 Example for df-po 5589. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
( < Po ℝ ∧ Β¬ ≀ Po ℝ)
 
Theoremex-xp 29689 Example for df-xp 5683. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
({1, 5} Γ— {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} βˆͺ {⟨5, 2⟩, ⟨5, 7⟩})
 
Theoremex-cnv 29690 Example for df-cnv 5685. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
β—‘{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}
 
Theoremex-co 29691 Example for df-co 5686. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((exp ∘ cos)β€˜0) = e
 
Theoremex-dm 29692 Example for df-dm 5687. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} β†’ dom 𝐹 = {2, 3})
 
Theoremex-rn 29693 Example for df-rn 5688. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} β†’ ran 𝐹 = {6, 9})
 
Theoremex-res 29694 Example for df-res 5689. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐡 = {1, 2}) β†’ (𝐹 β†Ύ 𝐡) = {⟨2, 6⟩})
 
Theoremex-ima 29695 Example for df-ima 5690. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐡 = {1, 2}) β†’ (𝐹 β€œ 𝐡) = {6})
 
Theoremex-fv 29696 Example for df-fv 6552. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} β†’ (πΉβ€˜3) = 9)
 
Theoremex-1st 29697 Example for df-1st 7975. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(1st β€˜βŸ¨3, 4⟩) = 3
 
Theoremex-2nd 29698 Example for df-2nd 7976. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(2nd β€˜βŸ¨3, 4⟩) = 4
 
Theorem1kp2ke3k 29699 Example for df-dec 12678, 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 12678 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

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

(1000 + 2000) = 3000
 
Theoremex-fl 29700 Example for df-fl 13757. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((βŒŠβ€˜(3 / 2)) = 1 ∧ (βŒŠβ€˜-(3 / 2)) = -2)
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