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Most recent proofs    These are the 100 (Unicode, GIF) or 1000 (Unicode, GIF) most recent proofs in the set.mm database for the Metamath Proof Explorer (and the Hilbert Space Explorer). The set.mm database is maintained on GitHub with master (stable) and develop (development) versions. This page was created from develop commit 212ee7d9, also available here: set.mm (43MB) or set.mm.bz2 (compressed, 13MB).

The original proofs of theorems with recently shortened proofs can often be found by appending "OLD" to the theorem name, for example 19.43OLD for 19.43. The "OLD" versions are usually deleted after a year.

Other links    Email: Norm Megill.    Mailing list: Metamath Google Group Updated 7-Dec-2021 .    Contributing: How can I contribute to Metamath?    Syndication: RSS feed (courtesy of Dan Getz)    Related wikis: Ghilbert site; Ghilbert Google Group.

Recent news items    (7-Aug-2021) Version 0.198 of the metamath program fixes a bug in "write source ... /rewrap" that prevented end-of-sentence punctuation from appearing in column 79, causing some rewrapped lines to be shorter than necessary. Because this affects about 2000 lines in set.mm, you should use version 0.198 or later for rewrapping before submitting to GitHub.

(7-May-2021) Mario Carneiro has written a Metamath verifier in Lean.

(5-May-2021) Marnix Klooster has written a Metamath verifier in Zig.

(24-Mar-2021) Metamath was mentioned in a couple of articles about OpenAI: Researchers find that large language models struggle with math and What Is GPT-F?.

(26-Dec-2020) Version 0.194 of the metamath program adds the keyword "htmlexturl" to the $t comment to specify external versions of theorem pages. This keyward has been added to set.mm, and you must update your local copy of set.mm for "verify markup" to pass with the new program version.

(19-Dec-2020) Aleksandr A. Adamov has translated the Wikipedia Metamath page into Russian.

(19-Nov-2020) Eric Schmidt's checkmm.cpp was used as a test case for C'est, "a non-standard version of the C++20 standard library, with enhanced support for compile-time evaluation." See C++20 Compile-time Metamath Proof Verification using C'est.

(10-Nov-2020) Filip Cernatescu has updated the XPuzzle (Android app) to version 1.2. XPuzzle is a puzzle with math formulas derived from the Metamath system. At the bottom of the web page is a link to the Google Play Store, where the app can be found.

(7-Nov-2020) Richard Penner created a cross-reference guide between Frege's logic notation and the notation used by set.mm.

(4-Sep-2020) Version 0.192 of the metamath program adds the qualifier '/extract' to 'write source'. See 'help write source' and also this Google Group post.

(23-Aug-2020) Version 0.188 of the metamath program adds keywords Conclusion, Fact, Introduction, Paragraph, Scolia, Scolion, Subsection, and Table to bibliographic references. See 'help write bibliography' for the complete current list.

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Last updated on 1-May-2024 at 5:06 AM ET.
Recent Additions to the Metamath Proof Explorer   Notes (last updated 7-Dec-2020 )
DateLabelDescription
Theorem
 
30-Apr-2024lcmineqlem3 39159 Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 30-Apr-2024.)
𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑀 ≤ 𝑁)    ⇒   (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (1 / (𝑀 + 𝑘))))
 
30-Apr-2024ifpdfbi 1065 Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)
((𝜑 ↔ 𝜓) ↔ if-(𝜑, 𝜓, ¬ 𝜓))
 
29-Apr-2024lcmfunnnd 39160 Useful equation to calculate the least common multiple of 1 to n. (Contributed by metakunt, 29-Apr-2024.)
(𝜑 → 𝑁 ∈ ℕ)    ⇒   (𝜑 → (lcm‘(1...𝑁)) = ((lcm‘(1...(𝑁 − 1))) lcm 𝑁))
 
29-Apr-2024lcmineqlem2 39158 Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.)
𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑀 ≤ 𝑁)    ⇒   (𝜑 → 𝐹 = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · ∫(0[,]1)((𝑥↑(𝑀 − 1)) · (𝑥↑𝑘)) d𝑥))
 
29-Apr-2024lcmineqlem1 39157 Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024.)
𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥    &   (𝜑 → 𝑁 ∈ ℕ)    &   (𝜑 → 𝑀 ∈ ℕ)    &   (𝜑 → 𝑀 ≤ 𝑁)    ⇒   (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥)
 
29-Apr-2024ineq2pow2tnd 39156 Inequality of 2^{2n} (Contributed by metakunt, 29-Apr-2024.)
(𝜑 → 𝑁 ∈ ℕ0)    ⇒   (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
 
29-Apr-2024wl-had0 34779 If the first input is false, then the adder sum is equivalent to the exclusive disjunction of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 29-Apr-2024.)
(¬ 𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ⊻ 𝜒)))
 
28-Apr-2024ifpbi23d 1075 Equivalence deduction for conditional operator for propositions. Convenience theorem for a frequent case. (Contributed by Wolf Lammen, 28-Apr-2024.)
(𝜑 → (𝜒 ↔ 𝜂))    &   (𝜑 → (𝜃 ↔ 𝜁))    ⇒   (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜓, 𝜂, 𝜁)))
 
26-Apr-20243factsumint 39155 First equation in lcm inequality proof, Theorem 3.1 from https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 26-Apr-2024.)
𝐴 = (𝐿[,]𝑈)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))    ⇒   (𝜑 → ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥))
 
26-Apr-20243factsumint4 39154 Lemma for first equation in lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.)
(𝜑 → 𝐵 ∈ Fin)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)    ⇒   (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = ∫𝐴(𝐹 · Σ𝑘 ∈ 𝐵 (𝐺 · 𝐻)) d𝑥)
 
26-Apr-20243factsumint3 39153 Lemma for first equation in lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.)
𝐴 = (𝐿[,]𝑈)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))    ⇒   (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 (𝐺 · ∫𝐴(𝐹 · 𝐻) d𝑥))
 
26-Apr-20243factsumint2 39152 Lemma for first equation in lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.)
((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)    ⇒   (𝜑 → Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐺 · (𝐹 · 𝐻)) d𝑥)
 
26-Apr-20243factsumint1 39151 Lemma for first equation in lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.)
𝐴 = (𝐿[,]𝑈)    &   (𝜑 → 𝐵 ∈ Fin)    &   (𝜑 → 𝐿 ∈ ℝ)    &   (𝜑 → 𝑈 ∈ ℝ)    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ∈ ℂ)    &   (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐹) ∈ (𝐴–cn→ℂ))    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐺 ∈ ℂ)    &   ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐻 ∈ ℂ)    &   ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐻) ∈ (𝐴–cn→ℂ))    ⇒   (𝜑 → ∫𝐴Σ𝑘 ∈ 𝐵 (𝐹 · (𝐺 · 𝐻)) d𝑥 = Σ𝑘 ∈ 𝐵 ∫𝐴(𝐹 · (𝐺 · 𝐻)) d𝑥)
 
26-Apr-2024pwrssmgc 30680 Given a function 𝐹, exhibit a Galois connection between subsets of its domain and subsets of its range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐺 = (𝑛 ∈ 𝒫 𝑌 ↦ (◡𝐹 “ 𝑛))    &   𝐻 = (𝑚 ∈ 𝒫 𝑋 ↦ {𝑦 ∈ 𝑌 ∣ (◡𝐹 “ {𝑦}) ⊆ 𝑚})    &   𝑉 = (toInc‘𝒫 𝑌)    &   𝑊 = (toInc‘𝒫 𝑋)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝐹:𝑋⟶𝑌)    ⇒   (𝜑 → 𝐺(𝑉MGalConn𝑊)𝐻)
 
26-Apr-2024mcgcnv 30679 The inverse Galois connection is the Galois connection of the dual orders. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐻 = (𝑉MGalConn𝑊)    &   𝑀 = ((ODual‘𝑊)MGalConn(ODual‘𝑉))    ⇒   ((𝑉 ∈ Proset ∧ 𝑊 ∈ Proset ) → (𝐹𝐻𝐺 ↔ 𝐺𝑀𝐹))
 
26-Apr-2024dfmgc2 30678 Alternate definition of the monotone Galois connection. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    ⇒   (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ∧ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣))) ∧ (∀𝑢 ∈ 𝐵 (𝐹‘(𝐺‘𝑢)) ≲ 𝑢 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))))))
 
26-Apr-2024dfmgc2lem 30677 Lemma for dfmgc2, backwards direction. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹:𝐴⟶𝐵)    &   (𝜑 → 𝐺:𝐵⟶𝐴)    &   (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))    &   (𝜑 → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ≲ 𝑣 → (𝐺‘𝑢) ≤ (𝐺‘𝑣)))    &   ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ (𝐺‘(𝐹‘𝑥)))    &   ((𝜑 ∧ 𝑢 ∈ 𝐵) → (𝐹‘(𝐺‘𝑢)) ≲ 𝑢)    ⇒   (𝜑 → 𝐹𝐻𝐺)
 
26-Apr-2024mgcmntco 30676 A Galois connection like statement, for two functions with same range. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    &   𝐶 = (Base‘𝑋)    &    < = (le‘𝑋)    &   (𝜑 → 𝑋 ∈ Proset )    &   (𝜑 → 𝐾 ∈ (𝑉Monot𝑋))    &   (𝜑 → 𝐿 ∈ (𝑊Monot𝑋))    ⇒   (𝜑 → (∀𝑥 ∈ 𝐴 (𝐾‘𝑥) < (𝐿‘(𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝐵 (𝐾‘(𝐺‘𝑦)) < (𝐿‘𝑦)))
 
26-Apr-2024mcgmnt2 30675 The upper adjoint 𝐺 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑋 ∈ 𝐵)    &   (𝜑 → 𝑌 ∈ 𝐵)    &   (𝜑 → 𝑋 ≲ 𝑌)    ⇒   (𝜑 → (𝐺‘𝑋) ≤ (𝐺‘𝑌))
 
26-Apr-2024mcgmnt1 30674 The lower adjoint 𝐹 of a Galois connection is monotonically increasing. (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋 ≤ 𝑌)    ⇒   (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
 
26-Apr-2024mgccole2 30673 Inequality for the closure operator (𝐹 ∘ 𝐺) of the Gallois connection 𝐻 (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑌 ∈ 𝐵)    ⇒   (𝜑 → (𝐹‘(𝐺‘𝑌)) ≲ 𝑌)
 
26-Apr-2024mgccole1 30672 An inequality for the kernel operator 𝐺 ∘ 𝐹 (Contributed by Thierry Arnoux, 26-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    &   (𝜑 → 𝑋 ∈ 𝐴)    ⇒   (𝜑 → 𝑋 ≤ (𝐺‘(𝐹‘𝑋)))
 
26-Apr-2024isposd 17560 Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by AV, 26-Apr-2024.)
(𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → ≤ = (le‘𝐾))    &   ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)    &   ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))    ⇒   (𝜑 → 𝐾 ∈ Poset)
 
25-Apr-2024lcm8un 39168 Least common multiple of natural numbers up to 8 equals 840. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...8)) = 840
 
25-Apr-2024lcm7un 39167 Least common multiple of natural numbers up to 7 equals 420. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...7)) = 420
 
25-Apr-2024lcm6un 39166 Least common multiple of natural numbers up to 6 equals 60. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...6)) = 60
 
25-Apr-2024lcm5un 39165 Least common multiple of natural numbers up to 5 equals 60. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...5)) = 60
 
25-Apr-2024lcm4un 39164 Least common multiple of natural numbers up to 4 equals 12. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...4)) = 12
 
25-Apr-2024lcm3un 39163 Least common multiple of natural numbers up to 3 equals 6. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...3)) = 6
 
25-Apr-2024lcm2un 39162 Least common multiple of natural numbers up to 2 equals 2. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...2)) = 2
 
25-Apr-2024lcm1un 39161 Least common multiple of natural numbers up to 1 equals 1. (Contributed by metakunt, 25-Apr-2024.)
(lcm‘(1...1)) = 1
 
25-Apr-2024420lcm8e840 39150 The lcm of 420 and 8 is 840. (Contributed by metakunt, 25-Apr-2024.)
(420 lcm 8) = 840
 
25-Apr-202460lcm7e420 39149 The lcm of 60 and 7 is 420. (Contributed by metakunt, 25-Apr-2024.)
(60 lcm 7) = 420
 
25-Apr-202460lcm6e60 39148 The lcm of 60 and 6 is 60. (Contributed by metakunt, 25-Apr-2024.)
(60 lcm 6) = 60
 
25-Apr-202412lcm5e60 39147 The lcm of 12 and 5 is 60. (Contributed by metakunt, 25-Apr-2024.)
(12 lcm 5) = 60
 
25-Apr-2024lcmeprodgcdi 39146 Calculate the least common multiple of two natural numbers. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝐺 ∈ ℕ    &   𝐻 ∈ ℕ    &   (𝑀 gcd 𝑁) = 𝐺    &   (𝐺 · 𝐻) = 𝐴    &   (𝑀 · 𝑁) = 𝐴    ⇒   (𝑀 lcm 𝑁) = 𝐻
 
25-Apr-2024420gcd8e4 39145 The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024.)
(420 gcd 8) = 4
 
25-Apr-202460gcd7e1 39144 The gcd of 60 and 7 is 1. (Contributed by metakunt, 25-Apr-2024.)
(60 gcd 7) = 1
 
25-Apr-202460gcd6e6 39143 The gcd of 60 and 6 is 6. (Contributed by metakunt, 25-Apr-2024.)
(60 gcd 6) = 6
 
25-Apr-202412gcd5e1 39142 The gcd of 12 and 5 is 1. (Contributed by metakunt, 25-Apr-2024.)
(12 gcd 5) = 1
 
25-Apr-2024gcdnncli 39141 Closure of the gcd operator. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    ⇒   (𝑀 gcd 𝑁) ∈ ℕ
 
25-Apr-2024gcdaddmzz2nncomi 39140 Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝐾 ∈ ℤ    ⇒   (𝑀 gcd 𝑁) = (𝑀 gcd ((𝐾 · 𝑀) + 𝑁))
 
25-Apr-2024gcdaddmzz2nni 39139 Adding a multiple of one operand of the gcd operator to the other does not alter the result. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    &   𝐾 ∈ ℤ    ⇒   (𝑀 gcd 𝑁) = (𝑀 gcd (𝑁 + (𝐾 · 𝑀)))
 
25-Apr-2024gcdmultiplei 39138 The GCD of a multiple of a positive integer is the positive integer itself. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    ⇒   (𝑀 gcd (𝑀 · 𝑁)) = 𝑀
 
25-Apr-2024neggcdnni 39137 Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    ⇒   (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁)
 
25-Apr-2024gcdnegnni 39136 Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    ⇒   (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁)
 
25-Apr-2024gcdcomnni 39135 Commutative law for gcd. (Contributed by metakunt, 25-Apr-2024.)
𝑀 ∈ ℕ    &   𝑁 ∈ ℕ    ⇒   (𝑀 gcd 𝑁) = (𝑁 gcd 𝑀)
 
25-Apr-2024mulcomnni 39134 Commutative law for multiplication. (Contributed by metakunt, 25-Apr-2024.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    ⇒   (𝐴 · 𝐵) = (𝐵 · 𝐴)
 
25-Apr-2024mulassnni 39133 Associative law for multiplication. (Contributed by metakunt, 25-Apr-2024.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   𝐶 ∈ ℕ    ⇒   ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
 
25-Apr-2024addcomnni 39132 Commutative law for addition. (Contributed by metakunt, 25-Apr-2024.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    ⇒   (𝐴 + 𝐵) = (𝐵 + 𝐴)
 
25-Apr-2024addassnni 39131 Associative law for addition. (Contributed by metakunt, 25-Apr-2024.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ    &   𝐶 ∈ ℕ    ⇒   ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 
25-Apr-2024frgpnabllem2 18989 Lemma for frgpnabl 18990. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    ∼ = ( ~FG ‘𝐼)    &    + = (+g‘𝐺)    &   𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   𝑈 = (varFGrp‘𝐼)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝐼)    &   (𝜑 → 𝐵 ∈ 𝐼)    &   (𝜑 → ((𝑈‘𝐴) + (𝑈‘𝐵)) = ((𝑈‘𝐵) + (𝑈‘𝐴)))    ⇒   (𝜑 → 𝐴 = 𝐵)
 
25-Apr-2024frgpnabllem1 18988 Lemma for frgpnabl 18990. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 25-Apr-2024.)
𝐺 = (freeGrp‘𝐼)    &   𝑊 = ( I ‘Word (𝐼 × 2o))    &    ∼ = ( ~FG ‘𝐼)    &    + = (+g‘𝐺)    &   𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)    &   𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))    &   𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥))    &   𝑈 = (varFGrp‘𝐼)    &   (𝜑 → 𝐼 ∈ 𝑉)    &   (𝜑 → 𝐴 ∈ 𝐼)    &   (𝜑 → 𝐵 ∈ 𝐼)    ⇒   (𝜑 → ⟨“⟨𝐴, ∅⟩⟨𝐵, ∅⟩”⟩ ∈ (𝐷 ∩ ((𝑈‘𝐴) + (𝑈‘𝐵))))
 
25-Apr-2024lsmpropd 18798 If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   (𝜑 → 𝐾 ∈ 𝑉)    &   (𝜑 → 𝐿 ∈ 𝑊)    ⇒   (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
 
24-Apr-2024wl-had1 34778 If the first input is true, then the adder sum is equivalent to the biconditionality of the other two inputs. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
(𝜑 → (hadd(𝜑, 𝜓, 𝜒) ↔ (𝜓 ↔ 𝜒)))
 
24-Apr-2024wl-hadcomb 34776 Commutative law for the adders sum. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜑, 𝜒, 𝜓))
 
24-Apr-2024wl-hadrot 34774 Rotation law for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(𝜑, 𝜓, 𝜒) ↔ hadd(𝜓, 𝜒, 𝜑))
 
24-Apr-2024wl-hadbi123d 34772 Equivalence theorem for the adder sum. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
(𝜑 → (𝜓 ↔ 𝜒))    &   (𝜑 → (𝜃 ↔ 𝜏))    &   (𝜑 → (𝜂 ↔ 𝜁))    ⇒   (𝜑 → (hadd(𝜓, 𝜃, 𝜂) ↔ hadd(𝜒, 𝜏, 𝜁)))
 
24-Apr-2024wl-dfhad3 34771 The adder sum is the same as the triple biconditional. (Contributed by Mario Carneiro, 4-Sep-2016.) Alternative definition. (Revised by Wolf Lammen, 24-Apr-2024.)
(hadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ↔ 𝜓) ↔ 𝜒))
 
24-Apr-2024wl-dfhad2 34770 Associative law for the adder sum. (Contributed by Wolf Lammen, 24-Apr-2024.)
(hadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ↔ (𝜓 ↔ 𝜒)))
 
24-Apr-2024wl-df-had 34769 Alternative definition of whad 1592 based on hadifp 1605. See df-had 1593 to learn how it is currently introduced.

Check whether an odd number of inputs is true. The oddness of two is expressed by the ⊻ operation, and for evenness we can use ↔. So, if the first input is true, the two remaining inputs need to amount to an even number, else an odd number is required. (Contributed by Wolf Lammen, 24-Apr-2024.)

(hadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ↔ 𝜒), (𝜓 ⊻ 𝜒)))
 
24-Apr-2024bj-elabtru 34212 This is as close as we can get to proving extensionality for "the" "universal" class without ax-ext 2792. (Contributed by BJ, 24-Apr-2024.)
(𝐴 ∈ {𝑥 ∣ ⊤} ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
 
24-Apr-2024bj-issettru 34211 Weak version of isset 3503 without ax-ext 2792. (Contributed by BJ, 24-Apr-2024.)
(∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
 
24-Apr-2024bj-denoteslem 34209 Lemma for bj-denotes 34210. (Contributed by BJ, 24-Apr-2024.)
(∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ {𝑦 ∣ ⊤})
 
24-Apr-2024mgcf2 30671 The upper adjoint 𝐺 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    ⇒   (𝜑 → 𝐺:𝐵⟶𝐴)
 
24-Apr-2024mgcf1 30670 The lower adjoint 𝐹 of a Galois connection is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    &   (𝜑 → 𝐹𝐻𝐺)    ⇒   (𝜑 → 𝐹:𝐴⟶𝐵)
 
24-Apr-2024mntf 30667 A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    ⇒   ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴⟶𝐵)
 
24-Apr-2024ismntd 30666 Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   (𝜑 → 𝑉 ∈ 𝐶)    &   (𝜑 → 𝑊 ∈ 𝐷)    &   (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))    &   (𝜑 → 𝑋 ∈ 𝐴)    &   (𝜑 → 𝑌 ∈ 𝐴)    &   (𝜑 → 𝑋 ≤ 𝑌)    ⇒   (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))
 
24-Apr-2024lbspropd 19866 If two structures have the same components (properties), they have the same set of bases. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   (𝜑 → 𝐵 ⊆ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   𝐹 = (Scalar‘𝐾)    &   𝐺 = (Scalar‘𝐿)    &   (𝜑 → 𝑃 = (Base‘𝐹))    &   (𝜑 → 𝑃 = (Base‘𝐺))    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘𝐹)𝑦) = (𝑥(+g‘𝐺)𝑦))    &   (𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝐿 ∈ 𝑌)    ⇒   (𝜑 → (LBasis‘𝐾) = (LBasis‘𝐿))
 
24-Apr-2024lsppropd 19785 If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)
(𝜑 → 𝐵 = (Base‘𝐾))    &   (𝜑 → 𝐵 = (Base‘𝐿))    &   (𝜑 → 𝐵 ⊆ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))    &   (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))    &   (𝜑 → 𝐾 ∈ 𝑋)    &   (𝜑 → 𝐿 ∈ 𝑌)    ⇒   (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
 
23-Apr-2024mgcval 30669 Monotone Galois connection between two functions 𝐹 and 𝐺. If this relation is satisfied, 𝐹 is called the lower adjoint of 𝐺, and 𝐺 is called the upper adjoint of 𝐹.

Technically, this is implemented as an operation taking a pair of structures 𝑉 and 𝑊, expected to be posets, which gives a relation between pairs of functions 𝐹 and 𝐺.

If such a relation exists, it can be proven to be unique.

Galois connections generalize the fundamental theorem of Galois theory about the correspondence between subgroups and subfields. (Contributed by Thierry Arnoux, 23-Apr-2024.)

𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    &   𝐻 = (𝑉MGalConn𝑊)    &   (𝜑 → 𝑉 ∈ Proset )    &   (𝜑 → 𝑊 ∈ Proset )    ⇒   (𝜑 → (𝐹𝐻𝐺 ↔ ((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝐺‘𝑦)))))
 
23-Apr-2024mgcoval 30668 Operation value of the monotone Galois connection. (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    ⇒   ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉MGalConn𝑊) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝐵 ↑m 𝐴) ∧ 𝑔 ∈ (𝐴 ↑m 𝐵)) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑓‘𝑥) ≲ 𝑦 ↔ 𝑥 ≤ (𝑔‘𝑦)))})
 
23-Apr-2024ismnt 30665 Express the statement "𝐹 is monotone " (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    ⇒   ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
 
23-Apr-2024mntoval 30664 Operation value of the monotone function (Contributed by Thierry Arnoux, 23-Apr-2024.)
𝐴 = (Base‘𝑉)    &   𝐵 = (Base‘𝑊)    &    ≤ = (le‘𝑉)    &    ≲ = (le‘𝑊)    ⇒   ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
 
23-Apr-2024norcom 1521 The connector ⊽ is commutative. (Contributed by Remi, 25-Oct-2023.) (Proof shortened by Wolf Lammen, 23-Apr-2024.)
((𝜑 ⊽ 𝜓) ↔ (𝜓 ⊽ 𝜑))
 
21-Apr-20242xp3dxp2ge1d 39173 2x+3 is greater than or equal to x+2 for x >= -1, a deduction version (Contributed by metakunt, 21-Apr-2024.)
(𝜑 → 𝑋 ∈ (-1[,)+∞))    ⇒   (𝜑 → 1 ≤ (((2 · 𝑋) + 3) / (𝑋 + 2)))
 
21-Apr-2024prodsplit 39172 Product split into two factors, original by Steven Nguyen. (Contributed by metakunt, 21-Apr-2024.)
(𝜑 → 𝑀 ∈ ℤ)    &   (𝜑 → 𝑁 ∈ ℤ)    &   (𝜑 → 𝑀 ≤ 𝑁)    &   (𝜑 → 𝐾 ∈ ℕ0)    &   ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 + 𝐾))) → 𝐴 ∈ ℂ)    ⇒   (𝜑 → ∏𝑘 ∈ (𝑀...(𝑁 + 𝐾))𝐴 = (∏𝑘 ∈ (𝑀...𝑁)𝐴 · ∏𝑘 ∈ ((𝑁 + 1)...(𝑁 + 𝐾))𝐴))
 
21-Apr-2024xorbi12i 1514 Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
(𝜑 ↔ 𝜓)    &   (𝜒 ↔ 𝜃)    ⇒   ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))
 
21-Apr-2024xorcom 1503 The connector ⊻ is commutative. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
((𝜑 ⊻ 𝜓) ↔ (𝜓 ⊻ 𝜑))
 
21-Apr-2024an33rean 1478 Rearrange a 9-fold conjunction. (Contributed by Thierry Arnoux, 14-Apr-2019.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)
(((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏 ∧ 𝜂) ∧ (𝜁 ∧ 𝜎 ∧ 𝜌)) ↔ ((𝜑 ∧ 𝜏 ∧ 𝜌) ∧ ((𝜓 ∧ 𝜃) ∧ (𝜂 ∧ 𝜎) ∧ (𝜒 ∧ 𝜁))))
 
20-Apr-2024sn-elabg 39180 Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2144, ax-11 2160, ax-12 2176. It also avoids a disjoint variable condition on 𝑥 and 𝐴. Compare sbievw2 2106. (Contributed by SN, 20-Apr-2024.)
(𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))    ⇒   (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
 
20-Apr-2024factwoffsmonot 39174 A factorial with offset is monotonely increasing (Contributed by metakunt, 20-Apr-2024.)
(((𝑋 ∈ ℕ0 ∧ 𝑌 ∈ ℕ0 ∧ 𝑋 ≤ 𝑌) ∧ 𝑁 ∈ ℕ0) → (!‘(𝑋 + 𝑁)) ≤ (!‘(𝑌 + 𝑁)))
 
20-Apr-2024df-mgc 30663 Define monotone Galois connections. See mgcval 30669 for an expanded version. (Contributed by Thierry Arnoux, 20-Apr-2024.)
MGalConn = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌⦋(Base‘𝑤) / 𝑏⦌{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑏 ↑m 𝑎) ∧ 𝑔 ∈ (𝑎 ↑m 𝑏)) ∧ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 ((𝑓‘𝑥)(le‘𝑤)𝑦 ↔ 𝑥(le‘𝑣)(𝑔‘𝑦)))})
 
20-Apr-2024df-mnt 30662 Define a monotone function between two ordered sets. (Contributed by Thierry Arnoux, 20-Apr-2024.)
Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))})
 
20-Apr-2024vtoclg 3564 Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) Avoid ax-12 2176. (Revised by SN, 20-Apr-2024.)
(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   (𝐴 ∈ 𝑉 → 𝜓)
 
20-Apr-2024vtocl 3556 Implicit substitution of a class for a setvar variable. See also vtoclALT 3557. (Contributed by NM, 30-Aug-1993.) Remove dependency on ax-10 2144. (Revised by BJ, 29-Nov-2020.) (Proof shortened by SN, 20-Apr-2024.)
𝐴 ∈ V    &   (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   𝜑    ⇒   𝜓
 
20-Apr-2024mpjao3dan 1426 Eliminate a three-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.) (Proof shortened by Wolf Lammen, 20-Apr-2024.)
((𝜑 ∧ 𝜓) → 𝜒)    &   ((𝜑 ∧ 𝜃) → 𝜒)    &   ((𝜑 ∧ 𝜏) → 𝜒)    &   (𝜑 → (𝜓 ∨ 𝜃 ∨ 𝜏))    ⇒   (𝜑 → 𝜒)
 
19-Apr-2024facp2 39171 The factorial of a successor's successor. (Contributed by metakunt, 19-Apr-2024.)
(𝑁 ∈ ℕ0 → (!‘(𝑁 + 2)) = ((!‘𝑁) · ((𝑁 + 1) · (𝑁 + 2))))
 
19-Apr-2024fac2xp3 39170 Factorial of 2x+3, sublemma for sublemma for AKS. (Contributed by metakunt, 19-Apr-2024.)
(𝑥 ∈ ℕ0 → (!‘((2 · 𝑥) + 3)) = ((!‘((2 · 𝑥) + 1)) · (((2 · 𝑥) + 2) · ((2 · 𝑥) + 3))))
 
17-Apr-2024rnmptc 6962 Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   (𝜑 → 𝐴 ≠ ∅)    ⇒   (𝜑 → ran 𝐹 = {𝐵})
 
17-Apr-2024ifpbi123d 1073 Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.)
(𝜑 → (𝜓 ↔ 𝜏))    &   (𝜑 → (𝜒 ↔ 𝜂))    &   (𝜑 → (𝜃 ↔ 𝜁))    ⇒   (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁)))
 
16-Apr-2024cbvrabv2w 41468 A more general version of cbvrabv 3488. Version of cbvrabv2 41467 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Revised by Gino Giotto, 16-Apr-2024.)
(𝑥 = 𝑦 → 𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ 𝜓}
 
16-Apr-2024andiff 39169 Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.)
(𝜑 → (𝜒 → 𝜃))    &   (𝜓 → (𝜃 → 𝜒))    ⇒   ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
 
14-Apr-2024bj-pw0ALT 34366 Alternate proof of pw0 4738. The proofs have a similar structure: pw0 4738 uses the definitions of powerclass and singleton as class abstractions, whereas bj-pw0ALT 34366 uses characterizations of their elements. Both proofs then use transitivity of a congruence relation (equality for pw0 4738 and biconditional for bj-pw0ALT 34366) to translate the property ss0b 4344 into the wanted result. To translate a biconditional into a class equality, pw0 4738 uses abbii 2885 (which yields an equality of class abstractions), while bj-pw0ALT 34366 uses eqriv 2817 (which requires a biconditional of membership of a given setvar variable). Note that abbii 2885, through its closed form abbi1 2883, is proved from eqrdv 2818, which is the deduction form of eqriv 2817. In the other direction, velpw 4537 and velsn 4576 are proved from the definitions of powerclass and singleton using elabg 3662, which is a version of abbii 2885 suited for membership characterizations. (Contributed by BJ, 14-Apr-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
𝒫 ∅ = {∅}
 
14-Apr-2024pwundif 4558 Break up the power class of a union into a union of smaller classes. (Contributed by NM, 25-Mar-2007.) (Proof shortened by Thierry Arnoux, 20-Dec-2016.) Remove use of ax-sep 5196, ax-nul 5203, ax-pr 5323 and shorten proof. (Revised by BJ, 14-Apr-2024.)
𝒫 (𝐴 ∪ 𝐵) = ((𝒫 (𝐴 ∪ 𝐵) ∖ 𝒫 𝐴) ∪ 𝒫 𝐴)
 
13-Apr-2024idlmulssprm 30982 Let 𝑃 be a prime ideal containing the product (𝐼 × 𝐽) of two ideals 𝐼 and 𝐽. Then 𝐼 ⊆ 𝑃 or 𝐽 ⊆ 𝑃. (Contributed by Thierry Arnoux, 13-Apr-2024.)
× = (LSSum‘(mulGrp‘𝑅))    &   (𝜑 → 𝑅 ∈ Ring)    &   (𝜑 → 𝑃 ∈ (PrmIdeal‘𝑅))    &   (𝜑 → 𝐼 ∈ (LIdeal‘𝑅))    &   (𝜑 → 𝐽 ∈ (LIdeal‘𝑅))    &   (𝜑 → (𝐼 × 𝐽) ⊆ 𝑃)    ⇒   (𝜑 → (𝐼 ⊆ 𝑃 ∨ 𝐽 ⊆ 𝑃))
 
13-Apr-2024elringlsmd 30969 Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.)
𝐵 = (Base‘𝑅)    &    · = (.r‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    × = (LSSum‘𝐺)    &   (𝜑 → 𝐸 ⊆ 𝐵)    &   (𝜑 → 𝐹 ⊆ 𝐵)    &   (𝜑 → 𝑋 ∈ 𝐸)    &   (𝜑 → 𝑌 ∈ 𝐹)    ⇒   (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹))
 
13-Apr-2024pwel 5275 Quantitative version of pwexg 5272: the powerset of an element of a class is an element of the double powerclass of the union of that class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.) Remove use of ax-nul 5203 and ax-pr 5323 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 ∪ 𝐵)
 
13-Apr-2024unieq 4842 Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by BJ, 13-Apr-2024.)
(𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)
 
13-Apr-2024pwunss 4552 The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of [Mendelson] p. 235. (Contributed by NM, 23-Nov-2003.) Remove use of ax-sep 5196, ax-nul 5203, ax-pr 5323 and shorten proof. (Revised by BJ, 13-Apr-2024.)
(𝒫 𝐴 ∪ 𝒫 𝐵) ⊆ 𝒫 (𝐴 ∪ 𝐵)

Older news:

(29-Jul-2020) Mario Carneiro presented MM0 at the CICM conference. See this Google Group post which includes a YouTube link.

(20-Jul-2020) Rohan Ridenour found 5 shorter D-proofs in our Shortest known proofs... file. In particular, he reduced *4.39 from 901 to 609 steps. A note on the Metamath Solitaire page mentions a tool that he worked with.

(19-Jul-2020) David A. Wheeler posted a video (https://youtu.be/3R27Qx69jHc) on how to (re)prove Schwabh�user 4.6 for the Metamath Proof Explorer. See also his older videos.

(19-Jul-2020) In version 0.184 of the metamath program, "verify markup" now checks that mathboxes are independent i.e. do not cross-reference each other. To turn off this check, use "/mathbox_skip"

(30-Jun-2020) In version 0.183 of the metamath program, (1) "verify markup" now has checking for (i) underscores in labels, (ii) that *ALT and *OLD theorems have both discouragement tags, and (iii) that lines don't have trailing spaces. (2) "save proof.../rewrap" no longer left-aligns $p/$a comments that contain the string "<HTML>"; see this note.

(5-Apr-2020) Glauco Siliprandi added a new proof to the 100 theorem list, e is Transcendental etransc, bringing the Metamath total to 74.

(12-Feb-2020) A bug in the 'minimize' command of metamath.exe versions 0.179 (29-Nov-2019) and 0.180 (10-Dec-2019) may incorrectly bring in the use of new axioms. Version 0.181 fixes it.

(20-Jan-2020) David A. Wheeler created a video called Walkthrough of the tutorial in mmj2. See the Google Group announcement for more details. (All of his videos are listed on the Other Metamath-Related Topics page.)

(18-Jan-2020) The FOMM 2020 talks are on youtube now. Mario Carneiro's talk is Metamath Zero, or: How to Verify a Verifier. Since they are washed out in the video, the PDF slides are available separately.

(14-Dec-2019) Glauco Siliprandi added a new proof to the 100 theorem list, Fourier series convergence fourier, bringing the Metamath total to 73.

(25-Nov-2019) Alexander van der Vekens added a new proof to the 100 theorem list, The Cayley-Hamilton Theorem cayleyhamilton, bringing the Metamath total to 72.

(25-Oct-2019) Mario Carneiro's paper "Metamath Zero: The Cartesian Theorem Prover" (submitted to CPP 2020) is now available on arXiv: https://arxiv.org/abs/1910.10703. There is a related discussion on Hacker News.

(30-Sep-2019) Mario Carneiro's talk about MM0 at ITP 2019 is available on YouTube: x86 verification from scratch (24 minutes). Google Group discussion: Metamath Zero.

(29-Sep-2019) David Wheeler created a fascinating Gource video that animates the construction of set.mm, available on YouTube: Metamath set.mm contributions viewed with Gource through 2019-09-26 (4 minutes). Google Group discussion: Gource video of set.mm contributions.

(24-Sep-2019) nLab added a page for Metamath. It mentions Stefan O'Rear's Busy Beaver work using the set.mm axiomatization (and fails to mention Mario's definitional soundness checker)

(1-Sep-2019) Xuanji Li published a Visual Studio Code extension to support metamath syntax highlighting.

(10-Aug-2019) (revised 21-Sep-2019) Version 0.178 of the metamath program has the following changes: (1) "minimize_with" will now prevent dependence on new $a statements unless the new qualifier "/allow_new_axioms" is specified. For routine usage, it is suggested that you use "minimize_with * /allow_new_axioms * /no_new_axioms_from ax-*" instead of just "minimize_with *". See "help minimize_with" and this Google Group post. Also note that the qualifier "/allow_growth" has been renamed to "/may_grow". (2) "/no_versioning" was added to "write theorem_list".

(8-Jul-2019) Jon Pennant announced the creation of a Metamath search engine. Try it and feel free to comment on it at https://groups.google.com/d/msg/metamath/cTeU5AzUksI/5GesBfDaCwAJ.

(16-May-2019) Set.mm now has a major new section on elementary geometry. This begins with definitions that implement Tarski's axioms of geometry (including concepts such as congruence and betweenness). This uses set.mm's extensible structures, making them easier to use for many circumstances. The section then connects Tarski geometry with geometry in Euclidean places. Most of the work in this section is due to Thierry Arnoux, with earlier work by Mario Carneiro and Scott Fenton. [Reported by DAW.]

(9-May-2019) We are sad to report that long-time contributor Alan Sare passed away on Mar. 23. There is some more information at the top of his mathbox (click on "Mathbox for Alan Sare") and his obituary. We extend our condolences to his family.

(10-Mar-2019) Jon Pennant and Mario Carneiro added a new proof to the 100 theorem list, Heron's formula heron, bringing the Metamath total to 71.

(22-Feb-2019) Alexander van der Vekens added a new proof to the 100 theorem list, Cramer's rule cramer, bringing the Metamath total to 70.

(6-Feb-2019) David A. Wheeler has made significant improvements and updates to the Metamath book. Any comments, errors found, or suggestions are welcome and should be turned into an issue or pull request at https://github.com/metamath/metamath-book (or sent to me if you prefer).

(26-Dec-2018) I added Appendix 8 to the MPE Home Page that cross-references new and old axiom numbers.

(20-Dec-2018) The axioms have been renumbered according to this Google Groups post.

(24-Nov-2018) Thierry Arnoux created a new page on topological structures. The page along with its SVG files are maintained on GitHub.

(11-Oct-2018) Alexander van der Vekens added a new proof to the 100 theorem list, the Friendship Theorem friendship, bringing the Metamath total to 69.

(1-Oct-2018) Naip Moro has written gramm, a Metamath proof verifier written in Antlr4/Java.

(16-Sep-2018) The definition df-riota has been simplified so that it evaluates to the empty set instead of an Undef value. This change affects a significant part of set.mm.

(2-Sep-2018) Thierry Arnoux added a new proof to the 100 theorem list, Euler's partition theorem eulerpart, bringing the Metamath total to 68.

(1-Sep-2018) The Kate editor now has Metamath syntax highlighting built in. (Communicated by Wolf Lammen.)

(15-Aug-2018) The Intuitionistic Logic Explorer now has a Most Recent Proofs page.

(4-Aug-2018) Version 0.163 of the metamath program now indicates (with an asterisk) which Table of Contents headers have associated comments.

(10-May-2018) George Szpiro, journalist and author of several books on popular mathematics such as Poincare's Prize and Numbers Rule, used a genetic algorithm to find shorter D-proofs of "*3.37" and "meredith" in our Shortest known proofs... file.

(19-Apr-2018) The EMetamath Eclipse plugin has undergone many improvements since its initial release as the change log indicates. Thierry uses it as his main proof assistant and writes, "I added support for mmj2's auto-transformations, which allows it to infer several steps when building proofs. This added a lot of comfort for writing proofs.... I can now switch back and forth between the proof assistant and editing the Metamath file.... I think no other proof assistant has this feature."

(11-Apr-2018) Benoît Jubin solved an open problem about the "Axiom of Twoness," showing that it is necessary for completeness. See item 14 on the "Open problems and miscellany" page.

(25-Mar-2018) Giovanni Mascellani has announced mmpp, a new proof editing environment for the Metamath language.

(27-Feb-2018) Bill Hale has released an app for the Apple iPad and desktop computer that allows you to browse Metamath theorems and their proofs.

(17-Jan-2018) Dylan Houlihan has kindly provided a new mirror site. He has also provided an rsync server; type "rsync uk.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(15-Jan-2018) The metamath program, version 0.157, has been updated to implement the file inclusion conventions described in the 21-Dec-2017 entry of mmnotes.txt.

(11-Dec-2017) I added a paragraph, suggested by Gérard Lang, to the distinct variable description here.

(10-Dec-2017) Per FL's request, his mathbox will be removed from set.mm. If you wish to export any of his theorems, today's version (master commit 1024a3a) is the last one that will contain it.

(11-Nov-2017) Alan Sare updated his completeusersproof program.

(3-Oct-2017) Sean B. Palmer created a web page that runs the metamath program under emulated Linux in JavaScript. He also wrote some programs to work with our shortest known proofs of the PM propositional calculus theorems.

(28-Sep-2017) Ivan Kuckir wrote a tutorial blog entry, Introduction to Metamath, that summarizes the language syntax. (It may have been written some time ago, but I was not aware of it before.)

(26-Sep-2017) The default directory for the Metamath Proof Explorer (MPE) has been changed from the GIF version (mpegif) to the Unicode version (mpeuni) throughout the site. Please let me know if you find broken links or other issues.

(24-Sep-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Ceva's Theorem cevath, bringing the Metamath total to 67.

(3-Sep-2017) Brendan Leahy added a new proof to the 100 theorem list, Area of a Circle areacirc, bringing the Metamath total to 66.

(7-Aug-2017) Mario Carneiro added a new proof to the 100 theorem list, Principle of Inclusion/Exclusion incexc, bringing the Metamath total to 65.

(1-Jul-2017) Glauco Siliprandi added a new proof to the 100 theorem list, Stirling's Formula stirling, bringing the Metamath total to 64. Related theorems include 2 versions of Wallis' formula for π (wallispi and wallispi2).

(7-May-2017) Thierry Arnoux added a new proof to the 100 theorem list, Betrand's Ballot Problem ballotth, bringing the Metamath total to 63.

(20-Apr-2017) Glauco Siliprandi added a new proof in the supplementary list on the 100 theorem list, Stone-Weierstrass Theorem stowei.

(28-Feb-2017) David Moews added a new proof to the 100 theorem list, Product of Segments of Chords chordthm, bringing the Metamath total to 62.

(1-Jan-2017) Saveliy Skresanov added a new proof to the 100 theorem list, Isosceles triangle theorem isosctr, bringing the Metamath total to 61.

(1-Jan-2017) Mario Carneiro added 2 new proofs to the 100 theorem list, L'Hôpital's Rule lhop and Taylor's Theorem taylth, bringing the Metamath total to 60.

(28-Dec-2016) David A. Wheeler is putting together a page on Metamath (specifically set.mm) conventions. Comments are welcome on the Google Group thread.

(24-Dec-2016) Mario Carneiro introduced the abbreviation "F/ x ph" (symbols: turned F, x, phi) in df-nf to represent the "effectively not free" idiom "A. x ( ph -> A. x ph )". Theorem nf2 shows a version without nested quantifiers.

(22-Dec-2016) Naip Moro has developed a Metamath database for G. Spencer-Brown's Laws of Form. You can follow the Google Group discussion here.

(20-Dec-2016) In metamath program version 0.137, 'verify markup *' now checks that ax-XXX $a matches axXXX $p when the latter exists, per the discussion at https://groups.google.com/d/msg/metamath/Vtz3CKGmXnI/Fxq3j1I_EQAJ.

(24-Nov-2016) Mingl Yuan has kindly provided a mirror site in Beijing, China. He has also provided an rsync server; type "rsync cn.metamath.org::" in a bash shell to check its status (it should return "metamath metamath").

(14-Aug-2016) All HTML pages on this site should now be mobile-friendly and pass the Mobile-Friendly Test. If you find one that does not, let me know.

(14-Aug-2016) Daniel Whalen wrote a paper describing the use of using deep learning to prove 14% of test theorems taken from set.mm: Holophrasm: a neural Automated Theorem Prover for higher-order logic. The associated program is called Holophrasm.

(14-Aug-2016) David A. Wheeler created a video called Metamath Proof Explorer: A Modern Principia Mathematica

(12-Aug-2016) A Gitter chat room has been created for Metamath.

(9-Aug-2016) Mario Carneiro wrote a Metamath proof verifier in the Scala language as part of the ongoing Metamath -> MMT import project

(9-Aug-2016) David A. Wheeler created a GitHub project called metamath-test (last execution run) to check that different verifiers both pass good databases and detect errors in defective ones.

(4-Aug-2016) Mario gave two presentations at CICM 2016.

(17-Jul-2016) Thierry Arnoux has written EMetamath, a Metamath plugin for the Eclipse IDE.

(16-Jul-2016) Mario recovered Chris Capel's collapsible proof demo.

(13-Jul-2016) FL sent me an updated version of PDF (LaTeX source) developed with Lamport's pf2 package. See the 23-Apr-2012 entry below.

(12-Jul-2016) David A. Wheeler produced a new video for mmj2 called "Creating functions in Metamath". It shows a more efficient approach than his previous recent video "Creating functions in Metamath" (old) but it can be of interest to see both approaches.

(10-Jul-2016) Metamath program version 0.132 changes the command 'show restricted' to 'show discouraged' and adds a new command, 'set discouragement'. See the mmnotes.txt entry of 11-May-2016 (updated 10-Jul-2016).

(12-Jun-2016) Dan Getz has written Metamath.jl, a Metamath proof verifier written in the Julia language.

(10-Jun-2016) If you are using metamath program versions 0.128, 0.129, or 0.130, please update to version 0.131. (In the bad versions, 'minimize_with' ignores distinct variable violations.)

(1-Jun-2016) Mario Carneiro added new proofs to the 100 theorem list, the Prime Number Theorem pnt and the Perfect Number Theorem perfect, bringing the Metamath total to 58.

(12-May-2016) Mario Carneiro added a new proof to the 100 theorem list, Dirichlet's theorem dirith, bringing the Metamath total to 56. (Added 17-May-2016) An informal exposition of the proof can be found at http://metamath-blog.blogspot.com/2016/05/dirichlets-theorem.html

(10-Mar-2016) Metamath program version 0.125 adds a new qualifier, /fast, to 'save proof'. See the mmnotes.txt entry of 10-Mar-2016.

(6-Mar-2016) The most recent set.mm has a large update converting variables from letters to symbols. See this Google Groups post.

(16-Feb-2016) Mario Carneiro's new paper "Models for Metamath" can be found here and on arxiv.org.

(6-Feb-2016) There are now 22 math symbols that can be used as variable names. See mmascii.html near the 50th table row, starting with "./\".

(29-Jan-2016) Metamath program version 0.123 adds /packed and /explicit qualifiers to 'save proof' and 'show proof'. See this Google Groups post.

(13-Jan-2016) The Unicode math symbols now provide for external CSS and use the XITS web font. Thanks to David A. Wheeler, Mario Carneiro, Cris Perdue, Jason Orendorff, and Frédéric Liné for discussions on this topic. Two commands, htmlcss and htmlfont, were added to the $t comment in set.mm and are recognized by Metamath program version 0.122.

(21-Dec-2015) Axiom ax-12, now renamed ax-12o, was replaced by a new shorter equivalent, ax-12. The equivalence is provided by theorems ax12o and ax12.

(13-Dec-2015) A new section on the theory of classes was added to the MPE Home Page. Thanks to Gérard Lang for suggesting this section and improvements to it.

(17-Nov-2015) Metamath program version 0.121: 'verify markup' was added to check comment markup consistency; see 'help verify markup'. You are encouraged to make sure 'verify markup */f' has no warnings prior to mathbox submissions. The date consistency rules are given in this Google Groups post.

(23-Sep-2015) Drahflow wrote, "I am currently working on yet another proof assistant, main reason being: I understand stuff best if I code it. If anyone is interested: https://github.com/Drahflow/Igor (but in my own programming language, so expect a complicated build process :P)"

(23-Aug-2015) Ivan Kuckir created MM Tool, a Metamath proof verifier and editor written in JavaScript that runs in a browser.

(25-Jul-2015) Axiom ax-10 is shown to be redundant by theorem ax10 , so it was removed from the predicate calculus axiom list.

(19-Jul-2015) Mario Carneiro gave two talks related to Metamath at CICM 2015, which are linked to at Other Metamath-Related Topics.

(18-Jul-2015) The metamath program has been updated to version 0.118. 'show trace_back' now has a '/to' qualifier to show the path back to a specific axiom such as ax-ac. See 'help show trace_back'.

(12-Jul-2015) I added the HOL Explorer for Mario Carneiro's hol.mm database. Although the home page needs to be filled out, the proofs can be accessed.

(11-Jul-2015) I started a new page, Other Metamath-Related Topics, that will hold miscellaneous material that doesn't fit well elsewhere (or is hard to find on this site). Suggestions welcome.

(23-Jun-2015) Metamath's mascot, Penny the cat (2007 photo), passed away today. She was 18 years old.

(21-Jun-2015) Mario Carneiro added 3 new proofs to the 100 theorem list: All Primes (1 mod 4) Equal the Sum of Two Squares 2sq, The Law of Quadratic Reciprocity lgsquad and the AM-GM theorem amgm, bringing the Metamath total to 55.

(13-Jun-2015) Stefan O'Rear's smm, written in JavaScript, can now be used as a standalone proof verifier. This brings the total number of independent Metamath verifiers to 8, written in just as many languages (C, Java. JavaScript, Python, Haskell, Lua, C#, C++).

(12-Jun-2015) David A. Wheeler added 2 new proofs to the 100 theorem list: The Law of Cosines lawcos and Ptolemy's Theorem ptolemy, bringing the Metamath total to 52.

(30-May-2015) The metamath program has been updated to version 0.117. (1) David A. Wheeler provided an enhancement to speed up the 'improve' command by 28%; see README.TXT for more information. (2) In web pages with proofs, local hyperlinks on step hypotheses no longer clip the Expression cell at the top of the page.

(9-May-2015) Stefan O'Rear has created an archive of older set.mm releases back to 1998: https://github.com/sorear/set.mm-history/.

(7-May-2015) The set.mm dated 7-May-2015 is a major revision, updated by Mario, that incorporates the new ordered pair definition df-op that was agreed upon. There were 700 changes, listed at the top of set.mm. Mathbox users are advised to update their local mathboxes. As usual, if any mathbox user has trouble incorporating these changes into their mathbox in progress, Mario or I will be glad to do them for you.

(7-May-2015) Mario has added 4 new theorems to the 100 theorem list: Ramsey's Theorem ramsey, The Solution of a Cubic cubic, The Solution of the General Quartic Equation quart, and The Birthday Problem birthday. In the Supplementary List, Stefan O'Rear added the Hilbert Basis Theorem hbt.

(28-Apr-2015) A while ago, Mario Carneiro wrote up a proof of the unambiguity of set.mm's grammar, which has now been added to this site: grammar-ambiguity.txt.

(22-Apr-2015) The metamath program has been updated to version 0.114. In MM-PA, 'show new_proof/unknown' now shows the relative offset (-1, -2,...) used for 'assign' arguments, suggested by Stefan O'Rear.

(20-Apr-2015) I retrieved an old version of the missing "Metamath 100" page from archive.org and updated it to what I think is the current state: mm_100.html. Anyone who wants to edit it can email updates to this page to me.

(19-Apr-2015) The metamath program has been updated to version 0.113, mostly with patches provided by Stefan O'Rear. (1) 'show statement %' (or any command allowing label wildcards) will select statements whose proofs were changed in current session. ('help search' will show all wildcard matching rules.) (2) 'show statement =' will select the statement being proved in MM-PA. (3) The proof date stamp is now created only if the proof is complete.

(18-Apr-2015) There is now a section for Scott Fenton's NF database: New Foundations Explorer.

(16-Apr-2015) Mario describes his recent additions to set.mm at https://groups.google.com/forum/#!topic/metamath/VAGNmzFkHCs. It include 2 new additions to the Formalizing 100 Theorems list, Leibniz' series for pi (leibpi) and the Konigsberg Bridge problem (konigsberg)

(10-Mar-2015) Mario Carneiro has written a paper, "Arithmetic in Metamath, Case Study: Bertrand's Postulate," for CICM 2015. A preprint is available at arXiv:1503.02349.

(23-Feb-2015) Scott Fenton has created a Metamath formalization of NF set theory: https://github.com/sctfn/metamath-nf/. For more information, see the Metamath Google Group posting.

(28-Jan-2015) Mario Carneiro added Wilson's Theorem (wilth), Ascending or Descending Sequences (erdsze, erdsze2), and Derangements Formula (derangfmla, subfaclim), bringing the Metamath total for Formalizing 100 Theorems to 44.

(19-Jan-2015) Mario Carneiro added Sylow's Theorem (sylow1, sylow2, sylow2b, sylow3), bringing the Metamath total for Formalizing 100 Theorems to 41.

(9-Jan-2015) The hypothesis order of mpbi*an* was changed. See the Notes entry of 9-Jan-2015.

(1-Jan-2015) Mario Carneiro has written a paper, "Conversion of HOL Light proofs into Metamath," that has been submitted to the Journal of Formalized Reasoning. A preprint is available on arxiv.org.

(22-Nov-2014) Stefan O'Rear added the Solutions to Pell's Equation (rmxycomplete) and Liouville's Theorem and the Construction of Transcendental Numbers (aaliou), bringing the Metamath total for Formalizing 100 Theorems to 40.

(22-Nov-2014) The metamath program has been updated with version 0.111. (1) Label wildcards now have a label range indicator "~" so that e.g. you can show or search all of the statements in a mathbox. See 'help search'. (Stefan O'Rear added this to the program.) (2) A qualifier was added to 'minimize_with' to prevent the use of any axioms not already used in the proof e.g. 'minimize_with * /no_new_axioms_from ax-*' will prevent the use of ax-ac if the proof doesn't already use it. See 'help minimize_with'.

(10-Oct-2014) Mario Carneiro has encoded the axiomatic basis for the HOL theorem prover into a Metamath source file, hol.mm.

(24-Sep-2014) Mario Carneiro added the Sum of the Angles of a Triangle (ang180), bringing the Metamath total for Formalizing 100 Theorems to 38.

(15-Sep-2014) Mario Carneiro added the Fundamental Theorem of Algebra (fta), bringing the Metamath total for Formalizing 100 Theorems to 37.

(3-Sep-2014) Mario Carneiro added the Fundamental Theorem of Integral Calculus (ftc1, ftc2). This brings the Metamath total for Formalizing 100 Theorems to 35. (added 14-Sep-2014) Along the way, he added the Mean Value Theorem (mvth), bringing the total to 36.

(16-Aug-2014) Mario Carneiro started a Metamath blog at http://metamath-blog.blogspot.com/.

(10-Aug-2014) Mario Carneiro added Erdős's proof of the divergence of the inverse prime series (prmrec). This brings the Metamath total for Formalizing 100 Theorems to 34.

(31-Jul-2014) Mario Carneiro added proofs for Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + .... (basel) and The Factor and Remainder Theorems (facth, plyrem). This brings the Metamath total for Formalizing 100 Theorems to 33.

(16-Jul-2014) Mario Carneiro added proofs for Four Squares Theorem (4sq), Formula for the Number of Combinations (hashbc), and Divisibility by 3 Rule (3dvds). This brings the Metamath total for Formalizing 100 Theorems to 31.

(11-Jul-2014) Mario Carneiro added proofs for Divergence of the Harmonic Series (harmonic), Order of a Subgroup (lagsubg), and Lebesgue Measure and Integration (itgcl). This brings the Metamath total for Formalizing 100 Theorems to 28.

(7-Jul-2014) Mario Carneiro presented a talk, "Natural Deduction in the Metamath Proof Language," at the 6PCM conference. Slides Audio

(25-Jun-2014) In version 0.108 of the metamath program, the 'minimize_with' command is now more automated. It now considers compressed proof length; it scans the statements in forward and reverse order and chooses the best; and it avoids $d conflicts. The '/no_distinct', '/brief', and '/reverse' qualifiers are obsolete, and '/verbose' no longer lists all statements scanned but gives more details about decision criteria.

(12-Jun-2014) To improve naming uniformity, theorems about operation values now use the abbreviation "ov". For example, df-opr, opreq1, oprabval5, and oprvres are now called df-ov, oveq1, ov5, and ovres respectively.

(11-Jun-2014) Mario Carneiro finished a major revision of set.mm. His notes are under the 11-Jun-2014 entry in the Notes

(4-Jun-2014) Mario Carneiro provided instructions and screenshots for syntax highlighting for the jEdit editor for use with Metamath and mmj2 source files.

(19-May-2014) Mario Carneiro added a feature to mmj2, in the build at https://github.com/digama0/mmj2/raw/dev-build/mmj2jar/mmj2.jar, which tests all but 5 definitions in set.mm for soundness. You can turn on the test by adding
SetMMDefinitionsCheckWithExclusions,ax-*,df-bi,df-clab,df-cleq,df-clel,df-sbc
to your RunParms.txt file.

(17-May-2014) A number of labels were changed in set.mm, listed at the top of set.mm as usual. Note in particular that the heavily-used visset, elisseti, syl11anc, syl111anc were changed respectively to vex, elexi, syl2anc, syl3anc.

(16-May-2014) Scott Fenton formalized a proof for "Sum of kth powers": fsumkthpow. This brings the Metamath total for Formalizing 100 Theorems to 25.

(9-May-2014) I (Norm Megill) presented an overview of Metamath at the "Formalization of mathematics in proof assistants" workshop at the Institut Henri Poincar� in Paris. The slides for this talk are here.

(22-Jun-2014) Version 0.107 of the metamath program adds a "PART" indention level to the Statement List table of contents, adds 'show proof ... /size' to show source file bytes used, and adds 'show elapsed_time'. The last one is helpful for measuring the run time of long commands. See 'help write theorem_list', 'help show proof', and 'help show elapsed_time' for more information.

(2-May-2014) Scott Fenton formalized a proof of Sum of the Reciprocals of the Triangular Numbers: trirecip. This brings the Metamath total for Formalizing 100 Theorems to 24.

(19-Apr-2014) Scott Fenton formalized a proof of the Formula for Pythagorean Triples: pythagtrip. This brings the Metamath total for Formalizing 100 Theorems to 23.

(11-Apr-2014) David A. Wheeler produced a much-needed and well-done video for mmj2, called "Introduction to Metamath & mmj2". Thanks, David!

(15-Mar-2014) Mario Carneiro formalized a proof of Bertrand's postulate: bpos. This brings the Metamath total for Formalizing 100 Theorems to 22.

(18-Feb-2014) Mario Carneiro proved that complex number axiom ax-cnex is redundant (theorem cnex). See also Real and Complex Numbers.

(11-Feb-2014) David A. Wheeler has created a theorem compilation that tracks those theorems in Freek Wiedijk's Formalizing 100 Theorems list that have been proved in set.mm. If you find a error or omission in this list, let me know so it can be corrected. (Update 1-Mar-2014: Mario has added eulerth and bezout to the list.)

(4-Feb-2014) Mario Carneiro writes:

The latest commit on the mmj2 development branch introduced an exciting new feature, namely syntax highlighting for mmp files in the main window. (You can pick up the latest mmj2.jar at https://github.com/digama0/mmj2/blob/develop/mmj2jar/mmj2.jar .) The reason I am asking for your help at this stage is to help with design for the syntax tokenizer, which is responsible for breaking down the input into various tokens with names like "comment", "set", and "stephypref", which are then colored according to the user's preference. As users of mmj2 and metamath, what types of highlighting would be useful to you?

One limitation of the tokenizer is that since (for performance reasons) it can be started at any line in the file, highly contextual coloring, like highlighting step references that don't exist previously in the file, is difficult to do. Similarly, true parsing of the formulas using the grammar is possible but likely to be unmanageably slow. But things like checking theorem labels against the database is quite simple to do under the current setup.

That said, how can this new feature be optimized to help you when writing proofs?

(13-Jan-2014) Mathbox users: the *19.21a*, *19.23a* series of theorems have been renamed to *alrim*, *exlim*. You can update your mathbox with a global replacement of string '19.21a' with 'alrim' and '19.23a' with 'exlim'.

(5-Jan-2014) If you downloaded mmj2 in the past 3 days, please update it with the current version, which fixes a bug introduced by the recent changes that made it unable to read in most of the proofs in the textarea properly.

(4-Jan-2014) I added a list of "Allowed substitutions" under the "Distinct variable groups" list on the theorem web pages, for example axsep. This is an experimental feature and comments are welcome.

(3-Jan-2014) Version 0.102 of the metamath program produces more space-efficient compressed proofs (still compatible with the specification in Appendix B of the Metamath book) using an algorithm suggested by Mario Carneiro. See 'help save proof' in the program. Also, mmj2 now generates proofs in the new format. The new mmj2 also has a mandatory update that fixes a bug related to the new format; you must update your mmj2 copy to use it with the latest set.mm.

(23-Dec-2013) Mario Carneiro has updated many older definitions to use the maps-to notation. If you have difficulty updating your local mathbox, contact him or me for assistance.

(1-Nov-2013) 'undo' and 'redo' commands were added to the Proof Assistant in metamath program version 0.07.99. See 'help undo' in the program.

(8-Oct-2013) Today's Notes entry describes some proof repair techniques.

(5-Oct-2013) Today's Notes entry explains some recent extensible structure improvements.

(8-Sep-2013) Mario Carneiro has revised the square root and sequence generator definitions. See today's Notes entry.

(3-Aug-2013) Mario Carneiro writes: "I finally found enough time to create a GitHub repository for development at https://github.com/digama0/mmj2. A permalink to the latest version plus source (akin to mmj2.zip) is https://github.com/digama0/mmj2/zipball/, and the jar file on its own (mmj2.jar) is at https://github.com/digama0/mmj2/blob/master/mmj2jar/mmj2.jar?raw=true. Unfortunately there is no easy way to automatically generate mmj2jar.zip, but this is available as part of the zip distribution for mmj2.zip. History tracking will be handled by the repository now. Do you have old versions of the mmj2 directory? I could add them as historical commits if you do."

(18-Jun-2013) Mario Carneiro has done a major revision and cleanup of the construction of real and complex numbers. In particular, rather than using equivalence classes as is customary for the construction of the temporary rationals, he used only "reduced fractions", so that the use of the axiom of infinity is avoided until it becomes necessary for the construction of the temporary reals.

(18-May-2013) Mario Carneiro has added the ability to produce compressed proofs to mmj2. This is not an official release but can be downloaded here if you want to try it: mmj2.jar. If you have any feedback, send it to me (NM), and I will forward it to Mario. (Disclaimer: this release has not been endorsed by Mel O'Cat. If anyone has been in contact with him, please let me know.)

(29-Mar-2013) Charles Greathouse reduced the size of our PNG symbol images using the pngout program.

(8-Mar-2013) Wolf Lammen has reorganized the theorems in the "Logical negation" section of set.mm into a more orderly, less scattered arrangement.

(27-Feb-2013) Scott Fenton has done a large cleanup of set.mm, eliminating *OLD references in 144 proofs. See the Notes entry for 27-Feb-2013.

(21-Feb-2013) *ATTENTION MATHBOX USERS* The order of hypotheses of many syl* theorems were changed, per a suggestion of Mario Carneiro. You need to update your local mathbox copy for compatibility with the new set.mm, or I can do it for you if you wish. See the Notes entry for 21-Feb-2013.

(16-Feb-2013) Scott Fenton shortened the direct-from-axiom proofs of *3.1, *3.43, *4.4, *4.41, *4.5, *4.76, *4.83, *5.33, *5.35, *5.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(27-Jan-2013) Scott Fenton writes, "I've updated Ralph Levien's mmverify.py. It's now a Python 3 program, and supports compressed proofs and file inclusion statements. This adds about fifty lines to the original program. Enjoy!"

(10-Jan-2013) A new mathbox was added for Mario Carneiro, who has contributed a number of cardinality theorems without invoking the Axiom of Choice. This is nice work, and I will be using some of these (those suffixed with "NEW") to replace the existing ones in the main part of set.mm that currently invoke AC unnecessarily.

(4-Jan-2013) As mentioned in the 19-Jun-2012 item below, Eric Schmidt discovered that the complex number axioms axaddcom (now addcom) and ax0id (now addid1) are redundant (schmidt-cnaxioms.pdf, .tex). In addition, ax1id (now mulid1) can be weakened to ax1rid. Scott Fenton has now formalized this work, so that now there are 23 instead of 25 axioms for real and complex numbers in set.mm. The Axioms for Complex Numbers page has been updated with these results. An interesting part of the proof, showing how commutativity of addition follows from other laws, is in addcomi.

(27-Nov-2012) The frequently-used theorems "an1s", "an1rs", "ancom13s", "ancom31s" were renamed to "an12s", "an32s", "an13s", "an31s" to conform to the convention for an12 etc.

(4-Nov-2012) The changes proposed in the Notes, renaming Grp to GrpOp etc., have been incorporated into set.mm. See the list of changes at the top of set.mm. If you want me to update your mathbox with these changes, send it to me along with the version of set.mm that it works with.

(20-Sep-2012) Mel O'Cat updated http://us2.metamath.org:88/ocat/mmj2/TESTmmj2jar.zip. See the README.TXT for a description of the new features.

(21-Aug-2012) Mel O'Cat has uploaded SearchOptionsMockup9.zip, a mockup for the new search screen in mmj2. See the README.txt file for instructions. He will welcome feedback via x178g243 at yahoo.com.

(19-Jun-2012) Eric Schmidt has discovered that in our axioms for complex numbers, axaddcom and ax0id are redundant. (At some point these need to be formalized for set.mm.) He has written up these and some other nice results, including some independence results for the axioms, in schmidt-cnaxioms.pdf (schmidt-cnaxioms.tex).

(23-Apr-2012) Frédéric Liné sent me a PDF (LaTeX source) developed with Lamport's pf2 package. He wrote: "I think it works well with Metamath since the proofs are in a tree form. I use it to have a sketch of a proof. I get this way a better understanding of the proof and I can cut down its size. For instance, inpreima5 was reduced by 50% when I wrote the corresponding proof with pf2."

(5-Mar-2012) I added links to Wikiproofs and its recent changes in the "Wikis" list at the top of this page.

(12-Jan-2012) Thanks to William Hoza who sent me a ZFC T-shirt, and thanks to the ZFC models (courtesy of the Inaccessible Cardinals agency).

FrontBackDetail
ZFC T-shirt front ZFC T-shirt back ZFC T-shirt detail

(24-Nov-2011) In metamath program version 0.07.71, the 'minimize_with' command by default now scans from bottom to top instead of top to bottom, since empirically this often (although not always) results in a shorter proof. A top to bottom scan can be specified with a new qualifier '/reverse'. You can try both methods (starting from the same original proof, of course) and pick the shorter proof.

(15-Oct-2011) From Mel O'Cat:
I just uploaded mmj2.zip containing the 1-Nov-2011 (20111101) release: http://us2.metamath.org:88/ocat/mmj2/mmj2.zip http://us2.metamath.org:88/ocat/mmj2/mmj2.md5
A few last minute tweaks:
1. I now bless double-click starting of mmj2.bat (MacMMJ2.command in Mac OS-X)! See mmj2\QuickStart.html
2. Much improved support of Mac OS-X systems. See mmj2\QuickStart.html
3. I tweaked the Command Line Argument Options report to
a) print every time;
b) print as much as possible even if there are errors in the command line arguments -- and the last line printed corresponds to the argument in error;
c) removed Y/N argument on the command line to enable/disable the report. this simplifies things.
4) Documentation revised, including the PATutorial.
See CHGLOG.TXT for list of all changes. Good luck. And thanks for all of your help!

(15-Sep-2011) MATHBOX USERS: I made a large number of label name changes to set.mm to improve naming consistency. There is a script at the top of the current set.mm that you can use to update your mathbox or older set.mm. Or if you wish, I can do the update on your next mathbox submission - in that case, please include a .zip of the set.mm version you used.

(30-Aug-2011) Scott Fenton shortened the direct-from-axiom proofs of *3.33, *3.45, *4.36, and meredith in the "Shortest known proofs of the propositional calculus theorems from Principia Mathematica" (pmproofs.txt).

(21-Aug-2011) A post on reddit generated 60,000 hits (and a TOS violation notice from my provider...),

(18-Aug-2011) The Metamath Google Group has a discussion of my canonical conjunctions proposal. Any feedback directly to me (Norm Megill) is also welcome.

(4-Jul-2011) John Baker has provided (metamath_kindle.zip) "a modified version of [the] metamath.tex [Metamath] book source that is formatted for the Kindle. If you compile the document the resulting PDF can be loaded into into a Kindle and easily read." (Update: the PDF file is now included also.)

(3-Jul-2011) Nested 'submit' calls are now allowed, in metamath program version 0.07.68. Thus you can create or modify a command file (script) from within a command file then 'submit' it. While 'submit' cannot pass arguments (nor are there plans to add this feature), you can 'substitute' strings in the 'submit' target file before calling it in order to emulate this.

(28-Jun-2011)The metamath program version 0.07.64 adds the '/include_mathboxes' qualifier to 'minimize_with'; by default, 'minimize_with *' will now skip checking user mathboxes. Since mathboxes should be independent from each other, this will help prevent accidental cross-"contamination". Also, '/rewrap' was added to 'write source' to automatically wrap $a and $p comments so as to conform to the current formatting conventions used in set.mm. This means you no longer have to be concerned about line length < 80 etc.

(19-Jun-2011) ATTENTION MATHBOX USERS: The wff variables et, ze, si, and rh are now global. This change was made primarily to resolve some conflicts between mathboxes, but it will also let you avoid having to constantly redeclare these locally in the future. Unfortunately, this change can affect the $f hypothesis order, which can cause proofs referencing theorems that use these variables to fail. All mathbox proofs currently in set.mm have been corrected for this, and you should refresh your local copy for further development of your mathbox. You can correct your proofs that are not in set.mm as follows. Only the proofs that fail under the current set.mm (using version 0.07.62 or later of the metamath program) need to be modified.

To fix a proof that references earlier theorems using et, ze, si, and rh, do the following (using a hypothetical theorem 'abc' as an example): 'prove abc' (ignore error messages), 'delete floating', 'initialize all', 'unify all/interactive', 'improve all', 'save new_proof/compressed'. If your proof uses dummy variables, these must be reassigned manually.

To fix a proof that uses et, ze, si, and rh as local variables, make sure the proof is saved in 'compressed' format. Then delete the local declarations ($v and $f statements) and follow the same steps above to correct the proof.

I apologize for the inconvenience. If you have trouble fixing your proofs, you can contact me for assistance.

Note: Versions of the metamath program before 0.07.62 did not flag an error when global variables were redeclared locally, as it should have according to the spec. This caused these spec violations to go unnoticed in some older set.mm versions. The new error messages are in fact just informational and can be ignored when working with older set.mm versions.

(7-Jun-2011) The metamath program version 0.07.60 fixes a bug with the 'minimize_with' command found by Andrew Salmon.

(12-May-2010) Andrew Salmon shortened many proofs, shown above. For comparison, I have temporarily kept the old version, which is suffixed with OLD, such as oridmOLD for oridm.

(9-Dec-2010) Eric Schmidt has written a Metamath proof verifier in C++, called checkmm.cpp.

(3-Oct-2010) The following changes were made to the tokens in set.mm. The subset and proper subset symbol changes to C_ and C. were made to prevent defeating the parenthesis matching in Emacs. Other changes were made so that all letters a-z and A-Z are now available for variable names. One-letter constants such as _V, _e, and _i are now shown on the web pages with Roman instead of italic font, to disambiguate italic variable names. The new convention is that a prefix of _ indicates Roman font and a prefix of ~ indicates a script (curly) font. Thanks to Stefan Allan and Frédéric Liné for discussions leading to this change.

OldNewDescription
C. _C binomial coefficient
E _E epsilon relation
e _e Euler's constant
I _I identity relation
i _i imaginary unit
V _V universal class
(_ C_ subset
(. C. proper subset
P~ ~P power class
H~ ~H Hilbert space

(25-Sep-2010) The metamath program (version 0.07.54) now implements the current Metamath spec, so footnote 2 on p. 92 of the Metamath book can be ignored.

(24-Sep-2010) The metamath program (version 0.07.53) fixes bug 2106, reported by Michal Burger.

(14-Sep-2010) The metamath program (version 0.07.52) has a revamped LaTeX output with 'show statement xxx /tex', which produces the combined statement, description, and proof similar to the web page generation. Also, 'show proof xxx /lemmon/renumber' now matches the web page step numbers. ('show proof xxx/renumber' still has the indented form conforming to the actual RPN proof, with slightly different numbering.)

(9-Sep-2010) The metamath program (version 0.07.51) was updated with a modification by Stefan Allan that adds hyperlinks the the Ref column of proofs.

(12-Jun-2010) Scott Fenton contributed a D-proof (directly from axioms) of Meredith's single axiom (see the end of pmproofs.txt). A description of Meredith's axiom can be found in theorem meredith.

(11-Jun-2010) A new Metamath mirror was added in Austria, courtesy of Kinder-Enduro.

(28-Feb-2010) Raph Levien's Ghilbert project now has a new Ghilbert site and a Google Group.

(26-Jan-2010) Dmitri Vlasov writes, "I admire the simplicity and power of the metamath language, but still I see its great disadvantage - the proofs in metamath are completely non-manageable by humans without proof assistants. Therefore I decided to develop another language, which would be a higher-level superstructure language towards metamath, and which will support human-readable/writable proofs directly, without proof assistants. I call this language mdl (acronym for 'mathematics development language')." The latest version of Dmitri's translators from metamath to mdl and back can be downloaded from http://mathdevlanguage.sourceforge.net/. Currently only Linux is supported, but Dmitri says is should not be difficult to port it to other platforms that have a g++ compiler.

(11-Sep-2009) The metamath program (version 0.07.48) has been updated to enforce the whitespace requirement of the current spec.

(10-Sep-2009) Matthew Leitch has written an nice article, "How to write mathematics clearly", that briefly mentions Metamath. Overall it makes some excellent points. (I have written to him about a few things I disagree with.)

(28-May-2009) AsteroidMeta is back on-line. Note the URL change.

(12-May-2009) Charles Greathouse wrote a Greasemonkey script to reformat the axiom list on Metamath web site proof pages. This is a beta version; he will appreciate feedback.

(11-May-2009) Stefan Allan modified the metamath program to add the command "show statement xxx /mnemonics", which produces the output file Mnemosyne.txt for use with the Mnemosyne project. The current Metamath program download incorporates this command. Instructions: Create the file mnemosyne.txt with e.g. "show statement ax-* /mnemonics". In the Mnemosyne program, load the file by choosing File->Import then file format "Q and A on separate lines". Notes: (1) Don't try to load all of set.mm, it will crash the program due to a bug in Mnemosyne. (2) On my computer, the arrows in ax-1 don't display. Stefan reports that they do on his computer. (Both are Windows XP.)

(3-May-2009) Steven Baldasty wrote a Metamath syntax highlighting file for the gedit editor. Screenshot.

(1-May-2009) Users on a gaming forum discuss our 2+2=4 proof. Notable comments include "Ew math!" and "Whoever wrote this has absolutely no life."

(12-Mar-2009) Chris Capel has created a Javascript theorem viewer demo that (1) shows substitutions and (2) allows expanding and collapsing proof steps. You are invited to take a look and give him feedback at his Metablog.

(28-Feb-2009) Chris Capel has written a Metamath proof verifier in C#, available at http://pdf23ds.net/bzr/MathEditor/Verifier/Verifier.cs and weighing in at 550 lines. Also, that same URL without the file on it is a Bazaar repository.

(2-Dec-2008) A new section was added to the Deduction Theorem page, called Logic, Metalogic, Metametalogic, and Metametametalogic.

(24-Aug-2008) (From ocat): The 1-Aug-2008 version of mmj2 is ready (mmj2.zip), size = 1,534,041 bytes. This version contains the Theorem Loader enhancement which provides a "sandboxing" capability for user theorems and dynamic update of new theorems to the Metamath database already loaded in memory by mmj2. Also, the new "mmj2 Service" feature enables calling mmj2 as a subroutine, or having mmj2 call your program, and provides access to the mmj2 data structures and objects loaded in memory (i.e. get started writing those Jython programs!) See also mmj2 on AsteroidMeta.

(23-May-2008) Gérard Lang pointed me to Bob Solovay's note on AC and strongly inaccessible cardinals. One of the eventual goals for set.mm is to prove the Axiom of Choice from Grothendieck's axiom, like Mizar does, and this note may be helpful for anyone wanting to attempt that. Separately, I also came across a history of the size reduction of grothprim (viewable in Firefox and some versions of Internet Explorer).

(14-Apr-2008) A "/join" qualifier was added to the "search" command in the metamath program (version 0.07.37). This qualifier will join the $e hypotheses to the $a or $p for searching, so that math tokens in the $e's can be matched as well. For example, "search *com* +v" produces no results, but "search *com* +v /join" yields commutative laws involving vector addition. Thanks to Stefan Allan for suggesting this idea.

(8-Apr-2008) The 8,000th theorem, hlrel, was added to the Metamath Proof Explorer part of the database.

(2-Mar-2008) I added a small section to the end of the Deduction Theorem page.

(17-Feb-2008) ocat has uploaded the "1-Mar-2008" mmj2: mmj2.zip. See the description.

(16-Jan-2008) O'Cat has written mmj2 Proof Assistant Quick Tips.

(30-Dec-2007) "How to build a library of formalized mathematics".

(22-Dec-2007) The Metamath Proof Explorer was included in the top 30 science resources for 2007 by the University at Albany Science Library.

(17-Dec-2007) Metamath's Wikipedia entry says, "This article may require cleanup to meet Wikipedia's quality standards" (see its discussion page). Volunteers are welcome. :) (In the interest of objectivity, I don't edit this entry.)

(20-Nov-2007) Jeff Hoffman created nicod.mm and posted it to the Google Metamath Group.

(19-Nov-2007) Reinder Verlinde suggested adding tooltips to the hyperlinks on the proof pages, which I did for proof step hyperlinks. Discussion.

(5-Nov-2007) A Usenet challenge. :)

(4-Aug-2007) I added a "Request for comments on proposed 'maps to' notation" at the bottom of the AsteroidMeta set.mm discussion page.

(21-Jun-2007) A preprint (PDF file) describing Kurt Maes' axiom of choice with 5 quantifiers, proved in set.mm as ackm.

(20-Jun-2007) The 7,000th theorem, ifpr, was added to the Metamath Proof Explorer part of the database.

(29-Apr-2007) Blog mentions of Metamath: here and here.

(21-Mar-2007) Paul Chapman is working on a new proof browser, which has highlighting that allows you to see the referenced theorem before and after the substitution was made. Here is a screenshot of theorem 0nn0 and a screenshot of theorem 2p2e4.

(15-Mar-2007) A picture of Penny the cat guarding the us2.metamath.org:8888 server and making the rounds.

(16-Feb-2007) For convenience, the program "drule.c" (pronounced "D-rule", not "drool") mentioned in pmproofs.txt can now be downloaded (drule.c) without having to ask me for it. The same disclaimer applies: even though this program works and has no known bugs, it was not intended for general release. Read the comments at the top of the program for instructions.

(28-Jan-2007) Jason Orendorff set up a new mailing list for Metamath: http://groups.google.com/group/metamath.

(20-Jan-2007) Bob Solovay provided a revised version of his Metamath database for Peano arithmetic, peano.mm.

(2-Jan-2007) Raph Levien has set up a wiki called Barghest for the Ghilbert language and software.

(26-Dec-2006) I posted an explanation of theorem ecoprass on Usenet.

(2-Dec-2006) Berislav Žarnić translated the Metamath Solitaire applet to Croatian.

(26-Nov-2006) Dan Getz has created an RSS feed for new theorems as they appear on this page.

(6-Nov-2006) The first 3 paragraphs in Appendix 2: Note on the Axioms were rewritten to clarify the connection between Tarski's axiom system and Metamath.

(31-Oct-2006) ocat asked for a do-over due to a bug in mmj2 -- if you downloaded the mmj2.zip version dated 10/28/2006, then download the new version dated 10/30.

(29-Oct-2006) ocat has announced that the long-awaited 1-Nov-2006 release of mmj2 is available now.
     The new "Unify+Get Hints" is quite useful, and any proof can be generated as follows. With "?" in the Hyp field and Ref field blank, select "Unify+Get Hints". Select a hint from the list and put it in the Ref field. Edit any $n dummy variables to become the desired wffs. Rinse and repeat for the new proof steps generated, until the proof is done.
     The new tutorial, mmj2PATutorial.bat, explains this in detail. One way to reduce or avoid dummy $n's is to fill in the Hyp field with a comma-separated list of any known hypothesis matches to earlier proof steps, keeping a "?" in the list to indicate that the remaining hypotheses are unknown. Then "Unify+Get Hints" can be applied. The tutorial page \mmj2\data\mmp\PATutorial\Page405.mmp has an example.
     Don't forget that the eimm export/import program lets you go back and forth between the mmj2 and the metamath program proof assistants, without exiting from either one, to exploit the best features of each as required.

(21-Oct-2006) Martin Kiselkov has written a Metamath proof verifier in the Lua scripting language, called verify.lua. While it is not practical as an everyday verifier - he writes that it takes about 40 minutes to verify set.mm on a a Pentium 4 - it could be useful to someone learning Lua or Metamath, and importantly it provides another independent way of verifying the correctness of Metamath proofs. His code looks like it is nicely structured and very readable. He is currently working on a faster version in C++.

(19-Oct-2006) New AsteroidMeta page by Raph, Distinctors_vs_binders.

(13-Oct-2006) I put a simple Metamath browser on my PDA (Palm Tungsten E) so that I don't have to lug around my laptop. Here is a screenshot. It isn't polished, but I'll provide the file + instructions if anyone wants it.

(3-Oct-2006) A blog entry, Principia for Reverse Mathematics.

(28-Sep-2006) A blog entry, Metamath responds.

(26-Sep-2006) A blog entry, Metamath isn't hygienic.

(11-Aug-2006) A blog entry, Metamath and the Peano Induction Axiom.

(26-Jul-2006) A new open problem in predicate calculus was added.

(18-Jun-2006) The 6,000th theorem, mt4d, was added to the Metamath Proof Explorer part of the database.

(9-May-2006) Luca Ciciriello has upgraded the t2mf program, which is a C program used to create the MIDI files on the Metamath Music Page, so that it works on MacOS X. This is a nice accomplishment, since the original program was written before C was standardized by ANSI and will not compile on modern compilers.
      Unfortunately, the original program source states no copyright terms. The main author, Tim Thompson, has kindly agreed to release his code to public domain, but two other authors have also contributed to the code, and so far I have been unable to contact them for copyright clearance. Therefore I cannot offer the MacOS X version for public download on this site until this is resolved. Update 10-May-2006: Another author, M. Czeiszperger, has released his contribution to public domain.
      If you are interested in Luca's modified source code, please contact me directly.

(18-Apr-2006) Incomplete proofs in progress can now be interchanged between the Metamath program's CLI Proof Assistant and mmj2's GUI Proof Assistant, using a new export-import program called eimm. This can be done without exiting either proof assistant, so that the strengths of each approach can be exploited during proof development. See "Use Case 5a" and "Use Case 5b" at mmj2ProofAssistantFeedback.

(28-Mar-2006) Scott Fenton updated his second version of Metamath Solitaire (the one that uses external axioms). He writes: "I've switched to making it a standalone program, as it seems silly to have an applet that can't be run in a web browser. Check the README file for further info." The download is mmsol-0.5.tar.gz.

(27-Mar-2006) Scott Fenton has updated the Metamath Solitaire Java applet to Java 1.5: (1) QSort has been stripped out: its functionality is in the Collections class that Sun ships; (2) all Vectors have been replaced by ArrayLists; (3) generic types have been tossed in wherever they fit: this cuts back drastically on casting; and (4) any warnings Eclipse spouted out have been dealt with. I haven't yet updated it officially, because I don't know if it will work with Microsoft's JVM in older versions of Internet Explorer. The current official version is compiled with Java 1.3, because it won't work with Microsoft's JVM if it is compiled with Java 1.4. (As distasteful as that seems, I will get complaints from users if it doesn't work with Microsoft's JVM.) If anyone can verify that Scott's new version runs on Microsoft's JVM, I would be grateful. Scott's new version is mm.java-1.5.gz; after uncompressing it, rename it to mm.java, use it to replace the existing mm.java file in the Metamath Solitaire download, and recompile according to instructions in the mm.java comments.
      Scott has also created a second version, mmsol-0.2.tar.gz, that reads the axioms from ASCII files, instead of having the axioms hard-coded in the program. This can be very useful if you want to play with custom axioms, and you can also add a collection of starting theorems as "axioms" to work from. However, it must be run from the local directory with appletviewer, since the default Java security model doesn't allow reading files from a browser. It works with the JDK 5 Update 6 Java download.
To compile (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\javac.exe mm.java
To run (from Windows Command Prompt): C:\Program Files\Java\jdk1.5.0_06\bin\appletviewer.exe mms.html

(21-Jan-2006) Juha Arpiainen proved the independence of axiom ax-11 from the others. This was published as an open problem in my 1995 paper (Remark 9.5 on PDF page 17). See Item 9a on the Workshop Miscellany for his seven-line proof. See also the Asteroid Meta metamathMathQuestions page under the heading "Axiom of variable substitution: ax-11". Congratulations, Juha!

(20-Oct-2005) Juha Arpiainen is working on a proof verifier in Common Lisp called Bourbaki. Its proof language has its roots in Metamath, with the goal of providing a more powerful syntax and definitional soundness checking. See its documentation and related discussion.

(17-Oct-2005) Marnix Klooster has written a Metamath proof verifier in Haskell, called Hmm. Also see his Announcement. The complete program (Hmm.hs, HmmImpl.hs, and HmmVerify.hs) has only 444 lines of code, excluding comments and blank lines. It verifies compressed as well as regular proofs; moreover, it transparently verifies both per-spec compressed proofs and the flawed format he uncovered (see comment below of 16-Oct-05).

(16-Oct-2005) Marnix Klooster noticed that for large proofs, the compressed proof format did not match the spec in the book. His algorithm to correct the problem has been put into the Metamath program (version 0.07.6). The program still verifies older proofs with the incorrect format, but the user will be nagged to update them with 'save proof *'. In set.mm, 285 out of 6376 proofs are affected. (The incorrect format did not affect proof correctness or verification, since the compression and decompression algorithms matched each other.)

(13-Sep-2005) Scott Fenton found an interesting axiom, ax46, which could be used to replace both ax-4 and ax-6.

(29-Jul-2005) Metamath was selected as site of the week by American Scientist Online.

(8-Jul-2005) Roy Longton has contributed 53 new theorems to the Quantum Logic Explorer. You can see them in the Theorem List starting at lem3.3.3lem1. He writes, "If you want, you can post an open challenge to see if anyone can find shorter proofs of the theorems I submitted."

(10-May-2005) A Usenet post I posted about the infinite prime proof; another one about indexed unions.

(3-May-2005) The theorem divexpt is the 5,000th theorem added to the Metamath Proof Explorer database.

(12-Apr-2005) Raph Levien solved the open problem in item 16 on the Workshop Miscellany page and as a corollary proved that axiom ax-9 is independent from the other axioms of predicate calculus and equality. This is the first such independence proof so far; a goal is to prove all of them independent (or to derive any redundant ones from the others).

(8-Mar-2005) I added a paragraph above our complex number axioms table, summarizing the construction and indicating where Dedekind cuts are defined. Thanks to Andrew Buhr for comments on this.

(16-Feb-2005) The Metamath Music Page is mentioned as a reference or resource for a university course called Math, Mind, and Music. .

(28-Jan-2005) Steven Cullinane parodied the Metamath Music Page in his blog.

(18-Jan-2005) Waldek Hebisch upgraded the Metamath program to run on the AMD64 64-bit processor.

(17-Jan-2005) A symbol list summary was added to the beginning of the Hilbert Space Explorer Home Page. Thanks to Mladen Pavicic for suggesting this.

(6-Jan-2005) Someone assembled an amazon.com list of some of the books in the Metamath Proof Explorer Bibliography.

(4-Jan-2005) The definition of ordinal exponentiation was decided on after this Usenet discussion.

(19-Dec-2004) A bit of trivia: my Erdös number is 2, as you can see from this list.

(20-Oct-2004) I started this Usenet discussion about the "reals are uncountable" proof (127 comments; last one on Nov. 12).

(12-Oct-2004) gch-kn shows the equivalence of the Generalized Continuum Hypothesis and Prof. Nambiar's Axiom of Combinatorial Sets. This proof answers his Open Problem 2 (PDF file).

(5-Aug-2004) I gave a talk on "Hilbert Lattice Equations" at the Argonne workshop.

(25-Jul-2004) The theorem nthruz is the 4,000th theorem added to the Metamath Proof Explorer database.

(27-May-2004) Josiah Burroughs contributed the proofs u1lemn1b, u1lem3var1, oi3oa3lem1, and oi3oa3 to the Quantum Logic Explorer database ql.mm.

(23-May-2004) Some minor typos found by Josh Purinton were corrected in the Metamath book. In addition, Josh simplified the definition of the closure of a pre-statement of a formal system in Appendix C.

(5-May-2004) Gregory Bush has found shorter proofs for 67 of the 193 propositional calculus theorems listed in Principia Mathematica, thus establishing 67 new records. (This was challenge #4 on the open problems page.)

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