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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | bgoldbachlt 41701* | The binary Goldbach conjecture is valid for small even numbers (i.e. for all even numbers less than or equal to a fixed big 𝑚). This is verified for m = 4 x 10^18 by Oliveira e Silva, see ax-bgbltosilva 41698. (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑚 ∈ ℕ ((4 · (;10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) | ||
Axiom | ax-hgprmladder 41702 | There is a partition ("ladder") of primes from 7 to 8.8 x 10^30 with parts ("rungs") having lengths of at least 4 and at most N - 4, see section 1.2.2 in [Helfgott] p. 4. Temporarily provided as "axiom". (Contributed by AV, 3-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑑 ∈ (ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (;10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (;10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)))) | ||
Theorem | tgblthelfgott 41703 | The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 41701, ax-hgprmladder 41702 and bgoldbtbnd 41697. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (;10↑;29))) → 𝑁 ∈ GoldbachOdd ) | ||
Theorem | tgoldbachlt 41704* | The ternary Goldbach conjecture is valid for small odd numbers (i.e. for all odd numbers less than a fixed big 𝑚 greater than 8 x 10^30). This is verified for m = 8.875694 x 10^30 by Helfgott, see tgblthelfgott 41703. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∃𝑚 ∈ ℕ ((8 · (;10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )) | ||
Theorem | tgoldbach 41705 | The ternary Goldbach conjecture is valid. Main theorem in [Helfgott] p. 2. This follows from tgoldbachlt 41704 and ax-tgoldbachgt 41699. (Contributed by AV, 2-Aug-2020.) (Revised by AV, 9-Sep-2021.) |
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) | ||
Axiom | ax-bgbltosilvaOLD 41706 | Obsolete version of ax-bgbltosilva 41698 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.) |
⊢ ((𝑁 ∈ Even ∧ 4 < 𝑁 ∧ 𝑁 ≤ (4 · (10↑;18))) → 𝑁 ∈ GoldbachEven ) | ||
Theorem | bgoldbachltOLD 41707* | Obsolete version of bgoldbachlt 41701 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑚 ∈ ℕ ((4 · (10↑;18)) ≤ 𝑚 ∧ ∀𝑛 ∈ Even ((4 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachEven )) | ||
Axiom | ax-hgprmladderOLD 41708 | Obsolete version of ax-hgprmladder 41702 as of 9-Sep-2021. (Contributed by AV, 3-Aug-2020.) (New usage is discouraged.) |
⊢ ∃𝑑 ∈ (ℤ≥‘3)∃𝑓 ∈ (RePart‘𝑑)(((𝑓‘0) = 7 ∧ (𝑓‘1) = ;13 ∧ (𝑓‘𝑑) = (;89 · (10↑;29))) ∧ ∀𝑖 ∈ (0..^𝑑)((𝑓‘𝑖) ∈ (ℙ ∖ {2}) ∧ ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)) < ((4 · (10↑;18)) − 4) ∧ 4 < ((𝑓‘(𝑖 + 1)) − (𝑓‘𝑖)))) | ||
Theorem | tgblthelfgottOLD 41709 | Obsolete version of tgblthelfgott 41703 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (;88 · (10↑;29))) → 𝑁 ∈ GoldbachOdd ) | ||
Theorem | tgoldbachltOLD 41710* | Obsolete version of tgoldbachlt 41704 as of 9-Sep-2021. (Contributed by AV, 4-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∃𝑚 ∈ ℕ ((8 · (10↑;30)) < 𝑚 ∧ ∀𝑛 ∈ Odd ((7 < 𝑛 ∧ 𝑛 < 𝑚) → 𝑛 ∈ GoldbachOdd )) | ||
Axiom | ax-tgoldbachgtOLD 41711* | Obsolete version of ax-tgoldbachgt 41699 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (New usage is discouraged.) |
⊢ ∃𝑚 ∈ ℕ (𝑚 ≤ (10↑;27) ∧ ∀𝑛 ∈ Odd (𝑚 < 𝑛 → 𝑛 ∈ GoldbachOdd )) | ||
Theorem | tgoldbachOLD 41712 | Obsolete version of tgoldbach 41705 as of 9-Sep-2021. (Contributed by AV, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ∀𝑛 ∈ Odd (7 < 𝑛 → 𝑛 ∈ GoldbachOdd ) | ||
Theorem | 1hegrlfgr 41713* | A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ≠ 𝐶) & ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑉) & ⊢ (𝜑 → (iEdg‘𝐺) = {〈𝐴, 𝐸〉}) & ⊢ (𝜑 → {𝐵, 𝐶} ⊆ 𝐸) ⇒ ⊢ (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}) | ||
Syntax | cupwlks 41714 | Extend class notation with walks (of a pseudograph). |
class UPWalks | ||
Definition | df-upwlks 41715* |
Define the set of all walks (in a pseudograph), called "simple walks"
in
the following.
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudograhs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ UPWalks = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | upwlksfval 41716* | The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (UPWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) | ||
Theorem | isupwlk 41717* | Properties of a pair of functions to be a simple walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | isupwlkg 41718* | Generalisation of isupwlk 41717: Conditions for two classes to represent a simple walk. (Contributed by AV, 5-Nov-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐺 ∈ 𝑊 → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | upwlkbprop 41719 | Basic properties of a simple walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V)) | ||
Theorem | upwlkwlk 41720 | A simple walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.) |
⊢ (𝐹(UPWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlk 41721 | In a pseudograph, a walk is a simple walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.) |
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃) | ||
Theorem | upgrwlkupwlkb 41722 | In a pseudograph, the definitions for a walk and a simple walk are equivalent. (Contributed by AV, 30-Dec-2020.) |
⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ 𝐹(UPWalks‘𝐺)𝑃)) | ||
Theorem | upgrisupwlkALT 41723* | Alternate proof of upgriswlk 26537 using the definition of UPGraph and related theorems. (Contributed by AV, 2-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) | ||
Theorem | sprid 41724 | Two identical representations of the class of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
⊢ {𝑝 ∣ ∃𝑎 ∈ V ∃𝑏 ∈ V 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
Theorem | elsprel 41725* | An unordered pair is an element of all unordered pairs. At least one of the two elements of the unordered pair must be a set. Otherwise, the unordered pair would be the empty set, see prprc 4302, which is not an element of all unordered pairs, see spr0nelg 41726. (Contributed by AV, 21-Nov-2021.) |
⊢ ((𝐴 ∈ 𝑉 ∨ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | spr0nelg 41726* | The empty set is not an element of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
⊢ ∅ ∉ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
Syntax | cspr 41727 | Extend class notation with set of pairs. |
class Pairs | ||
Definition | df-spr 41728* | Define the function which maps a set 𝑣 to the set of pairs consisting of elements of the set 𝑣. (Contributed by AV, 21-Nov-2021.) |
⊢ Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprval 41729* | The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprvalpw 41730* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ 𝒫 𝑉 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprssspr 41731* | The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.) |
⊢ (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎∃𝑏 𝑝 = {𝑎, 𝑏}} | ||
Theorem | spr0el 41732 | The empty set is not an unordered pair over any set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ ∅ ∉ (Pairs‘𝑉) | ||
Theorem | sprvalpwn0 41733* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 21-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑝 = {𝑎, 𝑏}}) | ||
Theorem | sprel 41734* | An element of the set of all unordered pairs over a given set 𝑉 is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
⊢ (𝑋 ∈ (Pairs‘𝑉) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
Theorem | prssspr 41735* | An element of a subset of the set of all unordered pairs over a given set 𝑉, is a pair of elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ 𝑋 ∈ 𝑃) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑋 = {𝑎, 𝑏}) | ||
Theorem | prelspr 41736 | An unordered pair of elements of a fixed set 𝑉 belongs to the set of all unordered pairs over the set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ ((𝑉 ∈ 𝑊 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} ∈ (Pairs‘𝑉)) | ||
Theorem | prsprel 41737 | The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.) |
⊢ (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
Theorem | prsssprel 41738 | The elements of a pair from a subset of the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 21-Nov-2021.) |
⊢ ((𝑃 ⊆ (Pairs‘𝑉) ∧ {𝑋, 𝑌} ∈ 𝑃 ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑊)) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) | ||
Theorem | sprvalpwle2 41739* | The set of all unordered pairs over a given set 𝑉, expressed by a restricted class abstraction. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑝) ≤ 2}) | ||
Theorem | sprsymrelfvlem 41740* | Lemma for sprsymrelf 41745 and sprsymrelfv 41744. (Contributed by AV, 19-Nov-2021.) |
⊢ (𝑃 ⊆ (Pairs‘𝑉) → {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑃 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉)) | ||
Theorem | sprsymrelf1lem 41741* | Lemma for sprsymrelf1 41746. (Contributed by AV, 22-Nov-2021.) |
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏)) | ||
Theorem | sprsymrelfolem1 41742* | Lemma 1 for sprsymrelfo 41747. (Contributed by AV, 22-Nov-2021.) |
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ 𝑄 ∈ 𝒫 (Pairs‘𝑉) | ||
Theorem | sprsymrelfolem2 41743* | Lemma 2 for sprsymrelfo 41747. (Contributed by AV, 23-Nov-2021.) |
⊢ 𝑄 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑅𝑏)} ⇒ ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑅 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑅𝑦 ↔ 𝑦𝑅𝑥)) → (𝑥𝑅𝑦 ↔ ∃𝑐 ∈ 𝑄 𝑐 = {𝑥, 𝑦})) | ||
Theorem | sprsymrelfv 41744* | The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (Contributed by AV, 19-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}}) | ||
Theorem | sprsymrelf 41745* | The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃⟶𝑅 | ||
Theorem | sprsymrelf1 41746* | The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ 𝐹:𝑃–1-1→𝑅 | ||
Theorem | sprsymrelfo 41747* | The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅) | ||
Theorem | sprsymrelf1o 41748* | The mapping 𝐹 is a bijection between the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} & ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {〈𝑥, 𝑦〉 ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}}) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–1-1-onto→𝑅) | ||
Theorem | sprbisymrel 41749* | There is a bijection between the subsets of the set of pairs over a fixed set 𝑉 and the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝑃–1-1-onto→𝑅) | ||
Theorem | sprsymrelen 41750* | The class 𝑃 of subsets of the set of pairs over a fixed set 𝑉 and the class 𝑅 of symmetric relations on the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅) | ||
Theorem | upgredgssspr 41751 | The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021.) |
⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ (Pairs‘(Vtx‘𝐺))) | ||
Theorem | uspgropssxp 41752* | The set 𝐺 of "simple pseudographs" for a fixed set 𝑉 of vertices is a subset of an Cartesian product. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 41762. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ⊆ (𝑊 × 𝑃)) | ||
Theorem | uspgrsprfv 41753* | The value of the function 𝐹 which maps a "simple pseudograph" for a fixed set 𝑉 of vertices to the set of edges (i.e. range of the edge function) of the graph. Solely for 𝐺 as defined here, the function 𝐹 is a bijection between the "simple pseudographs" and the subsets of the set of pairs 𝑃 over the the fixed set 𝑉 of vertices, see uspgrbispr 41759. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑋 ∈ 𝐺 → (𝐹‘𝑋) = (2nd ‘𝑋)) | ||
Theorem | uspgrsprf 41754* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺⟶𝑃 | ||
Theorem | uspgrsprf1 41755* | The mapping 𝐹 is a one-to-one function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ 𝐹:𝐺–1-1→𝑃 | ||
Theorem | uspgrsprfo 41756* | The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 onto the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) | ||
Theorem | uspgrsprf1o 41757* | The mapping 𝐹 is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. See also the comments on uspgrbisymrel 41762. (Contributed by AV, 25-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrex 41758* | The class 𝐺 of all "simple pseudographs" with a fixed set of vertices 𝑉 is a set. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ∈ V) | ||
Theorem | uspgrbispr 41759* | There is a bijection between the "simple pseudographs" with a fixed set of vertices 𝑉 and the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 26-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑃) | ||
Theorem | uspgrspren 41760* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑃 of subsets of the set of pairs over the fixed set 𝑉 are equinumerous. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) & ⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑃) | ||
Theorem | uspgrymrelen 41761* | The set 𝐺 of the "simple pseudographs" with a fixed set of vertices 𝑉 and the class 𝑅 of the symmetric relations on the fixed set 𝑉 are equinumerous. For more details about the class 𝐺 of all "simple pseudographs" see comments on uspgrbisymrel 41762. (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅) | ||
Theorem | uspgrbisymrel 41762* |
There is a bijection between the "simple pseudographs" for a fixed
set
𝑉 of vertices and the class 𝑅 of the
symmetric relations on the
fixed set 𝑉. The simple pseudographs, which are
graphs without
hyper- or multiedges, but which may contain loops, are expressed as
ordered pairs of the vertices and the edges (as proper or improper
unordered pairs of vertices, not as indexed edges!) in this theorem.
That class 𝐺 of such simple pseudographs is a set
(if 𝑉 is a
set, see uspgrex 41758) of equivalence classes of graphs
abstracting from
the index sets of their edge functions.
Solely for this abstraction, there is a bijection between the "simple pseudographs" as members of 𝐺 and the symmetric relations 𝑅 on the fixed set 𝑉 of vertices. This theorem would not hold for 𝐺 = {𝑔 ∈ USPGraph ∣ (Vtx‘𝑔) = 𝑉} and even not for 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ 〈𝑣, 𝑒〉 ∈ USPGraph )}, because these are much bigger classes. (Proposed by Gerard Lang, 16-Nov-2021.) (Contributed by AV, 27-Nov-2021.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | uspgrbisymrelALT 41763* | Alternate proof of uspgrbisymrel 41762 not using the definition of equinumerosity. (Contributed by AV, 26-Nov-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} & ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)} ⇒ ⊢ (𝑉 ∈ 𝑊 → ∃𝑓 𝑓:𝐺–1-1-onto→𝑅) | ||
Theorem | ovn0dmfun 41764 | If a class operation value for two operands is not the empty set, then the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6226. (Contributed by AV, 27-Jan-2020.) |
⊢ ((𝐴𝐹𝐵) ≠ ∅ → (〈𝐴, 𝐵〉 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {〈𝐴, 𝐵〉}))) | ||
Theorem | xpsnopab 41765* | A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.) |
⊢ ({𝑋} × 𝐶) = {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑋 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | xpiun 41766* | A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.) |
⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {〈𝑎, 𝑏〉 ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} | ||
Theorem | ovn0ssdmfun 41767* | If a class' operation value for two operands is not the empty set, the operands are contained in the domain of the class, and the class restricted to the operands is a function, analogous to fvfundmfvn0 6226. (Contributed by AV, 27-Jan-2020.) |
⊢ (∀𝑎 ∈ 𝐷 ∀𝑏 ∈ 𝐸 (𝑎𝐹𝑏) ≠ ∅ → ((𝐷 × 𝐸) ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ (𝐷 × 𝐸)))) | ||
Theorem | fnxpdmdm 41768 | The domain of the domain of a function over a Cartesian square. (Contributed by AV, 13-Jan-2020.) |
⊢ (𝐹 Fn (𝐴 × 𝐴) → dom dom 𝐹 = 𝐴) | ||
Theorem | cnfldsrngbas 41769 | The base set of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ⊆ ℂ → 𝑆 = (Base‘𝑅)) | ||
Theorem | cnfldsrngadd 41770 | The group addition operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → + = (+g‘𝑅)) | ||
Theorem | cnfldsrngmul 41771 | The ring multiplication operation of a subring of the field of complex numbers. (Contributed by AV, 31-Jan-2020.) |
⊢ 𝑅 = (ℂfld ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ 𝑉 → · = (.r‘𝑅)) | ||
Theorem | plusfreseq 41772 | If the empty set is not contained in the range of the group addition function of an extensible structure (not necessarily a magma), the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ (∅ ∉ ran ⨣ → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | mgmplusfreseq 41773 | If the empty set is not contained in the base set of a magma, the restriction of the addition operation to (the Cartesian square of) the base set is the functionalization of it. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ ⨣ = (+𝑓‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mgm ∧ ∅ ∉ 𝐵) → ( + ↾ (𝐵 × 𝐵)) = ⨣ ) | ||
Theorem | 0mgm 41774 | A set with an empty base set is always a magma". (Contributed by AV, 25-Feb-2020.) |
⊢ (Base‘𝑀) = ∅ ⇒ ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) | ||
Theorem | mgmpropd 41775* | If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) ⇒ ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm)) | ||
Theorem | ismgmd 41776* | Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐺)) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → + = (+g‘𝐺)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐺 ∈ Mgm) | ||
Syntax | cmgmhm 41777 | Hom-set generator class for magmas. |
class MgmHom | ||
Syntax | csubmgm 41778 | Class function taking a magma to its lattice of submagmas. |
class SubMgm | ||
Definition | df-mgmhm 41779* | A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.) |
⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))}) | ||
Definition | df-submgm 41780* | A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.) |
⊢ SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑠)𝑦) ∈ 𝑡}) | ||
Theorem | mgmhmrcl 41781 | Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm)) | ||
Theorem | submgmrcl 41782 | Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.) |
⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm) | ||
Theorem | ismgmhm 41783* | Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))) | ||
Theorem | mgmhmf 41784 | A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵⟶𝐶) | ||
Theorem | mgmhmpropd 41785* | Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐽)) & ⊢ (𝜑 → 𝐶 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐶 = (Base‘𝑀)) & ⊢ (𝜑 → 𝐵 ≠ ∅) & ⊢ (𝜑 → 𝐶 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦)) ⇒ ⊢ (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀)) | ||
Theorem | mgmhmlin 41786 | A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑆) & ⊢ + = (+g‘𝑆) & ⊢ ⨣ = (+g‘𝑇) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))) | ||
Theorem | mgmhmf1o 41787 | A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑅) & ⊢ 𝐶 = (Base‘𝑆) ⇒ ⊢ (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 MgmHom 𝑅))) | ||
Theorem | idmgmhm 41788 | The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀)) | ||
Theorem | issubmgm 41789* | Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))) | ||
Theorem | issubmgm2 41790 | Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 𝐻 ∈ Mgm))) | ||
Theorem | rabsubmgmd 41791* | Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ Mgm) & ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) & ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) & ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) & ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) ⇒ ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmss 41792 | Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 ⊆ 𝐵) | ||
Theorem | submgmid 41793 | Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀)) | ||
Theorem | submgmcl 41794 | Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.) |
⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | submgmmgm 41795 | Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm) | ||
Theorem | submgmbas 41796 | The base set of a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝑀 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻)) | ||
Theorem | subsubmgm 41797 | A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝐻 = (𝐺 ↾s 𝑆) ⇒ ⊢ (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴 ⊆ 𝑆))) | ||
Theorem | resmgmhm 41798 | Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑆 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 MgmHom 𝑇)) | ||
Theorem | resmgmhm2 41799 | One direction of resmgmhm2b 41800. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇)) | ||
Theorem | resmgmhm2b 41800 | Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.) |
⊢ 𝑈 = (𝑇 ↾s 𝑋) ⇒ ⊢ ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈))) |
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