P. 163 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 16201-16300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lgsquad2lem2 16201*	Lemma for lgsquad2 16202. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( ph -> M e. NN )   &    |- ( ph -> -. 2 || M )   &    |- ( ph -> N e. NN )   &    |- ( ph -> -. 2 || N )   &    |- ( ph -> ( M gcd N ) = 1 )   &    |- ( ( ph /\ ( m e. ( Prime \ { 2 } ) /\ ( m gcd N ) = 1 ) ) -> ( ( m /L N ) x. ( N /L m ) ) = ( -u 1 ^ ( ( ( m - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )   &    |- ( ps <-> A. x e. ( 1 ... k ) ( ( x gcd ( 2 x. N ) ) = 1 -> ( ( x /L N ) x. ( N /L x ) ) = ( -u 1 ^ ( ( ( x - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) ) )   =>    |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )
Theorem	lgsquad2 16202	Extend lgsquad 16199 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( ph -> M e. NN )   &    |- ( ph -> -. 2 || M )   &    |- ( ph -> N e. NN )   &    |- ( ph -> -. 2 || N )   &    |- ( ph -> ( M gcd N ) = 1 )   =>    |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )
Theorem	lgsquad3 16203	Extend lgsquad2 16202 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( ( ( M e. NN /\ -. 2 || M ) /\ ( N e. NN /\ -. 2 || N ) ) -> ( M /L N ) = ( ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) x. ( N /L M ) ) )
Theorem	m1lgs 16204	The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime P iff P == 1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( P e. ( Prime \ { 2 } ) -> ( ( -u 1 /L P ) = 1 <-> ( P mod 4 ) = 1 ) )
Theorem	2lgslem1a1 16205*	Lemma 1 for 2lgslem1a 16207. (Contributed by AV, 16-Jun-2021.)
|- ( ( P e. NN /\ -. 2 || P ) -> A. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( i x. 2 ) = ( ( i x. 2 ) mod P ) )
Theorem	2lgslem1a2 16206	Lemma 2 for 2lgslem1a 16207. (Contributed by AV, 18-Jun-2021.)
|- ( ( N e. ZZ /\ I e. ZZ ) -> ( ( |_ ` ( N / 4 ) ) < I <-> ( N / 2 ) < ( I x. 2 ) ) )
Theorem	2lgslem1a 16207*	Lemma 1 for 2lgslem1 16210. (Contributed by AV, 18-Jun-2021.)
|- ( ( P e. Prime /\ -. 2 || P ) -> { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } = { x e. ZZ | E. i e. ( ( ( |_ ` ( P / 4 ) ) + 1 ) ... ( ( P - 1 ) / 2 ) ) x = ( i x. 2 ) } )
Theorem	2lgslem1b 16208*	Lemma 2 for 2lgslem1 16210. (Contributed by AV, 18-Jun-2021.)
|- I = ( A ... B )   &    |- F = ( j e. I |-> ( j x. 2 ) )   =>    |- F : I -1-1-onto-> { x e. ZZ | E. i e. I x = ( i x. 2 ) }
Theorem	2lgslem1c 16209	Lemma 3 for 2lgslem1 16210. (Contributed by AV, 19-Jun-2021.)
|- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
Theorem	2lgslem1 16210*	Lemma 1 for 2lgs 16223. (Contributed by AV, 19-Jun-2021.)
|- ( ( P e. Prime /\ -. 2 || P ) -> (♯ ` { x e. ZZ | E. i e. ( 1 ... ( ( P - 1 ) / 2 ) ) ( x = ( i x. 2 ) /\ ( P / 2 ) < ( x mod P ) ) } ) = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) ) )
Theorem	2lgslem2 16211	Lemma 2 for 2lgs 16223. (Contributed by AV, 20-Jun-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. Prime /\ -. 2 || P ) -> N e. ZZ )
Theorem	2lgslem3a 16212	Lemma for 2lgslem3a1 16216. (Contributed by AV, 14-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 1 ) ) -> N = ( 2 x. K ) )
Theorem	2lgslem3b 16213	Lemma for 2lgslem3b1 16217. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 3 ) ) -> N = ( ( 2 x. K ) + 1 ) )
Theorem	2lgslem3c 16214	Lemma for 2lgslem3c1 16218. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 5 ) ) -> N = ( ( 2 x. K ) + 1 ) )
Theorem	2lgslem3d 16215	Lemma for 2lgslem3d1 16219. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( K e. NN0 /\ P = ( ( 8 x. K ) + 7 ) ) -> N = ( ( 2 x. K ) + 2 ) )
Theorem	2lgslem3a1 16216	Lemma 1 for 2lgslem3 16220. (Contributed by AV, 15-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. NN /\ ( P mod 8 ) = 1 ) -> ( N mod 2 ) = 0 )
Theorem	2lgslem3b1 16217	Lemma 2 for 2lgslem3 16220. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. NN /\ ( P mod 8 ) = 3 ) -> ( N mod 2 ) = 1 )
Theorem	2lgslem3c1 16218	Lemma 3 for 2lgslem3 16220. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. NN /\ ( P mod 8 ) = 5 ) -> ( N mod 2 ) = 1 )
Theorem	2lgslem3d1 16219	Lemma 4 for 2lgslem3 16220. (Contributed by AV, 15-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )
Theorem	2lgslem3 16220	Lemma 3 for 2lgs 16223. (Contributed by AV, 16-Jul-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. NN /\ -. 2 || P ) -> ( N mod 2 ) = if ( ( P mod 8 ) e. { 1 , 7 } , 0 , 1 ) )
Theorem	2lgs2 16221	The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.)
|- ( 2 /L 2 ) = 0
Theorem	2lgslem4 16222	Lemma 4 for 2lgs 16223: special case of 2lgs 16223 for P = 2. (Contributed by AV, 20-Jun-2021.)
|- ( ( 2 /L 2 ) = 1 <-> ( 2 mod 8 ) e. { 1 , 7 } )
Theorem	2lgs 16223	The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime P iff P == pm 1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of ( N /L 2 ) (lgs2 16136) to some degree, by demanding that reciprocity extend to the case Q = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.)
|- ( P e. Prime -> ( ( 2 /L P ) = 1 <-> ( P mod 8 ) e. { 1 , 7 } ) )
Theorem	2lgsoddprmlem1 16224	Lemma 1 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)
|- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )
Theorem	2lgsoddprmlem2 16225	Lemma 2 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ R = ( N mod 8 ) ) -> ( 2 || ( ( ( N ^ 2 ) - 1 ) / 8 ) <-> 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) ) )
Theorem	2lgsoddprmlem3a 16226	Lemma 1 for 2lgsoddprmlem3 16230. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 1 ^ 2 ) - 1 ) / 8 ) = 0
Theorem	2lgsoddprmlem3b 16227	Lemma 2 for 2lgsoddprmlem3 16230. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 3 ^ 2 ) - 1 ) / 8 ) = 1
Theorem	2lgsoddprmlem3c 16228	Lemma 3 for 2lgsoddprmlem3 16230. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 5 ^ 2 ) - 1 ) / 8 ) = 3
Theorem	2lgsoddprmlem3d 16229	Lemma 4 for 2lgsoddprmlem3 16230. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )
Theorem	2lgsoddprmlem3 16230	Lemma 3 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N /\ R = ( N mod 8 ) ) -> ( 2 || ( ( ( R ^ 2 ) - 1 ) / 8 ) <-> R e. { 1 , 7 } ) )
Theorem	2lgsoddprmlem4 16231	Lemma 4 for 2lgsoddprm . (Contributed by AV, 20-Jul-2021.)
|- ( ( N e. ZZ /\ -. 2 || N ) -> ( 2 || ( ( ( N ^ 2 ) - 1 ) / 8 ) <-> ( N mod 8 ) e. { 1 , 7 } ) )
Theorem	2lgsoddprm 16232	The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ( ( 2 /L P ) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.)
|- ( P e. ( Prime \ { 2 } ) -> ( 2 /L P ) = ( -u 1 ^ ( ( ( P ^ 2 ) - 1 ) / 8 ) ) )
11.4.7  All primes 4n+1 are the sum of two squares
Theorem	2sqlem1 16233*	Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( A e. S <-> E. x e. ZZ[_i] A = ( ( abs ` x ) ^ 2 ) )
Theorem	2sqlem2 16234*	Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( A e. S <-> E. x e. ZZ E. y e. ZZ A = ( ( x ^ 2 ) + ( y ^ 2 ) ) )
Theorem	mul2sq 16235	Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   =>    |- ( ( A e. S /\ B e. S ) -> ( A x. B ) e. S )
Theorem	2sqlem3 16236	Lemma for 2sqlem5 16238. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )   &    |- ( ph -> P || ( ( C x. B ) + ( A x. D ) ) )   =>    |- ( ph -> N e. S )
Theorem	2sqlem4 16237	Lemma for 2sqlem5 16238. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> C e. ZZ )   &    |- ( ph -> D e. ZZ )   &    |- ( ph -> ( N x. P ) = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- ( ph -> P = ( ( C ^ 2 ) + ( D ^ 2 ) ) )   =>    |- ( ph -> N e. S )
Theorem	2sqlem5 16238	Lemma for 2sq . If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> N e. NN )   &    |- ( ph -> P e. Prime )   &    |- ( ph -> ( N x. P ) e. S )   &    |- ( ph -> P e. S )   =>    |- ( ph -> N e. S )
Theorem	2sqlem6 16239*	Lemma for 2sq . If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- ( ph -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> A. p e. Prime ( p || B -> p e. S ) )   &    |- ( ph -> ( A x. B ) e. S )   =>    |- ( ph -> A e. S )
Theorem	2sqlem7 16240*	Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   =>    |- Y C_ ( S i^i NN )
Theorem	2sqlem8a 16241*	Lemma for 2sqlem8 16242. (Contributed by Mario Carneiro, 4-Jun-2016.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> ( A gcd B ) = 1 )   &    |- ( ph -> N = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- D = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   =>    |- ( ph -> ( C gcd D ) e. NN )
Theorem	2sqlem8 16242*	Lemma for 2sq . (Contributed by Mario Carneiro, 20-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> N e. NN )   &    |- ( ph -> M e. ( ZZ>= ` 2 ) )   &    |- ( ph -> A e. ZZ )   &    |- ( ph -> B e. ZZ )   &    |- ( ph -> ( A gcd B ) = 1 )   &    |- ( ph -> N = ( ( A ^ 2 ) + ( B ^ 2 ) ) )   &    |- C = ( ( ( A + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- D = ( ( ( B + ( M / 2 ) ) mod M ) - ( M / 2 ) )   &    |- E = ( C / ( C gcd D ) )   &    |- F = ( D / ( C gcd D ) )   =>    |- ( ph -> M e. S )
Theorem	2sqlem9 16243*	Lemma for 2sq . (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   &    |- ( ph -> A. b e. ( 1 ... ( M - 1 ) ) A. a e. Y ( b || a -> b e. S ) )   &    |- ( ph -> M || N )   &    |- ( ph -> M e. NN )   &    |- ( ph -> N e. Y )   =>    |- ( ph -> M e. S )
Theorem	2sqlem10 16244*	Lemma for 2sq . Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- S = ran ( w e. ZZ[_i] |-> ( ( abs ` w ) ^ 2 ) )   &    |- Y = { z | E. x e. ZZ E. y e. ZZ ( ( x gcd y ) = 1 /\ z = ( ( x ^ 2 ) + ( y ^ 2 ) ) ) }   =>    |- ( ( A e. Y /\ B e. NN /\ B || A ) -> B e. S )
PART 12  GRAPH THEORY
12.1  Vertices and edges
12.1.1  The edge function extractor for extensible structures
Syntax	cedgf 16245	Extend class notation with an edge function.
class .ef
Definition	df-edgf 16246	Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.) Use its index-independent form edgfid 16247 instead. (New usage is discouraged.)
|- .ef = Slot ; 1 8
Theorem	edgfid 16247	Utility theorem: index-independent form of df-edgf 16246. (Contributed by AV, 16-Nov-2021.)
|- .ef = Slot (.ef ` ndx )
Theorem	edgfndx 16248	Index value of the df-edgf 16246 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
|- (.ef ` ndx ) = ; 1 8
Theorem	edgfndxnn 16249	The index value of the edge function extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 13-Oct-2024.)
|- (.ef ` ndx ) e. NN
Theorem	edgfndxid 16250	The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 28-Oct-2024.)
|- ( G e. V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )
Theorem	basendxltedgfndx 16251	The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.) (Proof shortened by AV, 30-Oct-2024.)
|- ( Base ` ndx ) < (.ef ` ndx )
Theorem	basendxnedgfndx 16252	The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)
|- ( Base ` ndx ) =/= (.ef ` ndx )
12.1.2  Vertices and indexed edges
12.1.2.1  Definitions and basic properties
Syntax	cvtx 16253	Extend class notation with the vertices of "graphs".
class Vtx
Syntax	ciedg 16254	Extend class notation with the indexed edges of "graphs".
class iEdg
Definition	df-vtx 16255	Define the function mapping a graph to the set of its vertices. This definition is very general: It defines the set of vertices for any ordered pair as its first component, and for any other class as its "base set". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure representing a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 20-Sep-2020.)
|- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
Definition	df-iedg 16256	Define the function mapping a graph to its indexed edges. This definition is very general: It defines the indexed edges for any ordered pair as its second component, and for any other class as its "edge function". It is meaningful, however, only if the ordered pair represents a graph resp. the class is an extensible structure (containing a slot for "edge functions") representing a graph. (Contributed by AV, 20-Sep-2020.)
|- iEdg = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 2nd ` g ) , (.ef ` g ) ) )
Theorem	vtxvalg 16257	The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
|- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )
Theorem	iedgvalg 16258	The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
|- ( G e. V -> (iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) ) )
Theorem	vtxex 16259	Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (Vtx ` G ) e. _V )
Theorem	iedgex 16260	Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (iEdg ` G ) e. _V )
Theorem	1vgrex 16261	A graph with at least one vertex is a set. (Contributed by AV, 2-Mar-2021.)
|- V = (Vtx ` G )   =>    |- ( N e. V -> G e. _V )
12.1.2.2  The vertices and edges of a graph represented as ordered pair
Theorem	opvtxval 16262	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
|- ( G e. ( _V X. _V ) -> (Vtx ` G ) = ( 1st ` G ) )
Theorem	opvtxfv 16263	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
|- ( ( V e. X /\ E e. Y ) -> (Vtx ` <. V , E >. ) = V )
Theorem	opvtxov 16264	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
|- ( ( V e. X /\ E e. Y ) -> ( VVtx E ) = V )
Theorem	opiedgval 16265	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020.)
|- ( G e. ( _V X. _V ) -> (iEdg ` G ) = ( 2nd ` G ) )
Theorem	opiedgfv 16266	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)
|- ( ( V e. X /\ E e. Y ) -> (iEdg ` <. V , E >. ) = E )
Theorem	opiedgov 16267	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
|- ( ( V e. X /\ E e. Y ) -> ( ViEdg E ) = E )
Theorem	opvtxfvi 16268	The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
|- V e. _V   &    |- E e. _V   =>    |- (Vtx ` <. V , E >. ) = V
Theorem	opiedgfvi 16269	The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)
|- V e. _V   &    |- E e. _V   =>    |- (iEdg ` <. V , E >. ) = E
12.1.2.3  The vertices and edges of a graph represented as extensible structure
Theorem	funvtxdm2domval 16270	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )
Theorem	funiedgdm2domval 16271	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2o ~<_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
Theorem	funvtxdm2vald 16272	The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (Vtx ` G ) = ( Base ` G ) )
Theorem	funiedgdm2vald 16273	The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 12-Dec-2025.)
|- A e. _V   &    |- B e. _V   &    |- ( ph -> G e. X )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> A =/= B )   &    |- ( ph -> { A , B } C_ dom G )   =>    |- ( ph -> (iEdg ` G ) = (.ef ` G ) )
Theorem	funvtxval0d 16274	The set of vertices of an extensible structure with a base set and (at least) another slot. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
|- S e. _V   &    |- ( ph -> G e. V )   &    |- ( ph -> Fun ( G \ { (/) } ) )   &    |- ( ph -> S =/= ( Base ` ndx ) )   &    |- ( ph -> { ( Base ` ndx ) , S } C_ dom G )   =>    |- ( ph -> (Vtx ` G ) = ( Base ` G ) )
Theorem	basvtxval2dom 16275	The set of vertices of a graph represented as an extensible structure with the set of vertices as base set. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> 2o ~<_ dom G )   &    |- ( ph -> V e. Y )   &    |- ( ph -> <. ( Base ` ndx ) , V >. e. G )   =>    |- ( ph -> (Vtx ` G ) = V )
Theorem	edgfiedgval2dom 16276	The set of indexed edges of a graph represented as an extensible structure with the indexed edges in the slot for edge functions. (Contributed by AV, 14-Oct-2020.) (Revised by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> 2o ~<_ dom G )   &    |- ( ph -> E e. Y )   &    |- ( ph -> <. (.ef ` ndx ) , E >. e. G )   =>    |- ( ph -> (iEdg ` G ) = E )
Theorem	funvtxvalg 16277	The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ { ( Base ` ndx ) , (.ef ` ndx ) } C_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )
Theorem	funiedgvalg 16278	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 21-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)
|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ { ( Base ` ndx ) , (.ef ` ndx ) } C_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )
Theorem	struct2slots2dom 16279	There are at least two elements in an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Revised by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> 2o ~<_ dom G )
Theorem	structvtxval 16280	The set of vertices of an extensible structure with a base set and another slot. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> (Vtx ` G ) = V )
Theorem	structiedg0val 16281	The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- S e. NN   &    |- ( Base ` ndx ) < S   &    |- G = { <. ( Base ` ndx ) , V >. , <. S , E >. }   =>    |- ( ( V e. X /\ E e. Y /\ S =/= (.ef ` ndx ) ) -> (iEdg ` G ) = (/) )
Theorem	structgr2slots2dom 16282	There are at least two elements in a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> 2o ~<_ dom G )
Theorem	structgrssvtx 16283	The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> (Vtx ` G ) = V )
Theorem	structgrssiedg 16284	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 14-Oct-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- ( ph -> G Struct X )   &    |- ( ph -> V e. Y )   &    |- ( ph -> E e. Z )   &    |- ( ph -> { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. } C_ G )   =>    |- ( ph -> (iEdg ` G ) = E )
Theorem	struct2grstrg 16285	A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020.)
|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> G Struct <. ( Base ` ndx ) , (.ef ` ndx ) >. )
Theorem	struct2grvtx 16286	The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.)
|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> (Vtx ` G ) = V )
Theorem	struct2griedg 16287	The set of indexed edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.)
|- G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }   =>    |- ( ( V e. X /\ E e. Y ) -> (iEdg ` G ) = E )
Theorem	gropd 16288*	If any representation of a graph with vertices V and edges E has a certain property ps, then the ordered pair <. V , E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices V and edges E) has this property. (Contributed by AV, 11-Oct-2020.)
|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) )   &    |- ( ph -> V e. U )   &    |- ( ph -> E e. W )   =>    |- ( ph -> [. <. V , E >. / g ]. ps )
Theorem	grstructd2dom 16289*	If any representation of a graph with vertices V and edges E has a certain property ps, then any structure with base set V and value E in the slot for edge functions (which is such a representation of a graph with vertices V and edges E) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) )   &    |- ( ph -> V e. U )   &    |- ( ph -> E e. W )   &    |- ( ph -> S e. X )   &    |- ( ph -> Fun ( S \ { (/) } ) )   &    |- ( ph -> 2o ~<_ dom S )   &    |- ( ph -> ( Base ` S ) = V )   &    |- ( ph -> (.ef ` S ) = E )   =>    |- ( ph -> [. S / g ]. ps )
Theorem	gropeld 16290*	If any representation of a graph with vertices V and edges E is an element of an arbitrary class C, then the ordered pair <. V , E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices V and edges E) is an element of this class C. (Contributed by AV, 11-Oct-2020.)
|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )   &    |- ( ph -> V e. U )   &    |- ( ph -> E e. W )   =>    |- ( ph -> <. V , E >. e. C )
Theorem	grstructeld2dom 16291*	If any representation of a graph with vertices V and edges E is an element of an arbitrary class C, then any structure with base set V and value E in the slot for edge functions (which is such a representation of a graph with vertices V and edges E) is an element of this class C. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)
|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )   &    |- ( ph -> V e. U )   &    |- ( ph -> E e. W )   &    |- ( ph -> S e. X )   &    |- ( ph -> Fun ( S \ { (/) } ) )   &    |- ( ph -> 2o ~<_ dom S )   &    |- ( ph -> ( Base ` S ) = V )   &    |- ( ph -> (.ef ` S ) = E )   =>    |- ( ph -> S e. C )
Theorem	setsvtx 16292	The vertices of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 18-Jan-2020.) (Revised by AV, 16-Nov-2021.)
|- I = (.ef ` ndx )   &    |- ( ph -> G Struct X )   &    |- ( ph -> ( Base ` ndx ) e. dom G )   &    |- ( ph -> E e. W )   =>    |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = ( Base ` G ) )
Theorem	setsiedg 16293	The (indexed) edges of a structure with a base set and an inserted resp. replaced slot for the edge function. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)
|- I = (.ef ` ndx )   &    |- ( ph -> G Struct X )   &    |- ( ph -> ( Base ` ndx ) e. dom G )   &    |- ( ph -> E e. W )   =>    |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) = E )
12.1.2.4  Degenerated cases of representations of graphs
Theorem	vtxval0 16294	Degenerated case 1 for vertices: The set of vertices of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
|- (Vtx ` (/) ) = (/)
Theorem	iedgval0 16295	Degenerated case 1 for edges: The set of indexed edges of the empty set is the empty set. (Contributed by AV, 24-Sep-2020.)
|- (iEdg ` (/) ) = (/)
Theorem	vtxvalprc 16296	Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
|- ( C e/ _V -> (Vtx ` C ) = (/) )
Theorem	iedgvalprc 16297	Degenerated case 4 for edges: The set of indexed edges of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
|- ( C e/ _V -> (iEdg ` C ) = (/) )
12.1.3  Edges as range of the edge function
Syntax	cedg 16298	Extend class notation with the set of edges (of an undirected simple (hyper-/pseudo-)graph).
class Edg
Definition	df-edg 16299	Define the class of edges of a graph, see also definition "E = E(G)" in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges of a class as the range of its edge function (which does not even need to be a function). Therefore, this definition could also be used for hypergraphs, pseudographs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. In some cases, this is no problem, so theorems with Edg are meaningful nevertheless. Usually, however, this definition is used only for undirected simple (hyper-/pseudo-)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
|- Edg = ( g e. _V |-> ran (iEdg ` g ) )
Theorem	edgvalg 16300	The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.)
|- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )