P. 159 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15801-15900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	lgsquadlem3 15801*	Lemma for lgsquad 15802. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- ( ph -> P e. ( Prime \ { 2 } ) )   &    |- ( ph -> Q e. ( Prime \ { 2 } ) )   &    |- ( ph -> P =/= Q )   &    |- M = ( ( P - 1 ) / 2 )   &    |- N = ( ( Q - 1 ) / 2 )   &    |- S = { <. x , y >. | ( ( x e. ( 1 ... M ) /\ y e. ( 1 ... N ) ) /\ ( y x. P ) < ( x x. Q ) ) }   =>    |- ( ph -> ( ( P /L Q ) x. ( Q /L P ) ) = ( -u 1 ^ ( M x. N ) ) )
Theorem	lgsquad 15802	The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If P and Q are distinct odd primes, then the product of the Legendre symbols ( P /L Q ) and ( Q /L P ) is the parity of ( ( P - 1 ) / 2 ) x. ( ( Q - 1 ) / 2 ). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.)
|- ( ( P e. ( Prime \ { 2 } ) /\ Q e. ( Prime \ { 2 } ) /\ P =/= Q ) -> ( ( P /L Q ) x. ( Q /L P ) ) = ( -u 1 ^ ( ( ( P - 1 ) / 2 ) x. ( ( Q - 1 ) / 2 ) ) ) )
Theorem	lgsquad2lem1 15803	Lemma for lgsquad2 15805. (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 -> A e. NN )   &    |- ( ph -> B e. NN )   &    |- ( ph -> ( A x. B ) = M )   &    |- ( ph -> ( ( A /L N ) x. ( N /L A ) ) = ( -u 1 ^ ( ( ( A - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )   &    |- ( ph -> ( ( B /L N ) x. ( N /L B ) ) = ( -u 1 ^ ( ( ( B - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )   =>    |- ( ph -> ( ( M /L N ) x. ( N /L M ) ) = ( -u 1 ^ ( ( ( M - 1 ) / 2 ) x. ( ( N - 1 ) / 2 ) ) ) )
Theorem	lgsquad2lem2 15804*	Lemma for lgsquad2 15805. (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 15805	Extend lgsquad 15802 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 15806	Extend lgsquad2 15805 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 15807	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 15808*	Lemma 1 for 2lgslem1a 15810. (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 15809	Lemma 2 for 2lgslem1a 15810. (Contributed by AV, 18-Jun-2021.)
|- ( ( N e. ZZ /\ I e. ZZ ) -> ( ( |_ ` ( N / 4 ) ) < I <-> ( N / 2 ) < ( I x. 2 ) ) )
Theorem	2lgslem1a 15810*	Lemma 1 for 2lgslem1 15813. (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 15811*	Lemma 2 for 2lgslem1 15813. (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 15812	Lemma 3 for 2lgslem1 15813. (Contributed by AV, 19-Jun-2021.)
|- ( ( P e. Prime /\ -. 2 || P ) -> ( |_ ` ( P / 4 ) ) <_ ( ( P - 1 ) / 2 ) )
Theorem	2lgslem1 15813*	Lemma 1 for 2lgs 15826. (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 15814	Lemma 2 for 2lgs 15826. (Contributed by AV, 20-Jun-2021.)
|- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )   =>    |- ( ( P e. Prime /\ -. 2 || P ) -> N e. ZZ )
Theorem	2lgslem3a 15815	Lemma for 2lgslem3a1 15819. (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 15816	Lemma for 2lgslem3b1 15820. (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 15817	Lemma for 2lgslem3c1 15821. (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 15818	Lemma for 2lgslem3d1 15822. (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 15819	Lemma 1 for 2lgslem3 15823. (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 15820	Lemma 2 for 2lgslem3 15823. (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 15821	Lemma 3 for 2lgslem3 15823. (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 15822	Lemma 4 for 2lgslem3 15823. (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 15823	Lemma 3 for 2lgs 15826. (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 15824	The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.)
|- ( 2 /L 2 ) = 0
Theorem	2lgslem4 15825	Lemma 4 for 2lgs 15826: special case of 2lgs 15826 for P = 2. (Contributed by AV, 20-Jun-2021.)
|- ( ( 2 /L 2 ) = 1 <-> ( 2 mod 8 ) e. { 1 , 7 } )
Theorem	2lgs 15826	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 15739) 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 15827	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 15828	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 15829	Lemma 1 for 2lgsoddprmlem3 15833. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 1 ^ 2 ) - 1 ) / 8 ) = 0
Theorem	2lgsoddprmlem3b 15830	Lemma 2 for 2lgsoddprmlem3 15833. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 3 ^ 2 ) - 1 ) / 8 ) = 1
Theorem	2lgsoddprmlem3c 15831	Lemma 3 for 2lgsoddprmlem3 15833. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 5 ^ 2 ) - 1 ) / 8 ) = 3
Theorem	2lgsoddprmlem3d 15832	Lemma 4 for 2lgsoddprmlem3 15833. (Contributed by AV, 20-Jul-2021.)
|- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )
Theorem	2lgsoddprmlem3 15833	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 15834	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 15835	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 ) ) )
11.3.7  All primes 4n+1 are the sum of two squares
Theorem	2sqlem1 15836*	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 15837*	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 15838	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 15839	Lemma for 2sqlem5 15841. (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 15840	Lemma for 2sqlem5 15841. (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 15841	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 15842*	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 15843*	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 15844*	Lemma for 2sqlem8 15845. (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 15845*	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 15846*	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 15847*	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 15848	Extend class notation with an edge function.
class .ef
Definition	df-edgf 15849	Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.) Use its index-independent form edgfid 15850 instead. (New usage is discouraged.)
|- .ef = Slot ; 1 8
Theorem	edgfid 15850	Utility theorem: index-independent form of df-edgf 15849. (Contributed by AV, 16-Nov-2021.)
|- .ef = Slot (.ef ` ndx )
Theorem	edgfndx 15851	Index value of the df-edgf 15849 slot. (Contributed by AV, 13-Oct-2024.) (New usage is discouraged.)
|- (.ef ` ndx ) = ; 1 8
Theorem	edgfndxnn 15852	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 15853	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 15854	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 15855	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 15856	Extend class notation with the vertices of "graphs".
class Vtx
Syntax	ciedg 15857	Extend class notation with the indexed edges of "graphs".
class iEdg
Definition	df-vtx 15858	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 15859	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 15860	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 15861	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 15862	Applying the vertex function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (Vtx ` G ) e. _V )
Theorem	iedgex 15863	Applying the indexed edge function yields a set. (Contributed by Jim Kingdon, 29-Dec-2025.)
|- ( G e. V -> (iEdg ` G ) e. _V )
Theorem	1vgrex 15864	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 15865	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 15866	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 15867	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 15868	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 15869	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 15870	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 15871	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 15872	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 15873	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 15874	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 15875	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 15876	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 15877	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 15878	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 15879	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 15880	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 15881	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 15882	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 15883	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 15884	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 15885	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 15886	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 15887	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 15888	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 15889	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 15890	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 15891*	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 15892*	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 15893*	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 15894*	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 15895	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 15896	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 15897	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 15898	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 15899	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 15900	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 ) = (/) )