P. 259 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

axsegconlem5 25801*

Lemma for axsegcon 25807. Show that the distance between two points is nonnegative. (Contributed by Scott Fenton, 17-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   =>    |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> 0 <_ ( sqr ` S ) )

Theorem

axsegconlem6 25802*

Lemma for axsegcon 25807. Show that the distance between two distinct points is positive. (Contributed by Scott Fenton, 17-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   =>    |- ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) -> 0 < ( sqr ` S ) )

Theorem

axsegconlem7 25803*

Lemma for axsegcon 25807. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   &    |- T = sum_ p e. ( 1 ... N ) ( ( ( C ` p ) - ( D ` p ) ) ^ 2 )   =>    |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> ( ( sqr ` S ) / ( ( sqr ` S ) + ( sqr ` T ) ) ) e. ( 0 [,] 1 ) )

Theorem

axsegconlem8 25804*

Lemma for axsegcon 25807. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   &    |- T = sum_ p e. ( 1 ... N ) ( ( ( C ` p ) - ( D ` p ) ) ^ 2 )   &    |- F = ( k e. ( 1 ... N ) |-> ( ( ( ( ( sqr ` S ) + ( sqr ` T ) ) x. ( B ` k ) ) - ( ( sqr ` T ) x. ( A ` k ) ) ) / ( sqr ` S ) ) )   =>    |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> F e. ( EE ` N ) )

Theorem

axsegconlem9 25805*

Lemma for axsegcon 25807. Show that B F is congruent to C D. (Contributed by Scott Fenton, 19-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   &    |- T = sum_ p e. ( 1 ... N ) ( ( ( C ` p ) - ( D ` p ) ) ^ 2 )   &    |- F = ( k e. ( 1 ... N ) |-> ( ( ( ( ( sqr ` S ) + ( sqr ` T ) ) x. ( B ` k ) ) - ( ( sqr ` T ) x. ( A ` k ) ) ) / ( sqr ` S ) ) )   =>    |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> sum_ i e. ( 1 ... N ) ( ( ( B ` i ) - ( F ` i ) ) ^ 2 ) = sum_ i e. ( 1 ... N ) ( ( ( C ` i ) - ( D ` i ) ) ^ 2 ) )

Theorem

axsegconlem10 25806*

Lemma for axsegcon 25807. Show that the scaling constant from axsegconlem7 25803 produces the betweenness condition for A, B and F. (Contributed by Scott Fenton, 21-Sep-2013.)

|- S = sum_ p e. ( 1 ... N ) ( ( ( A ` p ) - ( B ` p ) ) ^ 2 )   &    |- T = sum_ p e. ( 1 ... N ) ( ( ( C ` p ) - ( D ` p ) ) ^ 2 )   &    |- F = ( k e. ( 1 ... N ) |-> ( ( ( ( ( sqr ` S ) + ( sqr ` T ) ) x. ( B ` k ) ) - ( ( sqr ` T ) x. ( A ` k ) ) ) / ( sqr ` S ) ) )   =>    |- ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ A =/= B ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - ( ( sqr ` S ) / ( ( sqr ` S ) + ( sqr ` T ) ) ) ) x. ( A ` i ) ) + ( ( ( sqr ` S ) / ( ( sqr ` S ) + ( sqr ` T ) ) ) x. ( F ` i ) ) ) )

Theorem

axsegcon 25807*

Any segment A B can be extended to a point x such that B x is congruent to C D. Axiom A4 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 4-Jun-2013.)

|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) -> E. x e. ( EE ` N ) ( B Btwn <. A , x >. /\ <. B , x >.Cgr <. C , D >. ) )

Theorem

ax5seglem1 25808*

Lemma for ax5seg 25818. Rexpress a one congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( B ` j ) ) ^ 2 ) = ( ( T ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )

Theorem

ax5seglem2 25809*

Lemma for ax5seg 25818. Rexpress another congruence sum given betweenness. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( C ` j ) ) ^ 2 ) = ( ( ( 1 - T ) ^ 2 ) x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) )

Theorem

ax5seglem3a 25810

Lemma for ax5seg 25818. (Contributed by Scott Fenton, 7-May-2015.)

|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ j e. ( 1 ... N ) ) -> ( ( ( A ` j ) - ( C ` j ) ) e. RR /\ ( ( D ` j ) - ( F ` j ) ) e. RR ) )

Theorem

ax5seglem3 25811*

Lemma for ax5seg 25818. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) /\ ( ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) = sum_ j e. ( 1 ... N ) ( ( ( D ` j ) - ( F ` j ) ) ^ 2 ) )

Theorem

ax5seglem4 25812*

Lemma for ax5seg 25818. Given two distinct points, the scaling constant in a betweenness statement is nonzero. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A =/= B ) -> T =/= 0 )

Theorem

ax5seglem5 25813*

Lemma for ax5seg 25818. If B is between A and C, and A is distinct from B, then A is distinct from C. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) ) /\ ( A =/= B /\ T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) =/= 0 )

Theorem

ax5seglem6 25814*

Lemma for ax5seg 25818. Given two line segments that are divided into pieces, if the pieces are congruent, then the scaling constant is the same. (Contributed by Scott Fenton, 12-Jun-2013.)

|- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) /\ F e. ( EE ` N ) ) ) ) /\ ( A =/= B /\ ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) /\ ( A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) /\ A. i e. ( 1 ... N ) ( E ` i ) = ( ( ( 1 - S ) x. ( D ` i ) ) + ( S x. ( F ` i ) ) ) ) ) /\ ( <. A , B >.Cgr <. D , E >. /\ <. B , C >.Cgr <. E , F >. ) ) -> T = S )

Theorem

ax5seglem7 25815

Lemma for ax5seg 25818. An algebraic calculation needed further down the line. (Contributed by Scott Fenton, 12-Jun-2013.)

|- A e. CC   &    |- T e. CC   &    |- C e. CC   &    |- D e. CC   =>    |- ( T x. ( ( C - D ) ^ 2 ) ) = ( ( ( ( ( ( 1 - T ) x. A ) + ( T x. C ) ) - D ) ^ 2 ) + ( ( 1 - T ) x. ( ( T x. ( ( A - C ) ^ 2 ) ) - ( ( A - D ) ^ 2 ) ) ) )

Theorem

ax5seglem8 25816

Lemma for ax5seg 25818. Use the weak deduction theorem to eliminate the hypotheses from ax5seglem7 25815. (Contributed by Scott Fenton, 11-Jun-2013.)

|- ( ( ( A e. CC /\ T e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( T x. ( ( C - D ) ^ 2 ) ) = ( ( ( ( ( ( 1 - T ) x. A ) + ( T x. C ) ) - D ) ^ 2 ) + ( ( 1 - T ) x. ( ( T x. ( ( A - C ) ^ 2 ) ) - ( ( A - D ) ^ 2 ) ) ) ) )

Theorem

ax5seglem9 25817*

Lemma for ax5seg 25818. Take the calculation in ax5seglem8 25816 and turn it into a series of measurements. (Contributed by Scott Fenton, 12-Jun-2013.) (Revised by Mario Carneiro, 22-May-2014.)

|- ( ( ( N e. NN /\ ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) ) ) ) /\ ( T e. ( 0 [,] 1 ) /\ A. i e. ( 1 ... N ) ( B ` i ) = ( ( ( 1 - T ) x. ( A ` i ) ) + ( T x. ( C ` i ) ) ) ) ) -> ( T x. sum_ j e. ( 1 ... N ) ( ( ( C ` j ) - ( D ` j ) ) ^ 2 ) ) = ( sum_ j e. ( 1 ... N ) ( ( ( B ` j ) - ( D ` j ) ) ^ 2 ) + ( ( 1 - T ) x. ( ( T x. sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( C ` j ) ) ^ 2 ) ) - sum_ j e. ( 1 ... N ) ( ( ( A ` j ) - ( D ` j ) ) ^ 2 ) ) ) ) )

Theorem

ax5seg 25818

The five segment axiom. Take two triangles A D C and E H G, a point B on A C, and a point F on E G. If all corresponding line segments except for C D and G H are congruent, then so are C D and G H. Axiom A5 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 12-Jun-2013.)

|- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ D e. ( EE ` N ) /\ E e. ( EE ` N ) ) /\ ( F e. ( EE ` N ) /\ G e. ( EE ` N ) /\ H e. ( EE ` N ) ) ) -> ( ( ( A =/= B /\ B Btwn <. A , C >. /\ F Btwn <. E , G >. ) /\ ( <. A , B >.Cgr <. E , F >. /\ <. B , C >.Cgr <. F , G >. ) /\ ( <. A , D >.Cgr <. E , H >. /\ <. B , D >.Cgr <. F , H >. ) ) -> <. C , D >.Cgr <. G , H >. ) )

Theorem

axbtwnid 25819

Points are indivisible. That is, if A lies between B and B, then A = B. Axiom A6 of [Schwabhauser] p. 11. (Contributed by Scott Fenton, 3-Jun-2013.)

|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( A Btwn <. B , B >. -> A = B ) )

Theorem

axpaschlem 25820*

Lemma for axpasch 25821. Set up coefficents used in the proof. (Contributed by Scott Fenton, 5-Jun-2013.)

|- ( ( T e. ( 0 [,] 1 ) /\ S e. ( 0 [,] 1 ) ) -> E. r e. ( 0 [,] 1 ) E. p e. ( 0 [,] 1 ) ( p = ( ( 1 - r ) x. ( 1 - T ) ) /\ r = ( ( 1 - p ) x. ( 1 - S ) ) /\ ( ( 1 - r ) x. T ) = ( ( 1 - p ) x. S ) ) )

Theorem

axpasch 25821*

The inner Pasch axiom. Take a triangle A C E, a point D on A C, and a point B extending C E. Then A E and D B intersect at some point x. Axiom A7 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 3-Jun-2013.)

|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ E e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , C >. /\ E Btwn <. B , C >. ) -> E. x e. ( EE ` N ) ( x Btwn <. D , B >. /\ x Btwn <. E , A >. ) ) )

Theorem

axlowdimlem1 25822

Lemma for axlowdim 25841. Establish a particular constant function as a function. (Contributed by Scott Fenton, 29-Jun-2013.)

|- ( ( 3 ... N ) X. { 0 } ) : ( 3 ... N ) --> RR

Theorem

axlowdimlem2 25823

Lemma for axlowdim 25841. Show that two sets are disjoint. (Contributed by Scott Fenton, 29-Jun-2013.)

|- ( ( 1 ... 2 ) i^i ( 3 ... N ) ) = (/)

Theorem

axlowdimlem3 25824

Lemma for axlowdim 25841. Set up a union property for an interval of integers. (Contributed by Scott Fenton, 29-Jun-2013.)

|- ( N e. ( ZZ>= ` 2 ) -> ( 1 ... N ) = ( ( 1 ... 2 ) u. ( 3 ... N ) ) )

Theorem

axlowdimlem4 25825

Lemma for axlowdim 25841. Set up a particular constant function. (Contributed by Scott Fenton, 17-Apr-2013.)

|- A e. RR   &    |- B e. RR   =>    |- { <. 1 , A >. , <. 2 , B >. } : ( 1 ... 2 ) --> RR

Theorem

axlowdimlem5 25826

Lemma for axlowdim 25841. Show that a particular union is a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

|- A e. RR   &    |- B e. RR   =>    |- ( N e. ( ZZ>= ` 2 ) -> ( { <. 1 , A >. , <. 2 , B >. } u. ( ( 3 ... N ) X. { 0 } ) ) e. ( EE ` N ) )

Theorem

axlowdimlem6 25827

Lemma for axlowdim 25841. Show that three points are non-colinear. (Contributed by Scott Fenton, 29-Jun-2013.)

|- A = ( { <. 1 , 0 >. , <. 2 , 0 >. } u. ( ( 3 ... N ) X. { 0 } ) )   &    |- B = ( { <. 1 , 1 >. , <. 2 , 0 >. } u. ( ( 3 ... N ) X. { 0 } ) )   &    |- C = ( { <. 1 , 0 >. , <. 2 , 1 >. } u. ( ( 3 ... N ) X. { 0 } ) )   =>    |- ( N e. ( ZZ>= ` 2 ) -> -. ( A Btwn <. B , C >. \/ B Btwn <. C , A >. \/ C Btwn <. A , B >. ) )

Theorem

axlowdimlem7 25828

Lemma for axlowdim 25841. Set up a point in Euclidean space. (Contributed by Scott Fenton, 29-Jun-2013.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   =>    |- ( N e. ( ZZ>= ` 3 ) -> P e. ( EE ` N ) )

Theorem

axlowdimlem8 25829

Lemma for axlowdim 25841. Calculate the value of P at three. (Contributed by Scott Fenton, 21-Apr-2013.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   =>    |- ( P ` 3 ) = -u 1

Theorem

axlowdimlem9 25830

Lemma for axlowdim 25841. Calculate the value of P away from three. (Contributed by Scott Fenton, 21-Apr-2013.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   =>    |- ( ( K e. ( 1 ... N ) /\ K =/= 3 ) -> ( P ` K ) = 0 )

Theorem

axlowdimlem10 25831

Lemma for axlowdim 25841. Set up a family of points in Euclidean space. (Contributed by Scott Fenton, 21-Apr-2013.)

|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   =>    |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> Q e. ( EE ` N ) )

Theorem

axlowdimlem11 25832

Lemma for axlowdim 25841. Calculate the value of Q at its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   =>    |- ( Q ` ( I + 1 ) ) = 1

Theorem

axlowdimlem12 25833

Lemma for axlowdim 25841. Calculate the value of Q away from its distinguished point. (Contributed by Scott Fenton, 21-Apr-2013.)

|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   =>    |- ( ( K e. ( 1 ... N ) /\ K =/= ( I + 1 ) ) -> ( Q ` K ) = 0 )

Theorem

axlowdimlem13 25834

Lemma for axlowdim 25841. Establish that P and Q are different points. (Contributed by Scott Fenton, 21-Apr-2013.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   &    |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   =>    |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) ) -> P =/= Q )

Theorem

axlowdimlem14 25835

Lemma for axlowdim 25841. Take two possible Q from axlowdimlem10 25831. They are the same iff their distinguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)

|- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   &    |- R = ( { <. ( J + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( J + 1 ) } ) X. { 0 } ) )   =>    |- ( ( N e. NN /\ I e. ( 1 ... ( N - 1 ) ) /\ J e. ( 1 ... ( N - 1 ) ) ) -> ( Q = R -> I = J ) )

Theorem

axlowdimlem15 25836*

Lemma for axlowdim 25841. Set up a one-to-one function of points. (Contributed by Scott Fenton, 21-Apr-2013.)

|- F = ( i e. ( 1 ... ( N - 1 ) ) |-> if ( i = 1 , ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) ) , ( { <. ( i + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( i + 1 ) } ) X. { 0 } ) ) ) )   =>    |- ( N e. ( ZZ>= ` 3 ) -> F : ( 1 ... ( N - 1 ) ) -1-1-> ( EE ` N ) )

Theorem

axlowdimlem16 25837*

Lemma for axlowdim 25841. Set up a summation that will help establish distance. (Contributed by Scott Fenton, 21-Apr-2013.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   &    |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   =>    |- ( ( N e. ( ZZ>= ` 3 ) /\ I e. ( 2 ... ( N - 1 ) ) ) -> sum_ i e. ( 3 ... N ) ( ( P ` i ) ^ 2 ) = sum_ i e. ( 3 ... N ) ( ( Q ` i ) ^ 2 ) )

Theorem

axlowdimlem17 25838

Lemma for axlowdim 25841. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)

|- P = ( { <. 3 , -u 1 >. } u. ( ( ( 1 ... N ) \ { 3 } ) X. { 0 } ) )   &    |- Q = ( { <. ( I + 1 ) , 1 >. } u. ( ( ( 1 ... N ) \ { ( I + 1 ) } ) X. { 0 } ) )   &    |- A = ( { <. 1 , X >. , <. 2 , Y >. } u. ( ( 3 ... N ) X. { 0 } ) )   &    |- X e. RR   &    |- Y e. RR   =>    |- ( ( N e. ( ZZ>= ` 3 ) /\ I e. ( 2 ... ( N - 1 ) ) ) -> <. P , A >.Cgr <. Q , A >. )

Theorem

axlowdim1 25839*

The lower dimension axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 25840. (Contributed by Scott Fenton, 22-Apr-2013.)

|- ( N e. NN -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) x =/= y )

Theorem

axlowdim2 25840*

The lower two-dimensional axiom. In any space where the dimension is greater than one, there are three non-colinear points. Axiom A8 of [Schwabhauser] p. 12. (Contributed by Scott Fenton, 15-Apr-2013.)

|- ( N e. ( ZZ>= ` 2 ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. z e. ( EE ` N ) -. ( x Btwn <. y , z >. \/ y Btwn <. z , x >. \/ z Btwn <. x , y >. ) )

Theorem

axlowdim 25841*

The general lower dimension axiom. Take a dimension N greater than or equal to three. Then, there are three non-colinear points in N dimensional space that are equidistant from N - 1 distinct points. Derived from remarks in Tarski's System of Geometry, Alfred Tarski and Steven Givant, Bulletin of Symbolic Logic, Volume 5, Number 2 (1999), 175-214. (Contributed by Scott Fenton, 22-Apr-2013.)

|- ( N e. ( ZZ>= ` 3 ) -> E. p E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. z e. ( EE ` N ) ( p : ( 1 ... ( N - 1 ) ) -1-1-> ( EE ` N ) /\ A. i e. ( 2 ... ( N - 1 ) ) ( <. ( p ` 1 ) , x >.Cgr <. ( p ` i ) , x >. /\ <. ( p ` 1 ) , y >.Cgr <. ( p ` i ) , y >. /\ <. ( p ` 1 ) , z >.Cgr <. ( p ` i ) , z >. ) /\ -. ( x Btwn <. y , z >. \/ y Btwn <. z , x >. \/ z Btwn <. x , y >. ) ) )

Theorem

axeuclidlem 25842*

Lemma for axeuclid 25843. Handle the algebraic aspects of the theorem. (Contributed by Scott Fenton, 9-Sep-2013.)

|- ( ( ( ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( C e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) /\ ( P e. ( 0 [,] 1 ) /\ Q e. ( 0 [,] 1 ) /\ P =/= 0 ) /\ A. i e. ( 1 ... N ) ( ( ( 1 - P ) x. ( A ` i ) ) + ( P x. ( T ` i ) ) ) = ( ( ( 1 - Q ) x. ( B ` i ) ) + ( Q x. ( C ` i ) ) ) ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) E. r e. ( 0 [,] 1 ) E. s e. ( 0 [,] 1 ) E. u e. ( 0 [,] 1 ) A. i e. ( 1 ... N ) ( ( B ` i ) = ( ( ( 1 - r ) x. ( A ` i ) ) + ( r x. ( x ` i ) ) ) /\ ( C ` i ) = ( ( ( 1 - s ) x. ( A ` i ) ) + ( s x. ( y ` i ) ) ) /\ ( T ` i ) = ( ( ( 1 - u ) x. ( x ` i ) ) + ( u x. ( y ` i ) ) ) ) )

Theorem

axeuclid 25843*

Euclid's axiom. Take an angle B A C and a point D between B and C. Now, if you extend the segment A D to a point T, then T lies between two points x and y that lie on the angle. Axiom A10 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 9-Sep-2013.)

|- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) /\ C e. ( EE ` N ) ) /\ ( D e. ( EE ` N ) /\ T e. ( EE ` N ) ) ) -> ( ( D Btwn <. A , T >. /\ D Btwn <. B , C >. /\ A =/= D ) -> E. x e. ( EE ` N ) E. y e. ( EE ` N ) ( B Btwn <. A , x >. /\ C Btwn <. A , y >. /\ T Btwn <. x , y >. ) ) )

Theorem

axcontlem1 25844*

Lemma for axcont 25856. Change bound variables for later use. (Contributed by Scott Fenton, 20-Jun-2013.)

|- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- F = { <. y , s >. | ( y e. D /\ ( s e. ( 0 [,) +oo ) /\ A. j e. ( 1 ... N ) ( y ` j ) = ( ( ( 1 - s ) x. ( Z ` j ) ) + ( s x. ( U ` j ) ) ) ) ) }

Theorem

axcontlem2 25845*

Lemma for axcont 25856. The idea here is to set up a mapping F that will allow us to transfer dedekind 10200 to two sets of points. Here, we set up F and show its domain and range. (Contributed by Scott Fenton, 17-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) -> F : D -1-1-onto-> ( 0 [,) +oo ) )

Theorem

axcontlem3 25846*

Lemma for axcont 25856. Given the separation assumption, B is a subset of D. (Contributed by Scott Fenton, 18-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   =>    |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( Z e. ( EE ` N ) /\ U e. A /\ Z =/= U ) ) -> B C_ D )

Theorem

axcontlem4 25847*

Lemma for axcont 25856. Given the separation assumption, A is a subset of D. (Contributed by Scott Fenton, 18-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   =>    |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( ( Z e. ( EE ` N ) /\ U e. A /\ B =/= (/) ) /\ Z =/= U ) ) -> A C_ D )

Theorem

axcontlem5 25848*

Lemma for axcont 25856. Compute the value of F. (Contributed by Scott Fenton, 18-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) = T <-> ( T e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - T ) x. ( Z ` i ) ) + ( T x. ( U ` i ) ) ) ) ) )

Theorem

axcontlem6 25849*

Lemma for axcont 25856. State the defining properties of the value of F. (Contributed by Scott Fenton, 19-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ P e. D ) -> ( ( F ` P ) e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( P ` i ) = ( ( ( 1 - ( F ` P ) ) x. ( Z ` i ) ) + ( ( F ` P ) x. ( U ` i ) ) ) ) )

Theorem

axcontlem7 25850*

Lemma for axcont 25856. Given two points in D, one preceeds the other iff its scaling constant is less than the other point's. (Contributed by Scott Fenton, 18-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ ( P e. D /\ Q e. D ) ) -> ( P Btwn <. Z , Q >. <-> ( F ` P ) <_ ( F ` Q ) ) )

Theorem

axcontlem8 25851*

Lemma for axcont 25856. A point in D is between two others if its function value falls in the middle. (Contributed by Scott Fenton, 18-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( ( N e. NN /\ Z e. ( EE ` N ) /\ U e. ( EE ` N ) ) /\ Z =/= U ) /\ ( P e. D /\ Q e. D /\ R e. D ) ) -> ( ( ( F ` P ) <_ ( F ` Q ) /\ ( F ` Q ) <_ ( F ` R ) ) -> Q Btwn <. P , R >. ) )

Theorem

axcontlem9 25852*

Lemma for axcont 25856. Given the separation assumption, all values of F over A are less than or equal to all values of F over B. (Contributed by Scott Fenton, 20-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( ( Z e. ( EE ` N ) /\ U e. A /\ B =/= (/) ) /\ Z =/= U ) ) -> A. n e. ( F " A ) A. m e. ( F " B ) n <_ m )

Theorem

axcontlem10 25853*

Lemma for axcont 25856. Given a handful of assumptions, derive the conclusion of the final theorem. (Contributed by Scott Fenton, 20-Jun-2013.)

|- D = { p e. ( EE ` N ) | ( U Btwn <. Z , p >. \/ p Btwn <. Z , U >. ) }   &    |- F = { <. x , t >. | ( x e. D /\ ( t e. ( 0 [,) +oo ) /\ A. i e. ( 1 ... N ) ( x ` i ) = ( ( ( 1 - t ) x. ( Z ` i ) ) + ( t x. ( U ` i ) ) ) ) ) }   =>    |- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( ( Z e. ( EE ` N ) /\ U e. A /\ B =/= (/) ) /\ Z =/= U ) ) -> E. b e. ( EE ` N ) A. x e. A A. y e. B b Btwn <. x , y >. )

Theorem

axcontlem11 25854*

Lemma for axcont 25856. Eliminate the hypotheses from axcontlem10 25853. (Contributed by Scott Fenton, 20-Jun-2013.)

|- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ ( ( Z e. ( EE ` N ) /\ U e. A /\ B =/= (/) ) /\ Z =/= U ) ) -> E. b e. ( EE ` N ) A. x e. A A. y e. B b Btwn <. x , y >. )

Theorem

axcontlem12 25855*

Lemma for axcont 25856. Eliminate the trivial cases from the previous lemmas. (Contributed by Scott Fenton, 20-Jun-2013.)

|- ( ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ A. x e. A A. y e. B x Btwn <. Z , y >. ) ) /\ Z e. ( EE ` N ) ) -> E. b e. ( EE ` N ) A. x e. A A. y e. B b Btwn <. x , y >. )

Theorem

axcont 25856*

The axiom of continuity. Take two sets of points A and B. If all the points in A come before the points of B on a line, then there is a point separating the two. Axiom A11 of [Schwabhauser] p. 13. (Contributed by Scott Fenton, 20-Jun-2013.)

|- ( ( N e. NN /\ ( A C_ ( EE ` N ) /\ B C_ ( EE ` N ) /\ E. a e. ( EE ` N ) A. x e. A A. y e. B x Btwn <. a , y >. ) ) -> E. b e. ( EE ` N ) A. x e. A A. y e. B b Btwn <. x , y >. )

15.4.2.3  EE^n fulfills Tarski's Axioms

Syntax

ceeng 25857

Extends class notation with the Tarski geometry structure for EE ^ N.

class EEG

Definition

df-eeng 25858*

Define the geometry structure for EE ^ N. (Contributed by Thierry Arnoux, 24-Aug-2017.)

|- EEG = ( n e. NN |-> ( { <. ( Base ` ndx ) , ( EE ` n ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` n ) , y e. ( EE ` n ) |-> sum_ i e. ( 1 ... n ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` n ) , y e. ( EE ` n ) |-> { z e. ( EE ` n ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` n ) , y e. ( ( EE ` n ) \ { x } ) |-> { z e. ( EE ` n ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )

Theorem

eengv 25859*

The value of the Euclidean geometry for dimension N. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( N e. NN -> (EEG ` N ) = ( { <. ( Base ` ndx ) , ( EE ` N ) >. , <. ( dist ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> sum_ i e. ( 1 ... N ) ( ( ( x ` i ) - ( y ` i ) ) ^ 2 ) ) >. } u. { <. (Itv ` ndx ) , ( x e. ( EE ` N ) , y e. ( EE ` N ) |-> { z e. ( EE ` N ) | z Btwn <. x , y >. } ) >. , <. (LineG ` ndx ) , ( x e. ( EE ` N ) , y e. ( ( EE ` N ) \ { x } ) |-> { z e. ( EE ` N ) | ( z Btwn <. x , y >. \/ x Btwn <. z , y >. \/ y Btwn <. x , z >. ) } ) >. } ) )

Theorem

eengstr 25860

The Euclidean geometry as a structure. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( N e. NN -> (EEG ` N ) Struct <. 1 , ; 1 7 >. )

Theorem

eengbas 25861

The Base of the Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( N e. NN -> ( EE ` N ) = ( Base ` (EEG ` N ) ) )

Theorem

ebtwntg 25862

The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( ph -> N e. NN )   &    |- P = ( Base ` (EEG ` N ) )   &    |- I = (Itv ` (EEG ` N ) )   &    |- ( ph -> X e. P )   &    |- ( ph -> Y e. P )   &    |- ( ph -> Z e. P )   =>    |- ( ph -> ( Z Btwn <. X , Y >. <-> Z e. ( X I Y ) ) )

Theorem

ecgrtg 25863

The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( ph -> N e. NN )   &    |- P = ( Base ` (EEG ` N ) )   &    |- .- = ( dist ` (EEG ` N ) )   &    |- ( ph -> A e. P )   &    |- ( ph -> B e. P )   &    |- ( ph -> C e. P )   &    |- ( ph -> D e. P )   =>    |- ( ph -> ( <. A , B >.Cgr <. C , D >. <-> ( A .- B ) = ( C .- D ) ) )

Theorem

elntg 25864*

The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)

|- P = ( Base ` (EEG ` N ) )   &    |- I = (Itv ` (EEG ` N ) )   =>    |- ( N e. NN -> (LineG ` (EEG ` N ) ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) )

Theorem

eengtrkg 25865

The geometry structure for EE ^ N is a Tarski geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( N e. NN -> (EEG ` N ) e. TarskiG )

Theorem

eengtrkge 25866

The geometry structure for EE ^ N is a Euclidean geometry. (Contributed by Thierry Arnoux, 15-Mar-2019.)

|- ( N e. NN -> (EEG ` N ) e. TarskiG_E )

PART 16  GRAPH THEORY

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic concepts:

Term	Reference	Definition	Remarks
Vertex	df-vtx 25876	A vertex of a graph G is an element of the set of vertices (Vtx ` G ) of the graph G.	The set of vertices (Vtx ` G ) (corresponding to V(G) in [Bollobas] p. 1) is usually the first component V of the graph G if it is represented by an ordered pair <. V , E >. (see opvtxfv 25884), or the base set ( Base ` G ) of the graph G if it is represented as extensible structure (see basvtxval 25901).
Edge	df-edg 25940	An edge of a graph G is a nonempty set of vertices of the graph. It is said that these vertices are "joined" or "connected" by the edge, see [Bollobas] p. 1.	The set of edges (Edg ` G ) (corresponding to E(G) in [Bollobas] p. 1) is usually the range ran E of the second component E of the graph G if it is represented by an ordered pair <. V , E >., or the range of the component (.ef ` G ) of the graph G if it is represented as extensible structure.
Loop		A loop in a graph G is an edge which connects a single vertex with itself (or, according to [Bollobas] p. 7 "joins a vertex to itself").	In other words, a loop is an edge e e. (Edg ` G ) which is a singleton consisting of a vertex v e. (Vtx ` G ): e = { v }
Edge function resp. indexed edges	df-iedg 25877	An edge function (or indexed set of edges) of a graph G is a mapping of an arbitrary index set to nonempty sets of vertices of the graph.	The edge function (iEdg ` G ) is usually the second component E of the graph G if it is represented by an ordered pair <. V , E > (see opiedgfv 25887), or the component (.ef ` G ) of the graph G if it is represented as extensible structure (see edgfiedgval 25902). The set of edges of a graph G is the range of its edge function: (Edg ` G ) = ran (iEdg ` G ), see edgval 25941. Whereas the concept of plain edges is sufficient for simple hypergraphs, indexed edges are required for e.g. multigraphs in which the same vertices may be connected by more than one edge.

Basic kinds of graphs:

Term	Reference	Definition	Remarks
Undirected hypergraph	df-uhgr 25953	a class G with an edge function E = (iEdg ` G ) which is a function into the power set of the vertices V = (Vtx ` G ): ran E C_ ( ~P V \ { (/) } ).	In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1. In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020) such a hypergraph is called a "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership". Notice that by using (Edg ` G ) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple hypergraphs.
Undirected simple hypergraph	df-ushgr 25954	a class G with an edge function E = (iEdg ` G ) which is a one-to-one function into the power set of the vertices V = (Vtx ` G ): ran E C_ ( ~P V \ { (/) } ).	See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a pseudograph needs not be a simple hypergraph. According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E_1, E_2, ..., E_m } such that E_i C_ E_j => i = j". By this definition, a simple hypergraph cannot contain the edges E_1 = { v_1 , v_2 } and E_2 = { v_1, v_2, v_3 }, because E_1 C_ E_2, but 1 =/= 2.
Undirected loop-free hypergraph	---	an undirected hypergraph without a loop, i.e. all edges connect at least two vertices.
Undirected pseudograph	df-upgr 25977	a class G with an edge function E = (iEdg ` G ) which is a function into the set of (proper or not proper) unordered pairs of vertices V = (Vtx ` G ).	A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4393). This means a pseudograph may contain loops. This defintion corresponds to the definition of a "multigraph" in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or in [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end(s).".
Undirected multigraph	df-umgr 25978	a class G with an edge function E = (iEdg ` G ) which is a function into the set of (proper!) unordered pairs of vertices V = (Vtx ` G ).	This definition is according to Chartrand, Gary and Zhang, Ping (2012): "A First Course in Graph Theory.", Dover, ISBN 978-0-486-48368-9, section 1.4, p. 26: "A multigraph M consists of a finite nonempty set V of vertices and a set E of edges, where every two vertices of M are joined by a finite number of edges (possibly zero). If two or more edges join the same pair of (distinct) vertices, then these edges are called parallel edges." A proper unordered pair contains two different elements, therefore a multigraph does not have loops.
Undirected simple pseudograph	df-uspgr 26045	a class G with an edge function E = (iEdg ` G ) which is a one-to-one function into the set of (proper or not proper) unordered pairs of vertices V = (Vtx ` G ).	This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph	df-usgr 26046	a class G with an edge function E = (iEdg ` G ) which is a one-to-one function into the set of (proper!) unordered pairs of vertices V = (Vtx ` G ).	An ordered pair <. V , E >. of two distinct sets V (the vertices) and E (the edges), the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2, can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see usgrausgrb 26064.
Finite graph	df-fusgr 26209	a graph G with a finite set of vertices V = (Vtx ` G ).	See definitions in [Bollobas] p. 1 or [Diestel] p. 2. In simple graphs, the set of (indexed) edges (iEdg ` G ) (and therefore also the set of edges (Edg ` G )) is finite if V = (Vtx ` G ) is finite, see fusgrfis 26222. The number of edges is limited by ( n _C 2 ) (or " n choose 2") with n = ( # ` V ), see fusgrmaxsize 26360. Analogously, the number of edges E = (iEdg ` G ) of an undirected simple pseudograph (which may have loops) is limited by ( ( n + 1 ) _C 2 ). In pseudographs or multigraphs, however, E can be infinite although V is finite.
Graph of finite size	---	a graph G with a finite set E = (iEdg ` G ), i.e. with a finite number of edges.	A graph can be of finite size although its set of vertices is infinite (most of the vertices would not be connected by an edge).

Terms and properties of graphs:

Term	Reference	Definition	Remarks
Edge joining resp. connecting (two) vertices	---	An edge e e. (Edg ` G ) joins resp. connects the vertices v_1, v_2, ... v_n ( n e. NN) if e = { v_1, v_2, ... v_n }.	If n = 1, e = { v_1 } is a loop, if n = 2, e = { v_1 , v_2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge	see definition in Section I.1 in [Bollobas] p. 1.	If an edge e e. (Edg ` G ) joins the vertices v_1, v_2, ... v_n ( n e. NN), then the vertices v_1, v_2, ... v_n are called the endvertices of the edge e.
(Two) Adjacent vertices	see definition in Section I.1 in [Bollobas] p. 1/2.	The vertices v_1, v_2, ... v_n ( n e. NN) are adjacent if there is an edge e = { v_1, v_2, ... v_n } joining these vertices. In this case, the vertices are incident with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or connected by the edge e.
Edge ending at a vertex		An edge e e. (Edg ` G ) is ending at a vertex v if the vertex is an endvertex of the edge: v e. e. In other words, the vertex v is incident with the edge e.
(Two) Adjacent edges		The edges e_0, e_1, ... e_n ( n e. NN) are adjacent if they have exactly one common endvertex.	Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph	see definition in Section I.1 in [Bollobas] p. 3	The order of a graph G is the number of vertices in the graph: ( # ` (Vtx ` G ) ).
Size of a graph	see definition in Section I.1 in [Bollobas] p. 3	The size of a graph G is the number of edges in the graph: ( # ` (iEdg ` G ) ). Or, for a simple graph G: ( # ` (Edg ` G ) )).
Neighborhood of a vertex	df-nbgr 26228 resp. definition in Section I.1 in [Bollobas] p. 3	A vertex connected with a vertex v by an edge is called a neighbor of the vertex v. The set of neighbors of a vertex v is called the neighborhood (or open neighborhood) of the vertex v. The closed neighborhood is the union of the (open) neighborhood of the vertex v with { v }.
Degree of a vertex	df-vtxdg 26362	The degree of a vertex is the number of the edges ending at this vertex.	In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex	usgrvd0nedg 26429	A vertex is called isolated if it is not an endvertex of any edge, thus having degree 0.
Universal vertex	df-uvtxa 26230	A vertex is called universal if it is connected with every other vertex of the graph by an edge, thus having degree ( ( # ` (Vtx ` G ) ) - ).

Special kinds of graphs:

Term	Reference	Definition	Remarks
Complete graph	df-cplgr 26231	A graph is called complete if each pair of vertices is connected by an edge.	The size of a complete undirected simple graph of order n is ( n _C 2 ) (or " n choose 2"), see cusgrsize 26350.
Empty graph	uhgr0e 25966	A graph is called empty if it has no edges.
Null graph	uhgr0 25968 and uhgr0vb 25967	A graph is called a null graph if it has no vertices (and therefore also no edges).
Trivial graph	usgr1v 26148	A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph	df-conngr 27047 resp. definition in Section I.1 in [Bollobas] p. 6	A graph is called connected if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

16.1  Vertices and edges

In the following, the vertices and (indexed) edges for an arbitrary class G (called "graph" in the following) are defined and examined. The main result of this section is to show that the set of vertices (Vtx ` G ) of a graph G is the first component V of the graph G if it is represented by an ordered pair <. V , E >. (see opvtxfv 25884), or the base set ( Base ` G ) of the graph G if it is represented as extensible structure (see basvtxval 25901), and that the set of indexed edges resp. the edge function (iEdg ` G ) is the second component E of the graph G if it is represented by an ordered pair <. V , E >. (see opiedgfv 25887), or the component (.ef ` G ) of the graph G if it is represented as extensible structure (see edgfiedgval 25902). Finally, it is shown that the set of edges of a graph G is the range of its edge function: (Edg ` G ) = ran (iEdg ` G ), see edgval 25941.

Usually, a graph G is a set. If G is a proper class, however, it represents the null graph (without vertices and edges), because (Vtx ` G ) = (/) and (iEdg ` G ) = (/) holds, see vtxvalprc 25937 and iedgvalprc 25938.

Up to the end of this section, the edges need not be related to the vertices. Once undirected hypergraphs are defined (see df-uhgr 25953), the edges become nonempty sets of vertices, and by this obtain their meaning as "connectors" of vertices.

16.1.1  The edge function extractor for extensible structures

Syntax

cedgf 25867

Extend class notation with an edge function.

class .ef

Definition

df-edgf 25868

Define the edge function (indexed edges) of a graph. (Contributed by AV, 18-Jan-2020.)

|- .ef = Slot ; 1 8

Theorem

edgfid 25869

Utility theorem: index-independent form of df-edgf 25868. (Contributed by AV, 16-Nov-2021.)

|- .ef = Slot (.ef ` ndx )

Theorem

edgfndxnn 25870

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.)

|- (.ef ` ndx ) e. NN

Theorem

edgfndxid 25871

The value of the edge function extractor is the value of the corresponding slot of the structure. (Contributed by AV, 21-Sep-2020.)

|- ( G e. V -> (.ef ` G ) = ( G ` (.ef ` ndx ) ) )

Theorem

baseltedgf 25872

The index value of the Base slot is less than the index value of the .ef slot. (Contributed by AV, 21-Sep-2020.)

|- ( Base ` ndx ) < (.ef ` ndx )

Theorem

slotsbaseefdif 25873

The slots Base and .ef are different. (Contributed by AV, 21-Sep-2020.)

|- ( Base ` ndx ) =/= (.ef ` ndx )

16.1.2  Vertices and indexed edges

The key concepts in graph theory are vertices and edges. In general, a graph "consists" (at least) of two sets: the set of vertices and the set of edges. The edges "connect" vertices. The meaning of "connect" is different for different kinds of graphs (directed/undirected graphs, hyper-/pseudo-/ multi-/simple graphs, etc.). The simplest way to represent a graph (of any kind) is to define a graph as "an ordered pair of disjoint sets (V, E)" (see section I.1 in [Bollobas] p. 1), or in the notation of Metamath: <. V , E >..

Another way is to regard a graph as a mathematical structure, which consistes at least of a set (of vertices) and a relation between the vertices (edge function), but which can be enhanced by additional features (see Wikipedia "Mathematical structure", 24-Sep-2020, https://en.wikipedia.org/wiki/Mathematical_structure): "In mathematics, a structure is a set endowed with some additional features on the set (e.g., operation, relation, metric, topology). Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or significance.". Such structures are provided as "extensible structures" in Metamath, see df-struct 15859.

To allow for expressing and proving most of the theorems for graphs independently from their representation, the functions Vtx and iEdg are defined (see df-vtx 25876 and df-iedg 25877), which provide the vertices resp. (indexed) edges of an arbitrary class G which represents a graph: (Vtx ` G ) resp. (iEdg ` G ). In literature, these functions are often denoted also by "V" and "E", see section I.1 in [Bollobas] p. 1 ("If G is a graph, then V = V(G) is the vertex set of G, and E = E(G) is the edge set.") or section 1.1 in [Diestel] p. 2 ("The vertex set of graph G is referred to as V(G), its edge set as E(G).").

Instead of providing edges themselves, iEdg is intended to provide a function as mapping of "indices" (the domain of the function) to the edges (therefore called "set of indexed edges"), which allows for hyper-/pseudo-/multigraphs with more than one edge between two (or more) vertices. For example, e₁ = e(1) = { a, b } and e₂ = e(2) = { a, b } are two different edges connecting the same two vertices a and b (in a pseudograph). In section 1.10 of [Diestel] p. 28, the edge function is defined differently: as "map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.". Here, the domain is the set of abstract edges: for two different edges e₁ and e₂ connecting the same two vertices a and b, we would have e(e₁) = e(e₂) = { a, b }. Since the set of abstract edges can be chosen as index set, these definitions are equivalent.

The result of these functions are as expected: for a graph represented as ordered pair ( G e. ( _V X. _V )), the set of vertices is (Vtx ` G ) = ( 1st ` G ) (see opvtxval 25883) and the set of (indexed) edges is (iEdg ` G ) = ( 2nd ` G ) (see opiedgval 25886), or if G is given as ordered pair G = <. V , E >., the set of vertices is (Vtx ` G ) = V (see opvtxfv 25884) and the set of (indexed) edges is (iEdg ` G ) = E (see opiedgfv 25887).

And for a graph represented as extensible structure ( G Struct <. ( Base ` ndx ) , (.ef ` ndx ) >.), the set of vertices is (Vtx ` G ) = ( Base ` G ) (see funvtxval 25905) and the set of (indexed) edges is (iEdg ` G ) = (.ef ` G ) (see funiedgval 25906), or if G is given in its simplest form as extensible structure with two slots ( G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }), the set of vertices is (Vtx ` G ) = V (see struct2grvtx 25919) and the set of (indexed) edges is (iEdg ` G ) = E (see struct2griedg 25920).

These two representations are convertible, see graop 25921 and grastruct 25922: If G is a graph (for example G = <. V , E >.), then H = { <. ( Base ` ndx ) , (Vtx ` G ) >. , <. (.ef ` ndx ) , (iEdg ` G ) >. } represents essentially the same graph, and if G is a graph (for example G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , E >. }), then H = <. (Vtx ` G ) , (iEdg ` G ) >. represents essentially the same graph. In both cases, (Vtx ` G ) = (Vtx ` H ) and (iEdg ` G ) = (iEdg ` H ) hold. Theorems gropd 25923 and gropeld 25925 show that if any representation of a graph with vertices V and edges E has a certain property, 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. Analogously, theorems grstructd 25924 and grstructeld 25926 show that if any representation of a graph with vertices V and edges E has a certain property, then any extensible structure with base set V and value E in the slot for edge functions (which is also such a representation of a graph with vertices V and edges E) has this property.

Besides the usual way to represent graphs without edges (consisting of unconnected vertices only), which would be G = <. V , (/) >. or G = { <. ( Base ` ndx ) , V >. , <. (.ef ` ndx ) , (/) >. }, a structure without a slot for edges can be used: G = { <. ( Base ` ndx ) , V >. }, see snstrvtxval 25929 and snstriedgval 25930. Analogously, the empty set (/) can be used to represent the null graph, see vtxval0 25931 and iedgval0 25932, which can also be represented by G = <. (/) , (/) >. or G = { <. ( Base ` ndx ) , (/) >. , <. (.ef ` ndx ) , (/) >. }. Even proper classes can be used to represent the null graph, see vtxvalprc 25937 and iedgvalprc 25938.

Other classes should not be used to represent graphs, because there could be a degenerated behavior of the vertex set and (indexed) edge functions, see vtxvalsnop 25933 resp. iedgvalsnop 25934, and vtxval3sn 25935 resp. iedgval3sn 25936.

16.1.2.1  Definitions and basic properties

Syntax

cvtx 25874

Extend class notation with the vertices of "graphs".

class Vtx

Syntax

ciedg 25875

Extend class notation with the indexed edges of "graphs".

class iEdg

Definition

df-vtx 25876

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 25877

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

vtxval 25878

The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

|- (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) )

Theorem

iedgval 25879

The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)

|- (iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) )

Theorem

vtxvalOLD 25880

Obsolete version of vtxval 25878 as of 11-Nov-2021. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

|- ( G e. V -> (Vtx ` G ) = if ( G e. ( _V X. _V ) , ( 1st ` G ) , ( Base ` G ) ) )

Theorem

iedgvalOLD 25881

Obsolete version of iedgval 25879 as of 11-Nov-2021. (Contributed by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

|- ( G e. V -> (iEdg ` G ) = if ( G e. ( _V X. _V ) , ( 2nd ` G ) , (.ef ` G ) ) )

Theorem

1vgrex 25882

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 )

16.1.2.2  The vertices and edges of a graph represented as ordered pair

Theorem

opvtxval 25883

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 25884

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 25885

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 25886

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 25887

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 25888

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 25889

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 25890

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

16.1.2.3  The vertices and edges of a graph represented as extensible structure

Theorem

funvtxdmge2val 25891

The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funiedgdmge2val 25892

The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

|- ( ( Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> (iEdg ` G ) = (.ef ` G ) )

Theorem

funvtxdm2val 25893

The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funiedgdm2val 25894

The set of indexed edges of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (Revised by AV, 12-Nov-2021.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( Fun ( G \ { (/) } ) /\ A =/= B /\ { A , B } C_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )

Theorem

funvtxdm2valOLD 25895

Obsolete version of funvtxdm2val 25893 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ A =/= B /\ { A , B } C_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funiedgdm2valOLD 25896

Obsolete version of funiedgdm2val 25894 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

|- A e. _V   &    |- B e. _V   =>    |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ A =/= B /\ { A , B } C_ dom G ) -> (iEdg ` G ) = (.ef ` G ) )

Theorem

funvtxval0 25897

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   =>    |- ( ( Fun ( G \ { (/) } ) /\ S =/= ( Base ` ndx ) /\ { ( Base ` ndx ) , S } C_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funvtxval0OLD 25898

Obsolete version of funvtxval0 25897 as of 11-Nov-2021. (Contributed by AV, 22-Sep-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

|- S e. _V   =>    |- ( ( ( G e. V /\ Fun ( G \ { (/) } ) ) /\ S =/= ( Base ` ndx ) /\ { ( Base ` ndx ) , S } C_ dom G ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funvtxdmge2valOLD 25899

Obsolete version of funvtxdmge2val 25891 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> (Vtx ` G ) = ( Base ` G ) )

Theorem

funiedgdmge2valOLD 25900

Obsolete version of funiedgdmge2val 25892 as of 11-Nov-2021. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 7-Jun-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

|- ( ( G e. V /\ Fun ( G \ { (/) } ) /\ 2 <_ ( # ` dom G ) ) -> (iEdg ` G ) = (.ef ` G ) )