Theorem constrcccl

Description: Constructible numbers are closed under circle-circle intersections. (Contributed by Thierry Arnoux, 2-Nov-2025)

	Ref	Expression
Hypotheses	constrcccl.a	|- ( ph -> A e. Constr )
	constrcccl.b	|- ( ph -> B e. Constr )
	constrcccl.c	|- ( ph -> C e. Constr )
	constrcccl.d	|- ( ph -> D e. Constr )
	constrcccl.e	|- ( ph -> E e. Constr )
	constrcccl.f	|- ( ph -> F e. Constr )
	constrcccl.x	|- ( ph -> X e. CC )
	constrcccl.1	|- ( ph -> A =/= D )
	constrcccl.2	|- ( ph -> ( abs ` ( X - A ) ) = ( abs ` ( B - C ) ) )
	constrcccl.3	|- ( ph -> ( abs ` ( X - D ) ) = ( abs ` ( E - F ) ) )
Assertion	constrcccl	|- ( ph -> X e. Constr )

Proof

Step	Hyp	Ref	Expression
1		constrcccl.a	|- ( ph -> A e. Constr )
2		constrcccl.b	|- ( ph -> B e. Constr )
3		constrcccl.c	|- ( ph -> C e. Constr )
4		constrcccl.d	|- ( ph -> D e. Constr )
5		constrcccl.e	|- ( ph -> E e. Constr )
6		constrcccl.f	|- ( ph -> F e. Constr )
7		constrcccl.x	|- ( ph -> X e. CC )
8		constrcccl.1	|- ( ph -> A =/= D )
9		constrcccl.2	|- ( ph -> ( abs ` ( X - A ) ) = ( abs ` ( B - C ) ) )
10		constrcccl.3	|- ( ph -> ( abs ` ( X - D ) ) = ( abs ` ( E - F ) ) )
11		constrcbvlem	|- rec ( ( z e. _V |-> { y e. CC | ( E. i e. z E. j e. z E. k e. z E. l e. z E. o e. RR E. p e. RR ( y = ( i + ( o x. ( j - i ) ) ) /\ y = ( k + ( p x. ( l - k ) ) ) /\ ( Im ` ( ( * ` ( j - i ) ) x. ( l - k ) ) ) =/= 0 ) \/ E. i e. z E. j e. z E. k e. z E. m e. z E. q e. z E. o e. RR ( y = ( i + ( o x. ( j - i ) ) ) /\ ( abs ` ( y - k ) ) = ( abs ` ( m - q ) ) ) \/ E. i e. z E. j e. z E. k e. z E. l e. z E. m e. z E. q e. z ( i =/= l /\ ( abs ` ( y - i ) ) = ( abs ` ( j - k ) ) /\ ( abs ` ( y - l ) ) = ( abs ` ( m - q ) ) ) ) } ) , { 0 , 1 } ) = rec ( ( s e. _V |-> { x e. CC | ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) =/= 0 ) \/ E. a e. s E. b e. s E. c e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s ( a =/= d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) } ) , { 0 , 1 } )
12	11 1 2 3 4 5 6 7 8 9 10	constrcccllem	|- ( ph -> X e. Constr )