Definition df-mir 25548

Description: Define the point inversion ("mirror") function. Definition 7.5 of [Schwabhauser] p. 49. See mirval 25550 and ismir 25554. (Contributed by Thierry Arnoux, 30-May-2019.)

Assertion

Ref	Expression
df-mir	|- pInvG = ( g e. _V |-> ( m e. ( Base ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) ) ) )

Distinct variable group:   a, b, g, m

Detailed syntax breakdown of Definition df-mir

Step	Hyp	Ref	Expression
1		cmir 25547	. 2 class pInvG
2		vg	. . 3 setvar g
3		cvv 3200	. . 3 class _V
4		vm	. . . 4 setvar m
5	2	cv 1482	. . . . 5 class g
6		cbs 15857	. . . . 5 class Base
7	5, 6	cfv 5888	. . . 4 class ( Base ` g )
8		va	. . . . 5 setvar a
9	4	cv 1482	. . . . . . . . 9 class m
10		vb	. . . . . . . . . 10 setvar b
11	10	cv 1482	. . . . . . . . 9 class b
12		cds 15950	. . . . . . . . . 10 class dist
13	5, 12	cfv 5888	. . . . . . . . 9 class ( dist ` g )
14	9, 11, 13	co 6650	. . . . . . . 8 class ( m ( dist ` g ) b )
15	8	cv 1482	. . . . . . . . 9 class a
16	9, 15, 13	co 6650	. . . . . . . 8 class ( m ( dist ` g ) a )
17	14, 16	wceq 1483	. . . . . . 7 wff ( m ( dist ` g ) b ) = ( m ( dist ` g ) a )
18		citv 25335	. . . . . . . . . 10 class Itv
19	5, 18	cfv 5888	. . . . . . . . 9 class (Itv ` g )
20	11, 15, 19	co 6650	. . . . . . . 8 class ( b (Itv ` g ) a )
21	9, 20	wcel 1990	. . . . . . 7 wff m e. ( b (Itv ` g ) a )
22	17, 21	wa 384	. . . . . 6 wff ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) )
23	22, 10, 7	crio 6610	. . . . 5 class ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) )
24	8, 7, 23	cmpt 4729	. . . 4 class ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) )
25	4, 7, 24	cmpt 4729	. . 3 class ( m e. ( Base ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) ) )
26	2, 3, 25	cmpt 4729	. 2 class ( g e. _V |-> ( m e. ( Base ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) ) ) )
27	1, 26	wceq 1483	1 wff pInvG = ( g e. _V |-> ( m e. ( Base ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( m ( dist ` g ) b ) = ( m ( dist ` g ) a ) /\ m e. ( b (Itv ` g ) a ) ) ) ) ) )

This definition is referenced by:  mirval  25550