Metamath Proof Explorer

Theorem mirbtwn

Description: Property of the image by the point inversion function. Definition 7.5 of Schwabhauser p. 49. (Contributed by Thierry Arnoux, 3-Jun-2019)

Ref Expression
Hypotheses mirval.p 
|- P = ( Base ` G )
mirval.d 
|- .- = ( dist ` G )
mirval.i 
|- I = ( Itv ` G )
mirval.l 
|- L = ( LineG ` G )
mirval.s 
|- S = ( pInvG ` G )
mirval.g 
|- ( ph -> G e. TarskiG )
mirval.a 
|- ( ph -> A e. P )
mirfv.m 
|- M = ( S ` A )
mirfv.b 
|- ( ph -> B e. P )
Assertion mirbtwn 
|- ( ph -> A e. ( ( M ` B ) I B ) )

Proof

Step Hyp Ref Expression
1 mirval.p 
 |-  P = ( Base ` G )
2 mirval.d 
 |-  .- = ( dist ` G )
3 mirval.i 
 |-  I = ( Itv ` G )
4 mirval.l 
 |-  L = ( LineG ` G )
5 mirval.s 
 |-  S = ( pInvG ` G )
6 mirval.g 
 |-  ( ph -> G e. TarskiG )
7 mirval.a 
 |-  ( ph -> A e. P )
8 mirfv.m 
 |-  M = ( S ` A )
9 mirfv.b 
 |-  ( ph -> B e. P )
10 1 2 3 4 5 6 7 8 9 mirfv 
 |-  ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
11 1 2 3 6 9 7 mirreu3 
 |-  ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
12 riotacl2 
 |-  ( E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) e. { z e. P | ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) } )
13 11 12 syl 
 |-  ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) e. { z e. P | ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) } )
14 10 13 eqeltrd 
 |-  ( ph -> ( M ` B ) e. { z e. P | ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) } )
15 oveq2 
 |-  ( z = ( M ` B ) -> ( A .- z ) = ( A .- ( M ` B ) ) )
16 15 eqeq1d 
 |-  ( z = ( M ` B ) -> ( ( A .- z ) = ( A .- B ) <-> ( A .- ( M ` B ) ) = ( A .- B ) ) )
17 oveq1 
 |-  ( z = ( M ` B ) -> ( z I B ) = ( ( M ` B ) I B ) )
18 17 eleq2d 
 |-  ( z = ( M ` B ) -> ( A e. ( z I B ) <-> A e. ( ( M ` B ) I B ) ) )
19 16 18 anbi12d 
 |-  ( z = ( M ` B ) -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- ( M ` B ) ) = ( A .- B ) /\ A e. ( ( M ` B ) I B ) ) ) )
20 19 elrab 
 |-  ( ( M ` B ) e. { z e. P | ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) } <-> ( ( M ` B ) e. P /\ ( ( A .- ( M ` B ) ) = ( A .- B ) /\ A e. ( ( M ` B ) I B ) ) ) )
21 14 20 sylib 
 |-  ( ph -> ( ( M ` B ) e. P /\ ( ( A .- ( M ` B ) ) = ( A .- B ) /\ A e. ( ( M ` B ) I B ) ) ) )
22 21 simprrd 
 |-  ( ph -> A e. ( ( M ` B ) I B ) )