Metamath Proof Explorer

Theorem ismir

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 )
ismir.1 
|- ( ph -> C e. P )
ismir.2 
|- ( ph -> ( A .- C ) = ( A .- B ) )
ismir.3 
|- ( ph -> A e. ( C I B ) )
Assertion ismir 
|- ( ph -> C = ( M ` 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 ismir.1 
 |-  ( ph -> C e. P )
11 ismir.2 
 |-  ( ph -> ( A .- C ) = ( A .- B ) )
12 ismir.3 
 |-  ( ph -> A e. ( C I B ) )
13 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 ) ) ) )
14 1 2 3 6 9 7 mirreu3 
 |-  ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15 oveq2 
 |-  ( z = C -> ( A .- z ) = ( A .- C ) )
16 15 eqeq1d 
 |-  ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17 oveq1 
 |-  ( z = C -> ( z I B ) = ( C I B ) )
18 17 eleq2d 
 |-  ( z = C -> ( A e. ( z I B ) <-> A e. ( C I B ) ) )
19 16 18 anbi12d 
 |-  ( z = C -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) ) )
20 19 riota2 
 |-  ( ( C e. P /\ E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) -> ( ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) <-> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = C ) )
21 10 14 20 syl2anc 
 |-  ( ph -> ( ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) <-> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = C ) )
22 11 12 21 mpbi2and 
 |-  ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = C )
23 13 22 eqtr2d 
 |-  ( ph -> C = ( M ` B ) )