Theorem mircgr 25552

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

Hypotheses

Ref	Expression
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

Ref	Expression
mircgr	|- ( ph -> ( A .- ( M ` B ) ) = ( A .- B ) )

Proof of Theorem mircgr

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirval.p	. . . . . 6 |- P = ( Base ` G )
2		mirval.d	. . . . . 6 |- .- = ( dist ` G )
3		mirval.i	. . . . . 6 |- I = (Itv ` G )
4		mirval.l	. . . . . 6 |- L = (LineG ` G )
5		mirval.s	. . . . . 6 |- S = (pInvG ` G )
6		mirval.g	. . . . . 6 |- ( ph -> G e. TarskiG )
7		mirval.a	. . . . . 6 |- ( ph -> A e. P )
8		mirfv.m	. . . . . 6 |- M = ( S ` A )
9		mirfv.b	. . . . . 6 |- ( ph -> B e. P )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mirfv 25551	. . . . 5 |- ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
11	1, 2, 3, 6, 9, 7	mirreu3 25549	. . . . . 6 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
12		riotacl2 6624	. . . . . 6 |- ( 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 17	. . . . 5 |- ( 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 2701	. . . 4 |- ( ph -> ( M ` B ) e. { z e. P | ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) } )
15		oveq2 6658	. . . . . . 7 |- ( z = ( M ` B ) -> ( A .- z ) = ( A .- ( M ` B ) ) )
16	15	eqeq1d 2624	. . . . . 6 |- ( z = ( M ` B ) -> ( ( A .- z ) = ( A .- B ) <-> ( A .- ( M ` B ) ) = ( A .- B ) ) )
17		oveq1 6657	. . . . . . 7 |- ( z = ( M ` B ) -> ( z I B ) = ( ( M ` B ) I B ) )
18	17	eleq2d 2687	. . . . . 6 |- ( z = ( M ` B ) -> ( A e. ( z I B ) <-> A e. ( ( M ` B ) I B ) ) )
19	16, 18	anbi12d 747	. . . . 5 |- ( 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 3363	. . . 4 |- ( ( 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 208	. . 3 |- ( ph -> ( ( M ` B ) e. P /\ ( ( A .- ( M ` B ) ) = ( A .- B ) /\ A e. ( ( M ` B ) I B ) ) ) )
22	21	simprd 479	. 2 |- ( ph -> ( ( A .- ( M ` B ) ) = ( A .- B ) /\ A e. ( ( M ` B ) I B ) ) )
23	22	simpld 475	1 |- ( ph -> ( A .- ( M ` B ) ) = ( A .- B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   {crab 2916   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  mirmir  25557  miriso  25565  mirmir2  25569  mircgrextend  25577  mirtrcgr  25578  mirauto  25579  miduniq  25580  krippenlem  25585  ragcol  25594  ragflat  25599  ragcgr  25602  footex  25613  colperpexlem1  25622  colperpexlem3  25624  mideulem2  25626  opphllem  25627  midcgr  25672  lmiisolem  25688  sacgr  25722