Theorem ismir 23905

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 )
ismir.1	|- ( ph -> C e. P )
ismir.2	|- ( ph -> ( A .- C ) = ( A .- B ) )
ismir.3	|- ( ph -> A e. ( C I B ) )

Assertion

Ref	Expression
ismir	|- ( ph -> C = ( M ` B ) )

Proof of Theorem ismir

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirval.p	. . 3 |- P = ( Base ` G )
2		mirval.d	. . 3 |- .- = ( dist ` G )
3		mirval.i	. . 3 |- I = (Itv ` G )
4		mirval.l	. . 3 |- L = (LineG ` G )
5		mirval.s	. . 3 |- S = (pInvG ` G )
6		mirval.g	. . 3 |- ( ph -> G e. TarskiG )
7		mirval.a	. . 3 |- ( ph -> A e. P )
8		mirfv.m	. . 3 |- M = ( S ` A )
9		mirfv.b	. . 3 |- ( ph -> B e. P )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mirfv 23902	. 2 |- ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
11		ismir.2	. . 3 |- ( ph -> ( A .- C ) = ( A .- B ) )
12		ismir.3	. . 3 |- ( ph -> A e. ( C I B ) )
13		ismir.1	. . . 4 |- ( ph -> C e. P )
14	1, 2, 3, 6, 9, 7	mirreu3 23900	. . . 4 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15		oveq2 6285	. . . . . . 7 |- ( z = C -> ( A .- z ) = ( A .- C ) )
16	15	eqeq1d 2443	. . . . . 6 |- ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17		oveq1 6284	. . . . . . 7 |- ( z = C -> ( z I B ) = ( C I B ) )
18	17	eleq2d 2511	. . . . . 6 |- ( z = C -> ( A e. ( z I B ) <-> A e. ( C I B ) ) )
19	16, 18	anbi12d 710	. . . . 5 |- ( z = C -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) ) )
20	19	riota2 6261	. . . 4 |- ( ( 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	13, 14, 20	syl2anc 661	. . 3 |- ( 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 919	. 2 |- ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = C )
23	10, 22	eqtr2d 2483	1 |- ( ph -> C = ( M ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1381   e. wcel 1802   E!wreu 2793   ` cfv 5574   iota_crio 6237  (class class class)co 6277   Basecbs 14504   distcds 14578  TarskiGcstrkg 23690  Itvcitv 23697  LineGclng 23698  pInvGcmir 23898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-trkgc 23709  df-trkgb 23710  df-trkgcb 23711  df-trkg 23715  df-mir 23899

This theorem is referenced by:  mirmir  23908  mireq  23911  mirinv  23912  miriso  23915  mirmir2  23919  mirauto  23926  colmid  23930  krippenlem  23932  midexlem  23934  mideulem2  23973  opphllem  23974  midcom  24013