Theorem ismir 23868

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 23865	. 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 23863	. . . 4 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15		oveq2 6302	. . . . . . 7 |- ( z = C -> ( A .- z ) = ( A .- C ) )
16	15	eqeq1d 2469	. . . . . 6 |- ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17		oveq1 6301	. . . . . . 7 |- ( z = C -> ( z I B ) = ( C I B ) )
18	17	eleq2d 2537	. . . . . 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 6278	. . . 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 2509	1 |- ( ph -> C = ( M ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   E!wreu 2819   ` cfv 5593   iota_crio 6254  (class class class)co 6294   Basecbs 14502   distcds 14576  TarskiGcstrkg 23668  Itvcitv 23675  LineGclng 23676  pInvGcmir 23861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-trkgc 23687  df-trkgb 23688  df-trkgcb 23689  df-trkg 23693  df-mir 23862

This theorem is referenced by:  mirmir  23871  mireq  23874  mirinv  23875  miriso  23878  mirmir2  23882  mirauto  23884  colmid  23888  krippenlem  23890  midexlem  23892  mideulem  23928  midcom  23940