Theorem ismir 24244

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 24241	. 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 24239	. . . 4 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15		oveq2 6278	. . . . . . 7 |- ( z = C -> ( A .- z ) = ( A .- C ) )
16	15	eqeq1d 2456	. . . . . 6 |- ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17		oveq1 6277	. . . . . . 7 |- ( z = C -> ( z I B ) = ( C I B ) )
18	17	eleq2d 2524	. . . . . 6 |- ( z = C -> ( A e. ( z I B ) <-> A e. ( C I B ) ) )
19	16, 18	anbi12d 708	. . . . 5 |- ( z = C -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) ) )
20	19	riota2 6254	. . . 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 659	. . 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 2496	1 |- ( ph -> C = ( M ` B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   E!wreu 2806   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14719   distcds 14796  TarskiGcstrkg 24026  Itvcitv 24033  LineGclng 24034  pInvGcmir 24237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-trkgc 24045  df-trkgb 24046  df-trkgcb 24047  df-trkg 24051  df-mir 24238

This theorem is referenced by:  mirmir  24247  mireq  24250  mirinv  24251  miriso  24254  mirmir2  24258  mirauto  24265  colmid  24269  krippenlem  24271  midexlem  24273  mideulem2  24312  opphllem  24313  midcom  24352