Theorem ismir 24697

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 24694	. 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 24692	. . . 4 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15		oveq2 6296	. . . . . . 7 |- ( z = C -> ( A .- z ) = ( A .- C ) )
16	15	eqeq1d 2452	. . . . . 6 |- ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17		oveq1 6295	. . . . . . 7 |- ( z = C -> ( z I B ) = ( C I B ) )
18	17	eleq2d 2513	. . . . . 6 |- ( z = C -> ( A e. ( z I B ) <-> A e. ( C I B ) ) )
19	16, 18	anbi12d 716	. . . . 5 |- ( z = C -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- C ) = ( A .- B ) /\ A e. ( C I B ) ) ) )
20	19	riota2 6272	. . . 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 666	. . 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 931	. 2 |- ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = C )
23	10, 22	eqtr2d 2485	1 |- ( ph -> C = ( M ` B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   E!wreu 2738   ` cfv 5581   iota_crio 6249  (class class class)co 6288   Basecbs 15114   distcds 15192  TarskiGcstrkg 24471  Itvcitv 24477  LineGclng 24478  pInvGcmir 24690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-trkgc 24489  df-trkgb 24490  df-trkgcb 24491  df-trkg 24494  df-mir 24691

This theorem is referenced by:  mirmir  24700  mireq  24703  mirinv  24704  miriso  24708  mirmir2  24712  mirauto  24722  colmid  24726  krippenlem  24728  midexlem  24730  mideulem2  24769  opphllem  24770  midcom  24817  trgcopyeulem  24840