Theorem ismir 23214

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 23211	. 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 23209	. . . 4 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
15		oveq2 6211	. . . . . . 7 |- ( z = C -> ( A .- z ) = ( A .- C ) )
16	15	eqeq1d 2456	. . . . . 6 |- ( z = C -> ( ( A .- z ) = ( A .- B ) <-> ( A .- C ) = ( A .- B ) ) )
17		oveq1 6210	. . . . . . 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 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 6187	. . . 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 912	. 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 369   = wceq 1370   e. wcel 1758   E!wreu 2801   ` cfv 5529   iota_crio 6163  (class class class)co 6203   Basecbs 14296   distcds 14370  TarskiGcstrkg 23032  Itvcitv 23039  LineGclng 23040  pInvGcmir 23207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-trkgc 23051  df-trkgb 23052  df-trkgcb 23053  df-trkg 23057  df-mir 23208

This theorem is referenced by:  mirmir  23217  mireq  23220  mirinv  23221  miriso  23224  mirmir2  23228  mirauto  23229  colmid  23233  krippenlem  23235  midexlem  23237  mideulem  23270  midcom  23282