Theorem mireq 25560

Description: Equality deduction for point inversion. Theorem 7.9 of [Schwabhauser] p. 50. (Contributed by Thierry Arnoux, 30-May-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 )
mirmir.b	|- ( ph -> B e. P )
mireq.c	|- ( ph -> C e. P )
mireq.d	|- ( ph -> ( M ` B ) = ( M ` C ) )

Assertion

Ref	Expression
mireq	|- ( ph -> B = C )

Proof of Theorem mireq

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		mireq.c	. . . 4 |- ( ph -> C e. P )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mircl 25556	. . 3 |- ( ph -> ( M ` C ) e. P )
11		mirmir.b	. . 3 |- ( ph -> B e. P )
12	1, 2, 3, 4, 5, 6, 7, 8, 11	mirfv 25551	. . . . . . 7 |- ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
13		mireq.d	. . . . . . 7 |- ( ph -> ( M ` B ) = ( M ` C ) )
14	12, 13	eqtr3d 2658	. . . . . 6 |- ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = ( M ` C ) )
15	1, 2, 3, 6, 11, 7	mirreu3 25549	. . . . . . 7 |- ( ph -> E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) )
16		oveq2 6658	. . . . . . . . . 10 |- ( z = ( M ` C ) -> ( A .- z ) = ( A .- ( M ` C ) ) )
17	16	eqeq1d 2624	. . . . . . . . 9 |- ( z = ( M ` C ) -> ( ( A .- z ) = ( A .- B ) <-> ( A .- ( M ` C ) ) = ( A .- B ) ) )
18		oveq1 6657	. . . . . . . . . 10 |- ( z = ( M ` C ) -> ( z I B ) = ( ( M ` C ) I B ) )
19	18	eleq2d 2687	. . . . . . . . 9 |- ( z = ( M ` C ) -> ( A e. ( z I B ) <-> A e. ( ( M ` C ) I B ) ) )
20	17, 19	anbi12d 747	. . . . . . . 8 |- ( z = ( M ` C ) -> ( ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) <-> ( ( A .- ( M ` C ) ) = ( A .- B ) /\ A e. ( ( M ` C ) I B ) ) ) )
21	20	riota2 6633	. . . . . . 7 |- ( ( ( M ` C ) e. P /\ E! z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) -> ( ( ( A .- ( M ` C ) ) = ( A .- B ) /\ A e. ( ( M ` C ) I B ) ) <-> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = ( M ` C ) ) )
22	10, 15, 21	syl2anc 693	. . . . . 6 |- ( ph -> ( ( ( A .- ( M ` C ) ) = ( A .- B ) /\ A e. ( ( M ` C ) I B ) ) <-> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) = ( M ` C ) ) )
23	14, 22	mpbird 247	. . . . 5 |- ( ph -> ( ( A .- ( M ` C ) ) = ( A .- B ) /\ A e. ( ( M ` C ) I B ) ) )
24	23	simpld 475	. . . 4 |- ( ph -> ( A .- ( M ` C ) ) = ( A .- B ) )
25	24	eqcomd 2628	. . 3 |- ( ph -> ( A .- B ) = ( A .- ( M ` C ) ) )
26	23	simprd 479	. . . 4 |- ( ph -> A e. ( ( M ` C ) I B ) )
27	1, 2, 3, 6, 10, 7, 11, 26	tgbtwncom 25383	. . 3 |- ( ph -> A e. ( B I ( M ` C ) ) )
28	1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 25, 27	ismir 25554	. 2 |- ( ph -> B = ( M ` ( M ` C ) ) )
29	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 25557	. 2 |- ( ph -> ( M ` ( M ` C ) ) = C )
30	28, 29	eqtrd 2656	1 |- ( ph -> B = C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  mirhl  25574  mirbtwnhl  25575  mirhl2  25576  colperpexlem3  25624