Theorem mirreu 25559

Description: Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (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 )
mirmir.b	|- ( ph -> B e. P )

Assertion

Ref	Expression
mirreu	|- ( ph -> E! a e. P ( M ` a ) = B )

Distinct variable groups:   B, a   M, a   P, a   ph, a

Allowed substitution hints:   A( a)   S( a)   G( a)   I( a)   L( a)   .- ( a)

Proof of Theorem mirreu

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		mirmir.b	. . 3 |- ( ph -> B e. P )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mircl 25556	. 2 |- ( ph -> ( M ` B ) e. P )
11	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 25557	. 2 |- ( ph -> ( M ` ( M ` B ) ) = B )
12	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> G e. TarskiG )
13	7	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> A e. P )
14		simplr 792	. . . . . 6 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> a e. P )
15	1, 2, 3, 4, 5, 12, 13, 8, 14	mirmir 25557	. . . . 5 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> ( M ` ( M ` a ) ) = a )
16		simpr 477	. . . . . 6 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> ( M ` a ) = B )
17	16	fveq2d 6195	. . . . 5 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> ( M ` ( M ` a ) ) = ( M ` B ) )
18	15, 17	eqtr3d 2658	. . . 4 |- ( ( ( ph /\ a e. P ) /\ ( M ` a ) = B ) -> a = ( M ` B ) )
19	18	ex 450	. . 3 |- ( ( ph /\ a e. P ) -> ( ( M ` a ) = B -> a = ( M ` B ) ) )
20	19	ralrimiva 2966	. 2 |- ( ph -> A. a e. P ( ( M ` a ) = B -> a = ( M ` B ) ) )
21		fveq2 6191	. . . 4 |- ( a = ( M ` B ) -> ( M ` a ) = ( M ` ( M ` B ) ) )
22	21	eqeq1d 2624	. . 3 |- ( a = ( M ` B ) -> ( ( M ` a ) = B <-> ( M ` ( M ` B ) ) = B ) )
23	22	eqreu 3398	. 2 |- ( ( ( M ` B ) e. P /\ ( M ` ( M ` B ) ) = B /\ A. a e. P ( ( M ` a ) = B -> a = ( M ` B ) ) ) -> E! a e. P ( M ` a ) = B )
24	10, 11, 20, 23	syl3anc 1326	1 |- ( ph -> E! a e. P ( M ` a ) = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   ` cfv 5888   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: (None)