Theorem mirfv 25551

Description: Value of the point inversion function M. Definition 7.5 of [Schwabhauser] p. 49. (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 )
mirfv.b	|- ( ph -> B e. P )

Assertion

Ref	Expression
mirfv	|- ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )

Distinct variable groups:   z, A   z, B   z, G   z, M   z, I   z, P   ph, z   z, .-

Allowed substitution hints:   S( z)   L( z)

Proof of Theorem mirfv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirfv.m	. . 3 |- M = ( S ` A )
2		mirval.p	. . . 4 |- P = ( Base ` G )
3		mirval.d	. . . 4 |- .- = ( dist ` G )
4		mirval.i	. . . 4 |- I = (Itv ` G )
5		mirval.l	. . . 4 |- L = (LineG ` G )
6		mirval.s	. . . 4 |- S = (pInvG ` G )
7		mirval.g	. . . 4 |- ( ph -> G e. TarskiG )
8		mirval.a	. . . 4 |- ( ph -> A e. P )
9	2, 3, 4, 5, 6, 7, 8	mirval 25550	. . 3 |- ( ph -> ( S ` A ) = ( y e. P |-> ( iota_ z e. P ( ( A .- z ) = ( A .- y ) /\ A e. ( z I y ) ) ) ) )
10	1, 9	syl5eq 2668	. 2 |- ( ph -> M = ( y e. P |-> ( iota_ z e. P ( ( A .- z ) = ( A .- y ) /\ A e. ( z I y ) ) ) ) )
11		simplr 792	. . . . . 6 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> y = B )
12	11	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> ( A .- y ) = ( A .- B ) )
13	12	eqeq2d 2632	. . . 4 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> ( ( A .- z ) = ( A .- y ) <-> ( A .- z ) = ( A .- B ) ) )
14	11	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> ( z I y ) = ( z I B ) )
15	14	eleq2d 2687	. . . 4 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> ( A e. ( z I y ) <-> A e. ( z I B ) ) )
16	13, 15	anbi12d 747	. . 3 |- ( ( ( ph /\ y = B ) /\ z e. P ) -> ( ( ( A .- z ) = ( A .- y ) /\ A e. ( z I y ) ) <-> ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
17	16	riotabidva 6627	. 2 |- ( ( ph /\ y = B ) -> ( iota_ z e. P ( ( A .- z ) = ( A .- y ) /\ A e. ( z I y ) ) ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )
18		mirfv.b	. 2 |- ( ph -> B e. P )
19		riotaex 6615	. . 3 |- ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) e. _V
20	19	a1i 11	. 2 |- ( ph -> ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) e. _V )
21	10, 17, 18, 20	fvmptd 6288	1 |- ( ph -> ( M ` B ) = ( iota_ z e. P ( ( A .- z ) = ( A .- B ) /\ A e. ( z I B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` 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-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-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-mir 25548

This theorem is referenced by:  mircgr  25552  mirbtwn  25553  ismir  25554  mirf  25555  mireq  25560