Theorem mircgrextend 25577

Description: Link congruence over a pair of mirror points. cf tgcgrextend 25380. (Contributed by Thierry Arnoux, 4-Oct-2020.)

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 )
mirtrcgr.e	|- .~ = (cgrG ` G )
mirtrcgr.m	|- M = ( S ` B )
mirtrcgr.n	|- N = ( S ` Y )
mirtrcgr.a	|- ( ph -> A e. P )
mirtrcgr.b	|- ( ph -> B e. P )
mirtrcgr.x	|- ( ph -> X e. P )
mirtrcgr.y	|- ( ph -> Y e. P )
mircgrextend.1	|- ( ph -> ( A .- B ) = ( X .- Y ) )

Assertion

Ref	Expression
mircgrextend	|- ( ph -> ( A .- ( M ` A ) ) = ( X .- ( N ` X ) ) )

Proof of Theorem mircgrextend

Step	Hyp	Ref	Expression
1		mirval.p	. 2 |- P = ( Base ` G )
2		mirval.d	. 2 |- .- = ( dist ` G )
3		mirval.i	. 2 |- I = (Itv ` G )
4		mirval.g	. 2 |- ( ph -> G e. TarskiG )
5		mirtrcgr.a	. 2 |- ( ph -> A e. P )
6		mirtrcgr.b	. 2 |- ( ph -> B e. P )
7		mirval.l	. . 3 |- L = (LineG ` G )
8		mirval.s	. . 3 |- S = (pInvG ` G )
9		mirtrcgr.m	. . 3 |- M = ( S ` B )
10	1, 2, 3, 7, 8, 4, 6, 9, 5	mircl 25556	. 2 |- ( ph -> ( M ` A ) e. P )
11		mirtrcgr.x	. 2 |- ( ph -> X e. P )
12		mirtrcgr.y	. 2 |- ( ph -> Y e. P )
13		mirtrcgr.n	. . 3 |- N = ( S ` Y )
14	1, 2, 3, 7, 8, 4, 12, 13, 11	mircl 25556	. 2 |- ( ph -> ( N ` X ) e. P )
15	1, 2, 3, 7, 8, 4, 6, 9, 5	mirbtwn 25553	. . 3 |- ( ph -> B e. ( ( M ` A ) I A ) )
16	1, 2, 3, 4, 10, 6, 5, 15	tgbtwncom 25383	. 2 |- ( ph -> B e. ( A I ( M ` A ) ) )
17	1, 2, 3, 7, 8, 4, 12, 13, 11	mirbtwn 25553	. . 3 |- ( ph -> Y e. ( ( N ` X ) I X ) )
18	1, 2, 3, 4, 14, 12, 11, 17	tgbtwncom 25383	. 2 |- ( ph -> Y e. ( X I ( N ` X ) ) )
19		mircgrextend.1	. 2 |- ( ph -> ( A .- B ) = ( X .- Y ) )
20	1, 2, 3, 4, 5, 6, 11, 12, 19	tgcgrcomlr 25375	. . 3 |- ( ph -> ( B .- A ) = ( Y .- X ) )
21	1, 2, 3, 7, 8, 4, 6, 9, 5	mircgr 25552	. . 3 |- ( ph -> ( B .- ( M ` A ) ) = ( B .- A ) )
22	1, 2, 3, 7, 8, 4, 12, 13, 11	mircgr 25552	. . 3 |- ( ph -> ( Y .- ( N ` X ) ) = ( Y .- X ) )
23	20, 21, 22	3eqtr4d 2666	. 2 |- ( ph -> ( B .- ( M ` A ) ) = ( Y .- ( N ` X ) ) )
24	1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23	tgcgrextend 25380	1 |- ( ph -> ( A .- ( M ` A ) ) = ( X .- ( N ` X ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  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:  mirtrcgr  25578