Theorem lmimid 24848

Description: If we have a right angle, then the mirror point is the point inversion. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
ismid.p	|- P = ( Base ` G )
ismid.d	|- .- = ( dist ` G )
ismid.i	|- I = (Itv ` G )
ismid.g	|- ( ph -> G e. TarskiG )
ismid.1	|- ( ph -> GDimTarskiG≥ 2 )
lmif.m	|- M = ( (lInvG ` G ) ` D )
lmif.l	|- L = (LineG ` G )
lmif.d	|- ( ph -> D e. ran L )
lmicl.1	|- ( ph -> A e. P )
lmimid.s	|- S = ( (pInvG ` G ) ` B )
lmimid.r	|- ( ph -> <" A B C "> e. (∟G ` G ) )
lmimid.a	|- ( ph -> A e. D )
lmimid.b	|- ( ph -> B e. D )
lmimid.c	|- ( ph -> C e. P )
lmimid.d	|- ( ph -> A =/= B )

Assertion

Ref	Expression
lmimid	|- ( ph -> ( M ` C ) = ( S ` C ) )

Proof of Theorem lmimid

Step	Hyp	Ref	Expression
1		lmimid.s	. . . . . . 7 |- S = ( (pInvG ` G ) ` B )
2	1	a1i 11	. . . . . 6 |- ( ph -> S = ( (pInvG ` G ) ` B ) )
3	2	fveq1d 5872	. . . . 5 |- ( ph -> ( S ` C ) = ( ( (pInvG ` G ) ` B ) ` C ) )
4		ismid.p	. . . . . 6 |- P = ( Base ` G )
5		ismid.d	. . . . . 6 |- .- = ( dist ` G )
6		ismid.i	. . . . . 6 |- I = (Itv ` G )
7		ismid.g	. . . . . 6 |- ( ph -> G e. TarskiG )
8		ismid.1	. . . . . 6 |- ( ph -> GDimTarskiG≥ 2 )
9		lmimid.c	. . . . . 6 |- ( ph -> C e. P )
10		lmif.l	. . . . . . 7 |- L = (LineG ` G )
11		eqid 2453	. . . . . . 7 |- (pInvG ` G ) = (pInvG ` G )
12		lmif.d	. . . . . . . 8 |- ( ph -> D e. ran L )
13		lmimid.b	. . . . . . . 8 |- ( ph -> B e. D )
14	4, 10, 6, 7, 12, 13	tglnpt 24606	. . . . . . 7 |- ( ph -> B e. P )
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 24718	. . . . . 6 |- ( ph -> ( S ` C ) e. P )
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 24832	. . . . 5 |- ( ph -> ( ( S ` C ) = ( ( (pInvG ` G ) ` B ) ` C ) <-> ( C (midG ` G ) ( S ` C ) ) = B ) )
17	3, 16	mpbid 214	. . . 4 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) = B )
18	17, 13	eqeltrd 2531	. . 3 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. D )
19		df-ne 2626	. . . . . 6 |- ( C =/= ( S ` C ) <-> -. C = ( S ` C ) )
20	7	adantr 467	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> G e. TarskiG )
21	12	adantr 467	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> D e. ran L )
22	9	adantr 467	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. P )
23	15	adantr 467	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( S ` C ) e. P )
24		simpr 463	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= ( S ` C ) )
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 24687	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( C L ( S ` C ) ) e. ran L )
26	13	adantr 467	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. D )
27	14	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. P )
28	4, 5, 6, 7, 8, 9, 15	midbtwn 24833	. . . . . . . . . . . 12 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. ( C I ( S ` C ) ) )
29	17, 28	eqeltrrd 2532	. . . . . . . . . . 11 |- ( ph -> B e. ( C I ( S ` C ) ) )
30	29	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C I ( S ` C ) ) )
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 24676	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C L ( S ` C ) ) )
32	26, 31	elind 3620	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( D i^i ( C L ( S ` C ) ) ) )
33		lmimid.a	. . . . . . . . 9 |- ( ph -> A e. D )
34	33	adantr 467	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A e. D )
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 24690	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. ( C L ( S ` C ) ) )
36		lmimid.d	. . . . . . . . 9 |- ( ph -> A =/= B )
37	36	adantr 467	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A =/= B )
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 24723	. . . . . . . . . . . . . 14 |- ( ph -> ( ( S ` C ) = C <-> B = C ) )
39		eqcom 2460	. . . . . . . . . . . . . 14 |- ( B = C <-> C = B )
40	38, 39	syl6bb 265	. . . . . . . . . . . . 13 |- ( ph -> ( ( S ` C ) = C <-> C = B ) )
41	40	biimpar 488	. . . . . . . . . . . 12 |- ( ( ph /\ C = B ) -> ( S ` C ) = C )
42	41	eqcomd 2459	. . . . . . . . . . 11 |- ( ( ph /\ C = B ) -> C = ( S ` C ) )
43	42	ex 436	. . . . . . . . . 10 |- ( ph -> ( C = B -> C = ( S ` C ) ) )
44	43	necon3d 2647	. . . . . . . . 9 |- ( ph -> ( C =/= ( S ` C ) -> C =/= B ) )
45	44	imp 431	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= B )
46		lmimid.r	. . . . . . . . 9 |- ( ph -> <" A B C "> e. (∟G ` G ) )
47	46	adantr 467	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> <" A B C "> e. (∟G ` G ) )
48	4, 5, 6, 10, 20, 21, 25, 32, 34, 35, 37, 45, 47	ragperp 24774	. . . . . . 7 |- ( ( ph /\ C =/= ( S ` C ) ) -> D (⟂G ` G ) ( C L ( S ` C ) ) )
49	48	ex 436	. . . . . 6 |- ( ph -> ( C =/= ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
50	19, 49	syl5bir 222	. . . . 5 |- ( ph -> ( -. C = ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
51	50	orrd 380	. . . 4 |- ( ph -> ( C = ( S ` C ) \/ D (⟂G ` G ) ( C L ( S ` C ) ) ) )
52	51	orcomd 390	. . 3 |- ( ph -> ( D (⟂G ` G ) ( C L ( S ` C ) ) \/ C = ( S ` C ) ) )
53		lmif.m	. . . 4 |- M = ( (lInvG ` G ) ` D )
54	4, 5, 6, 7, 8, 53, 10, 12, 9, 15	islmib 24841	. . 3 |- ( ph -> ( ( S ` C ) = ( M ` C ) <-> ( ( C (midG ` G ) ( S ` C ) ) e. D /\ ( D (⟂G ` G ) ( C L ( S ` C ) ) \/ C = ( S ` C ) ) ) ) )
55	18, 52, 54	mpbir2and 934	. 2 |- ( ph -> ( S ` C ) = ( M ` C ) )
56	55	eqcomd 2459	1 |- ( ph -> ( M ` C ) = ( S ` C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 370   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   class class class wbr 4405   ran crn 4838   ` cfv 5585  (class class class)co 6295   2c2 10666   <"cs3 12945   Basecbs 15133   distcds 15211  TarskiGcstrkg 24490  Dim_TarskiG≥cstrkgld 24494  Itvcitv 24496  LineGclng 24497  pInvGcmir 24709  ∟Gcrag 24750  ⟂Gcperpg 24752  midGcmid 24826  lInvGclmi 24827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-concat 12673  df-s1 12674  df-s2 12951  df-s3 12952  df-trkgc 24508  df-trkgb 24509  df-trkgcb 24510  df-trkgld 24512  df-trkg 24513  df-cgrg 24568  df-leg 24640  df-mir 24710  df-rag 24751  df-perpg 24753  df-mid 24828  df-lmi 24829

This theorem is referenced by:  hypcgrlem1  24853