Theorem lmimid 25686

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 6193	. . . . 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 2622	. . . . . . 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 25444	. . . . . . 7 |- ( ph -> B e. P )
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 25556	. . . . . 6 |- ( ph -> ( S ` C ) e. P )
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 25670	. . . . 5 |- ( ph -> ( ( S ` C ) = ( ( (pInvG ` G ) ` B ) ` C ) <-> ( C (midG ` G ) ( S ` C ) ) = B ) )
17	3, 16	mpbid 222	. . . 4 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) = B )
18	17, 13	eqeltrd 2701	. . 3 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. D )
19		df-ne 2795	. . . . . 6 |- ( C =/= ( S ` C ) <-> -. C = ( S ` C ) )
20	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> G e. TarskiG )
21	12	adantr 481	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> D e. ran L )
22	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. P )
23	15	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( S ` C ) e. P )
24		simpr 477	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= ( S ` C ) )
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 25525	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( C L ( S ` C ) ) e. ran L )
26	13	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. D )
27	14	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. P )
28	4, 5, 6, 7, 8, 9, 15	midbtwn 25671	. . . . . . . . . . . 12 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. ( C I ( S ` C ) ) )
29	17, 28	eqeltrrd 2702	. . . . . . . . . . 11 |- ( ph -> B e. ( C I ( S ` C ) ) )
30	29	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C I ( S ` C ) ) )
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 25514	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C L ( S ` C ) ) )
32	26, 31	elind 3798	. . . . . . . 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 481	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A e. D )
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 25528	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. ( C L ( S ` C ) ) )
36		lmimid.d	. . . . . . . . 9 |- ( ph -> A =/= B )
37	36	adantr 481	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A =/= B )
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 25561	. . . . . . . . . . . . . 14 |- ( ph -> ( ( S ` C ) = C <-> B = C ) )
39		eqcom 2629	. . . . . . . . . . . . . 14 |- ( B = C <-> C = B )
40	38, 39	syl6bb 276	. . . . . . . . . . . . 13 |- ( ph -> ( ( S ` C ) = C <-> C = B ) )
41	40	biimpar 502	. . . . . . . . . . . 12 |- ( ( ph /\ C = B ) -> ( S ` C ) = C )
42	41	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ph /\ C = B ) -> C = ( S ` C ) )
43	42	ex 450	. . . . . . . . . 10 |- ( ph -> ( C = B -> C = ( S ` C ) ) )
44	43	necon3d 2815	. . . . . . . . 9 |- ( ph -> ( C =/= ( S ` C ) -> C =/= B ) )
45	44	imp 445	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= B )
46		lmimid.r	. . . . . . . . 9 |- ( ph -> <" A B C "> e. (∟G ` G ) )
47	46	adantr 481	. . . . . . . 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 25612	. . . . . . 7 |- ( ( ph /\ C =/= ( S ` C ) ) -> D (⟂G ` G ) ( C L ( S ` C ) ) )
49	48	ex 450	. . . . . 6 |- ( ph -> ( C =/= ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
50	19, 49	syl5bir 233	. . . . 5 |- ( ph -> ( -. C = ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
51	50	orrd 393	. . . 4 |- ( ph -> ( C = ( S ` C ) \/ D (⟂G ` G ) ( C L ( S ` C ) ) ) )
52	51	orcomd 403	. . 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 25679	. . 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 957	. 2 |- ( ph -> ( S ` C ) = ( M ` C ) )
56	55	eqcomd 2628	1 |- ( ph -> ( M ` C ) = ( S ` C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   2c2 11070   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588  ⟂Gcperpg 25590  midGcmid 25664  lInvGclmi 25665

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkgld 25351  df-trkg 25352  df-cgrg 25406  df-leg 25478  df-mir 25548  df-rag 25589  df-perpg 25591  df-mid 25666  df-lmi 25667

This theorem is referenced by:  hypcgrlem1  25691