Theorem lmimid 24376

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 5874	. . . . 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 2457	. . . . . . 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 24153	. . . . . . 7 |- ( ph -> B e. P )
15	4, 5, 6, 10, 11, 7, 14, 1, 9	mircl 24259	. . . . . 6 |- ( ph -> ( S ` C ) e. P )
16	4, 5, 6, 7, 8, 9, 15, 11, 14	ismidb 24361	. . . . 5 |- ( ph -> ( ( S ` C ) = ( ( (pInvG ` G ) ` B ) ` C ) <-> ( C (midG ` G ) ( S ` C ) ) = B ) )
17	3, 16	mpbid 210	. . . 4 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) = B )
18	17, 13	eqeltrd 2545	. . 3 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. D )
19		df-ne 2654	. . . . . 6 |- ( C =/= ( S ` C ) <-> -. C = ( S ` C ) )
20	7	adantr 465	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> G e. TarskiG )
21	12	adantr 465	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> D e. ran L )
22	9	adantr 465	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. P )
23	15	adantr 465	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( S ` C ) e. P )
24		simpr 461	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= ( S ` C ) )
25	4, 6, 10, 20, 22, 23, 24	tgelrnln 24227	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> ( C L ( S ` C ) ) e. ran L )
26	13	adantr 465	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. D )
27	14	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. P )
28	4, 5, 6, 7, 8, 9, 15	midbtwn 24362	. . . . . . . . . . . 12 |- ( ph -> ( C (midG ` G ) ( S ` C ) ) e. ( C I ( S ` C ) ) )
29	17, 28	eqeltrrd 2546	. . . . . . . . . . 11 |- ( ph -> B e. ( C I ( S ` C ) ) )
30	29	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C I ( S ` C ) ) )
31	4, 6, 10, 20, 22, 23, 27, 24, 30	btwnlng1 24216	. . . . . . . . 9 |- ( ( ph /\ C =/= ( S ` C ) ) -> B e. ( C L ( S ` C ) ) )
32	26, 31	elind 3684	. . . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A e. D )
35	4, 6, 10, 20, 22, 23, 24	tglinerflx1 24230	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C e. ( C L ( S ` C ) ) )
36		lmimid.d	. . . . . . . . 9 |- ( ph -> A =/= B )
37	36	adantr 465	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> A =/= B )
38	4, 5, 6, 10, 11, 7, 14, 1, 9	mirinv 24264	. . . . . . . . . . . . . 14 |- ( ph -> ( ( S ` C ) = C <-> B = C ) )
39		eqcom 2466	. . . . . . . . . . . . . 14 |- ( B = C <-> C = B )
40	38, 39	syl6bb 261	. . . . . . . . . . . . 13 |- ( ph -> ( ( S ` C ) = C <-> C = B ) )
41	40	biimpar 485	. . . . . . . . . . . 12 |- ( ( ph /\ C = B ) -> ( S ` C ) = C )
42	41	eqcomd 2465	. . . . . . . . . . 11 |- ( ( ph /\ C = B ) -> C = ( S ` C ) )
43	42	ex 434	. . . . . . . . . 10 |- ( ph -> ( C = B -> C = ( S ` C ) ) )
44	43	necon3d 2681	. . . . . . . . 9 |- ( ph -> ( C =/= ( S ` C ) -> C =/= B ) )
45	44	imp 429	. . . . . . . 8 |- ( ( ph /\ C =/= ( S ` C ) ) -> C =/= B )
46		lmimid.r	. . . . . . . . 9 |- ( ph -> <" A B C "> e. (∟G ` G ) )
47	46	adantr 465	. . . . . . . 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 24311	. . . . . . 7 |- ( ( ph /\ C =/= ( S ` C ) ) -> D (⟂G ` G ) ( C L ( S ` C ) ) )
49	48	ex 434	. . . . . 6 |- ( ph -> ( C =/= ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
50	19, 49	syl5bir 218	. . . . 5 |- ( ph -> ( -. C = ( S ` C ) -> D (⟂G ` G ) ( C L ( S ` C ) ) ) )
51	50	orrd 378	. . . 4 |- ( ph -> ( C = ( S ` C ) \/ D (⟂G ` G ) ( C L ( S ` C ) ) ) )
52	51	orcomd 388	. . 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 24370	. . 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 922	. 2 |- ( ph -> ( S ` C ) = ( M ` C ) )
56	55	eqcomd 2465	1 |- ( ph -> ( M ` C ) = ( S ` C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   class class class wbr 4456   ran crn 5009   ` cfv 5594  (class class class)co 6296   2c2 10606   <"cs3 12819   Basecbs 14735   distcds 14812  TarskiGcstrkg 24042  Dim_TarskiG≥cstrkgld 24046  Itvcitv 24049  LineGclng 24050  pInvGcmir 24250  ∟Gcrag 24287  ⟂Gcperpg 24289  midGcmid 24355  lInvGclmi 24356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 24061  df-trkgb 24062  df-trkgcb 24063  df-trkgld 24065  df-trkg 24067  df-cgrg 24120  df-leg 24187  df-mir 24251  df-rag 24288  df-perpg 24290  df-mid 24357  df-lmi 24358

This theorem is referenced by:  hypcgrlem1  24381