Theorem islmib 25679

Description: Property of the line mirror. (Contributed by Thierry Arnoux, 11-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 )
islmib.b	|- ( ph -> B e. P )

Assertion

Ref	Expression
islmib	|- ( ph -> ( B = ( M ` A ) <-> ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) ) )

Proof of Theorem islmib

Dummy variables a b g d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmif.m	. . . . 5 |- M = ( (lInvG ` G ) ` D )
2		df-lmi 25667	. . . . . . . 8 |- lInvG = ( g e. _V |-> ( d e. ran (LineG ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) ) )
3	2	a1i 11	. . . . . . 7 |- ( ph -> lInvG = ( g e. _V |-> ( d e. ran (LineG ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) ) ) )
4		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> (LineG ` g ) = (LineG ` G ) )
5		lmif.l	. . . . . . . . . . 11 |- L = (LineG ` G )
6	4, 5	syl6eqr 2674	. . . . . . . . . 10 |- ( g = G -> (LineG ` g ) = L )
7	6	rneqd 5353	. . . . . . . . 9 |- ( g = G -> ran (LineG ` g ) = ran L )
8		fveq2 6191	. . . . . . . . . . 11 |- ( g = G -> ( Base ` g ) = ( Base ` G ) )
9		ismid.p	. . . . . . . . . . 11 |- P = ( Base ` G )
10	8, 9	syl6eqr 2674	. . . . . . . . . 10 |- ( g = G -> ( Base ` g ) = P )
11		fveq2 6191	. . . . . . . . . . . . . 14 |- ( g = G -> (midG ` g ) = (midG ` G ) )
12	11	oveqd 6667	. . . . . . . . . . . . 13 |- ( g = G -> ( a (midG ` g ) b ) = ( a (midG ` G ) b ) )
13	12	eleq1d 2686	. . . . . . . . . . . 12 |- ( g = G -> ( ( a (midG ` g ) b ) e. d <-> ( a (midG ` G ) b ) e. d ) )
14		eqidd 2623	. . . . . . . . . . . . . 14 |- ( g = G -> d = d )
15		fveq2 6191	. . . . . . . . . . . . . 14 |- ( g = G -> (⟂G ` g ) = (⟂G ` G ) )
16	6	oveqd 6667	. . . . . . . . . . . . . 14 |- ( g = G -> ( a (LineG ` g ) b ) = ( a L b ) )
17	14, 15, 16	breq123d 4667	. . . . . . . . . . . . 13 |- ( g = G -> ( d (⟂G ` g ) ( a (LineG ` g ) b ) <-> d (⟂G ` G ) ( a L b ) ) )
18	17	orbi1d 739	. . . . . . . . . . . 12 |- ( g = G -> ( ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) <-> ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) )
19	13, 18	anbi12d 747	. . . . . . . . . . 11 |- ( g = G -> ( ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) <-> ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) )
20	10, 19	riotaeqbidv 6614	. . . . . . . . . 10 |- ( g = G -> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) = ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) )
21	10, 20	mpteq12dv 4733	. . . . . . . . 9 |- ( g = G -> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) = ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) )
22	7, 21	mpteq12dv 4733	. . . . . . . 8 |- ( g = G -> ( d e. ran (LineG ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) ) = ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) )
23	22	adantl 482	. . . . . . 7 |- ( ( ph /\ g = G ) -> ( d e. ran (LineG ` g ) |-> ( a e. ( Base ` g ) |-> ( iota_ b e. ( Base ` g ) ( ( a (midG ` g ) b ) e. d /\ ( d (⟂G ` g ) ( a (LineG ` g ) b ) \/ a = b ) ) ) ) ) = ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) )
24		ismid.g	. . . . . . . 8 |- ( ph -> G e. TarskiG )
25		elex 3212	. . . . . . . 8 |- ( G e. TarskiG -> G e. _V )
26	24, 25	syl 17	. . . . . . 7 |- ( ph -> G e. _V )
27		fvex 6201	. . . . . . . . . 10 |- (LineG ` G ) e. _V
28	5, 27	eqeltri 2697	. . . . . . . . 9 |- L e. _V
29		rnexg 7098	. . . . . . . . 9 |- ( L e. _V -> ran L e. _V )
30		mptexg 6484	. . . . . . . . 9 |- ( ran L e. _V -> ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) e. _V )
31	28, 29, 30	mp2b 10	. . . . . . . 8 |- ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) e. _V
32	31	a1i 11	. . . . . . 7 |- ( ph -> ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) e. _V )
33	3, 23, 26, 32	fvmptd 6288	. . . . . 6 |- ( ph -> (lInvG ` G ) = ( d e. ran L |-> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) ) )
34		eleq2 2690	. . . . . . . . . 10 |- ( d = D -> ( ( a (midG ` G ) b ) e. d <-> ( a (midG ` G ) b ) e. D ) )
35		breq1 4656	. . . . . . . . . . 11 |- ( d = D -> ( d (⟂G ` G ) ( a L b ) <-> D (⟂G ` G ) ( a L b ) ) )
36	35	orbi1d 739	. . . . . . . . . 10 |- ( d = D -> ( ( d (⟂G ` G ) ( a L b ) \/ a = b ) <-> ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) )
37	34, 36	anbi12d 747	. . . . . . . . 9 |- ( d = D -> ( ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) <-> ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) )
38	37	riotabidv 6613	. . . . . . . 8 |- ( d = D -> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) = ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) )
39	38	mpteq2dv 4745	. . . . . . 7 |- ( d = D -> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) = ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) )
40	39	adantl 482	. . . . . 6 |- ( ( ph /\ d = D ) -> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. d /\ ( d (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) = ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) )
41		lmif.d	. . . . . 6 |- ( ph -> D e. ran L )
42		fvex 6201	. . . . . . . . 9 |- ( Base ` G ) e. _V
43	9, 42	eqeltri 2697	. . . . . . . 8 |- P e. _V
44	43	mptex 6486	. . . . . . 7 |- ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) e. _V
45	44	a1i 11	. . . . . 6 |- ( ph -> ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) e. _V )
46	33, 40, 41, 45	fvmptd 6288	. . . . 5 |- ( ph -> ( (lInvG ` G ) ` D ) = ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) )
47	1, 46	syl5eq 2668	. . . 4 |- ( ph -> M = ( a e. P |-> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) ) )
48		oveq1 6657	. . . . . . . 8 |- ( a = A -> ( a (midG ` G ) b ) = ( A (midG ` G ) b ) )
49	48	eleq1d 2686	. . . . . . 7 |- ( a = A -> ( ( a (midG ` G ) b ) e. D <-> ( A (midG ` G ) b ) e. D ) )
50		oveq1 6657	. . . . . . . . 9 |- ( a = A -> ( a L b ) = ( A L b ) )
51	50	breq2d 4665	. . . . . . . 8 |- ( a = A -> ( D (⟂G ` G ) ( a L b ) <-> D (⟂G ` G ) ( A L b ) ) )
52		eqeq1 2626	. . . . . . . 8 |- ( a = A -> ( a = b <-> A = b ) )
53	51, 52	orbi12d 746	. . . . . . 7 |- ( a = A -> ( ( D (⟂G ` G ) ( a L b ) \/ a = b ) <-> ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
54	49, 53	anbi12d 747	. . . . . 6 |- ( a = A -> ( ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) <-> ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) )
55	54	riotabidv 6613	. . . . 5 |- ( a = A -> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) )
56	55	adantl 482	. . . 4 |- ( ( ph /\ a = A ) -> ( iota_ b e. P ( ( a (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( a L b ) \/ a = b ) ) ) = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) )
57		lmicl.1	. . . 4 |- ( ph -> A e. P )
58		ismid.d	. . . . . 6 |- .- = ( dist ` G )
59		ismid.i	. . . . . 6 |- I = (Itv ` G )
60		ismid.1	. . . . . 6 |- ( ph -> GDimTarskiG≥ 2 )
61	9, 58, 59, 24, 60, 5, 41, 57	lmieu 25676	. . . . 5 |- ( ph -> E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) )
62		riotacl 6625	. . . . 5 |- ( E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) -> ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) e. P )
63	61, 62	syl 17	. . . 4 |- ( ph -> ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) e. P )
64	47, 56, 57, 63	fvmptd 6288	. . 3 |- ( ph -> ( M ` A ) = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) )
65	64	eqeq2d 2632	. 2 |- ( ph -> ( B = ( M ` A ) <-> B = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) ) )
66		islmib.b	. . . 4 |- ( ph -> B e. P )
67		oveq2 6658	. . . . . . 7 |- ( b = B -> ( A (midG ` G ) b ) = ( A (midG ` G ) B ) )
68	67	eleq1d 2686	. . . . . 6 |- ( b = B -> ( ( A (midG ` G ) b ) e. D <-> ( A (midG ` G ) B ) e. D ) )
69		oveq2 6658	. . . . . . . 8 |- ( b = B -> ( A L b ) = ( A L B ) )
70	69	breq2d 4665	. . . . . . 7 |- ( b = B -> ( D (⟂G ` G ) ( A L b ) <-> D (⟂G ` G ) ( A L B ) ) )
71		eqeq2 2633	. . . . . . 7 |- ( b = B -> ( A = b <-> A = B ) )
72	70, 71	orbi12d 746	. . . . . 6 |- ( b = B -> ( ( D (⟂G ` G ) ( A L b ) \/ A = b ) <-> ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) )
73	68, 72	anbi12d 747	. . . . 5 |- ( b = B -> ( ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) <-> ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) ) )
74	73	riota2 6633	. . . 4 |- ( ( B e. P /\ E! b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) -> ( ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) <-> ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) = B ) )
75	66, 61, 74	syl2anc 693	. . 3 |- ( ph -> ( ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) <-> ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) = B ) )
76		eqcom 2629	. . 3 |- ( B = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) <-> ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) = B )
77	75, 76	syl6bbr 278	. 2 |- ( ph -> ( ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) <-> B = ( iota_ b e. P ( ( A (midG ` G ) b ) e. D /\ ( D (⟂G ` G ) ( A L b ) \/ A = b ) ) ) ) )
78	65, 77	bitr4d 271	1 |- ( ph -> ( B = ( M ` A ) <-> ( ( A (midG ` G ) B ) e. D /\ ( D (⟂G ` G ) ( A L B ) \/ A = B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   E!wreu 2914   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888   iota_crio 6610  (class class class)co 6650   2c2 11070   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Dim_TarskiG≥cstrkgld 25333  Itvcitv 25335  LineGclng 25336  ⟂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:  lmicom  25680  lmiinv  25684  lmimid  25686  lmiisolem  25688  hypcgrlem1  25691  hypcgrlem2  25692  lmiopp  25694  trgcopyeulem  25697