Theorem mirbtwnhl 25575

Description: If the center of the point inversion A is between two points X and Y, then the half lines are mirrored. (Contributed by Thierry Arnoux, 3-Mar-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 )
mirhl.m	|- M = ( S ` A )
mirhl.k	|- K = (hlG ` G )
mirhl.a	|- ( ph -> A e. P )
mirhl.x	|- ( ph -> X e. P )
mirhl.y	|- ( ph -> Y e. P )
mirhl.z	|- ( ph -> Z e. P )
mirbtwnhl.1	|- ( ph -> X =/= A )
mirbtwnhl.2	|- ( ph -> Y =/= A )
mirbtwnhl.3	|- ( ph -> A e. ( X I Y ) )

Assertion

Ref	Expression
mirbtwnhl	|- ( ph -> ( Z ( K ` A ) X <-> ( M ` Z ) ( K ` A ) Y ) )

Proof of Theorem mirbtwnhl

Step	Hyp	Ref	Expression
1		mirval.p	. . . . . 6 |- P = ( Base ` G )
2		mirval.i	. . . . . 6 |- I = (Itv ` G )
3		mirhl.k	. . . . . 6 |- K = (hlG ` G )
4		mirhl.a	. . . . . 6 |- ( ph -> A e. P )
5		mirhl.x	. . . . . 6 |- ( ph -> X e. P )
6		mirval.g	. . . . . 6 |- ( ph -> G e. TarskiG )
7	1, 2, 3, 4, 5, 4, 6	hleqnid 25503	. . . . 5 |- ( ph -> -. A ( K ` A ) X )
8	7	adantr 481	. . . 4 |- ( ( ph /\ Z = A ) -> -. A ( K ` A ) X )
9		simpr 477	. . . . 5 |- ( ( ph /\ Z = A ) -> Z = A )
10	9	breq1d 4663	. . . 4 |- ( ( ph /\ Z = A ) -> ( Z ( K ` A ) X <-> A ( K ` A ) X ) )
11	8, 10	mtbird 315	. . 3 |- ( ( ph /\ Z = A ) -> -. Z ( K ` A ) X )
12		mirhl.y	. . . . . 6 |- ( ph -> Y e. P )
13	1, 2, 3, 4, 12, 4, 6	hleqnid 25503	. . . . 5 |- ( ph -> -. A ( K ` A ) Y )
14	13	adantr 481	. . . 4 |- ( ( ph /\ Z = A ) -> -. A ( K ` A ) Y )
15	9	fveq2d 6195	. . . . . 6 |- ( ( ph /\ Z = A ) -> ( M ` Z ) = ( M ` A ) )
16		mirval.d	. . . . . . . 8 |- .- = ( dist ` G )
17		mirval.l	. . . . . . . 8 |- L = (LineG ` G )
18		mirval.s	. . . . . . . 8 |- S = (pInvG ` G )
19		mirhl.m	. . . . . . . 8 |- M = ( S ` A )
20	1, 16, 2, 17, 18, 6, 4, 19	mircinv 25563	. . . . . . 7 |- ( ph -> ( M ` A ) = A )
21	20	adantr 481	. . . . . 6 |- ( ( ph /\ Z = A ) -> ( M ` A ) = A )
22	15, 21	eqtrd 2656	. . . . 5 |- ( ( ph /\ Z = A ) -> ( M ` Z ) = A )
23	22	breq1d 4663	. . . 4 |- ( ( ph /\ Z = A ) -> ( ( M ` Z ) ( K ` A ) Y <-> A ( K ` A ) Y ) )
24	14, 23	mtbird 315	. . 3 |- ( ( ph /\ Z = A ) -> -. ( M ` Z ) ( K ` A ) Y )
25	11, 24	2falsed 366	. 2 |- ( ( ph /\ Z = A ) -> ( Z ( K ` A ) X <-> ( M ` Z ) ( K ` A ) Y ) )
26		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> Z =/= A )
27	26	neneqd 2799	. . . . . . 7 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> -. Z = A )
28	6	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> G e. TarskiG )
29	4	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> A e. P )
30		mirhl.z	. . . . . . . . 9 |- ( ph -> Z e. P )
31	30	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> Z e. P )
32		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> ( M ` Z ) = A )
33	20	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> ( M ` A ) = A )
34	32, 33	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> ( M ` Z ) = ( M ` A ) )
35	1, 16, 2, 17, 18, 28, 29, 19, 31, 29, 34	mireq 25560	. . . . . . 7 |- ( ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) /\ ( M ` Z ) = A ) -> Z = A )
36	27, 35	mtand 691	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> -. ( M ` Z ) = A )
37	36	neqned 2801	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( M ` Z ) =/= A )
38		mirbtwnhl.2	. . . . . 6 |- ( ph -> Y =/= A )
39	38	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> Y =/= A )
40	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> G e. TarskiG )
41	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> X e. P )
42	4	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> A e. P )
43	1, 16, 2, 17, 18, 6, 4, 19, 30	mircl 25556	. . . . . . 7 |- ( ph -> ( M ` Z ) e. P )
44	43	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( M ` Z ) e. P )
45	12	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> Y e. P )
46		mirbtwnhl.1	. . . . . . 7 |- ( ph -> X =/= A )
47	46	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> X =/= A )
48	30	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> Z e. P )
49	1, 2, 3, 30, 5, 4, 6	ishlg 25497	. . . . . . . . . . 11 |- ( ph -> ( Z ( K ` A ) X <-> ( Z =/= A /\ X =/= A /\ ( Z e. ( A I X ) \/ X e. ( A I Z ) ) ) ) )
50	49	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ Z =/= A ) -> ( Z ( K ` A ) X <-> ( Z =/= A /\ X =/= A /\ ( Z e. ( A I X ) \/ X e. ( A I Z ) ) ) ) )
51	50	biimpa 501	. . . . . . . . 9 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( Z =/= A /\ X =/= A /\ ( Z e. ( A I X ) \/ X e. ( A I Z ) ) ) )
52	51	simp3d 1075	. . . . . . . 8 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( Z e. ( A I X ) \/ X e. ( A I Z ) ) )
53	52	orcomd 403	. . . . . . 7 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( X e. ( A I Z ) \/ Z e. ( A I X ) ) )
54	1, 16, 2, 17, 18, 40, 19, 42, 41, 48, 53	mirconn 25573	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> A e. ( X I ( M ` Z ) ) )
55		mirbtwnhl.3	. . . . . . 7 |- ( ph -> A e. ( X I Y ) )
56	55	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> A e. ( X I Y ) )
57	1, 2, 40, 41, 42, 44, 45, 47, 54, 56	tgbtwnconn2 25471	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) )
58	37, 39, 57	3jca 1242	. . . 4 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( ( M ` Z ) =/= A /\ Y =/= A /\ ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) ) )
59	1, 2, 3, 43, 12, 4, 6	ishlg 25497	. . . . . 6 |- ( ph -> ( ( M ` Z ) ( K ` A ) Y <-> ( ( M ` Z ) =/= A /\ Y =/= A /\ ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) ) ) )
60	59	adantr 481	. . . . 5 |- ( ( ph /\ Z =/= A ) -> ( ( M ` Z ) ( K ` A ) Y <-> ( ( M ` Z ) =/= A /\ Y =/= A /\ ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) ) ) )
61	60	adantr 481	. . . 4 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( ( M ` Z ) ( K ` A ) Y <-> ( ( M ` Z ) =/= A /\ Y =/= A /\ ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) ) ) )
62	58, 61	mpbird 247	. . 3 |- ( ( ( ph /\ Z =/= A ) /\ Z ( K ` A ) X ) -> ( M ` Z ) ( K ` A ) Y )
63		simplr 792	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> Z =/= A )
64	46	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> X =/= A )
65	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> G e. TarskiG )
66	12	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> Y e. P )
67	4	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> A e. P )
68	30	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> Z e. P )
69	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> X e. P )
70	38	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> Y =/= A )
71	20	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( M ` A ) = A )
72	43	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( M ` Z ) e. P )
73	1, 16, 2, 17, 18, 65, 67, 19, 66	mircl 25556	. . . . . . . . 9 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( M ` Y ) e. P )
74	60	biimpa 501	. . . . . . . . . . 11 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( ( M ` Z ) =/= A /\ Y =/= A /\ ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) ) )
75	74	simp3d 1075	. . . . . . . . . 10 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( ( M ` Z ) e. ( A I Y ) \/ Y e. ( A I ( M ` Z ) ) ) )
76	1, 16, 2, 17, 18, 65, 19, 67, 72, 66, 75	mirconn 25573	. . . . . . . . 9 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> A e. ( ( M ` Z ) I ( M ` Y ) ) )
77	1, 16, 2, 65, 72, 67, 73, 76	tgbtwncom 25383	. . . . . . . 8 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> A e. ( ( M ` Y ) I ( M ` Z ) ) )
78	71, 77	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( M ` A ) e. ( ( M ` Y ) I ( M ` Z ) ) )
79	1, 16, 2, 17, 18, 65, 67, 19, 66, 67, 68	mirbtwnb 25567	. . . . . . 7 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( A e. ( Y I Z ) <-> ( M ` A ) e. ( ( M ` Y ) I ( M ` Z ) ) ) )
80	78, 79	mpbird 247	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> A e. ( Y I Z ) )
81	1, 16, 2, 6, 5, 4, 12, 55	tgbtwncom 25383	. . . . . . 7 |- ( ph -> A e. ( Y I X ) )
82	81	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> A e. ( Y I X ) )
83	1, 2, 65, 66, 67, 68, 69, 70, 80, 82	tgbtwnconn2 25471	. . . . 5 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( Z e. ( A I X ) \/ X e. ( A I Z ) ) )
84	63, 64, 83	3jca 1242	. . . 4 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( Z =/= A /\ X =/= A /\ ( Z e. ( A I X ) \/ X e. ( A I Z ) ) ) )
85	50	adantr 481	. . . 4 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> ( Z ( K ` A ) X <-> ( Z =/= A /\ X =/= A /\ ( Z e. ( A I X ) \/ X e. ( A I Z ) ) ) ) )
86	84, 85	mpbird 247	. . 3 |- ( ( ( ph /\ Z =/= A ) /\ ( M ` Z ) ( K ` A ) Y ) -> Z ( K ` A ) X )
87	62, 86	impbida 877	. 2 |- ( ( ph /\ Z =/= A ) -> ( Z ( K ` A ) X <-> ( M ` Z ) ( K ` A ) Y ) )
88	25, 87	pm2.61dane 2881	1 |- ( ph -> ( Z ( K ` A ) X <-> ( M ` Z ) ( K ` A ) Y ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  hlGchlg 25495  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-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-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-trkg 25352  df-cgrg 25406  df-hlg 25496  df-mir 25548

This theorem is referenced by:  opphllem6  25644