Theorem mirconn 25573

Description: Point inversion of connectedness. (Contributed by Thierry Arnoux, 2-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 )
mirconn.m	|- M = ( S ` A )
mirconn.a	|- ( ph -> A e. P )
mirconn.x	|- ( ph -> X e. P )
mirconn.y	|- ( ph -> Y e. P )
mirconn.1	|- ( ph -> ( X e. ( A I Y ) \/ Y e. ( A I X ) ) )

Assertion

Ref	Expression
mirconn	|- ( ph -> A e. ( X I ( M ` Y ) ) )

Proof of Theorem mirconn

Step	Hyp	Ref	Expression
1		mirval.p	. . 3 |- P = ( Base ` G )
2		mirval.d	. . 3 |- .- = ( dist ` G )
3		mirval.i	. . 3 |- I = (Itv ` G )
4		mirval.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> G e. TarskiG )
6		mirconn.x	. . . 4 |- ( ph -> X e. P )
7	6	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> X e. P )
8		mirconn.a	. . . 4 |- ( ph -> A e. P )
9	8	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> A e. P )
10		mirval.l	. . . . 5 |- L = (LineG ` G )
11		mirval.s	. . . . 5 |- S = (pInvG ` G )
12		mirconn.m	. . . . 5 |- M = ( S ` A )
13		mirconn.y	. . . . 5 |- ( ph -> Y e. P )
14	1, 2, 3, 10, 11, 4, 8, 12, 13	mircl 25556	. . . 4 |- ( ph -> ( M ` Y ) e. P )
15	14	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> ( M ` Y ) e. P )
16	13	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> Y e. P )
17		simpr 477	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> X e. ( A I Y ) )
18	1, 2, 3, 10, 11, 4, 8, 12, 13	mirbtwn 25553	. . . 4 |- ( ph -> A e. ( ( M ` Y ) I Y ) )
19	18	adantr 481	. . 3 |- ( ( ph /\ X e. ( A I Y ) ) -> A e. ( ( M ` Y ) I Y ) )
20	1, 2, 3, 5, 7, 9, 15, 16, 17, 19	tgbtwnintr 25388	. 2 |- ( ( ph /\ X e. ( A I Y ) ) -> A e. ( X I ( M ` Y ) ) )
21	1, 2, 3, 4, 6, 8	tgbtwntriv2 25382	. . . . . 6 |- ( ph -> A e. ( X I A ) )
22	21	adantr 481	. . . . 5 |- ( ( ph /\ Y = A ) -> A e. ( X I A ) )
23		simpr 477	. . . . . . . 8 |- ( ( ph /\ Y = A ) -> Y = A )
24	23	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ Y = A ) -> ( M ` Y ) = ( M ` A ) )
25	1, 2, 3, 10, 11, 4, 8, 12	mircinv 25563	. . . . . . . 8 |- ( ph -> ( M ` A ) = A )
26	25	adantr 481	. . . . . . 7 |- ( ( ph /\ Y = A ) -> ( M ` A ) = A )
27	24, 26	eqtrd 2656	. . . . . 6 |- ( ( ph /\ Y = A ) -> ( M ` Y ) = A )
28	27	oveq2d 6666	. . . . 5 |- ( ( ph /\ Y = A ) -> ( X I ( M ` Y ) ) = ( X I A ) )
29	22, 28	eleqtrrd 2704	. . . 4 |- ( ( ph /\ Y = A ) -> A e. ( X I ( M ` Y ) ) )
30	29	adantlr 751	. . 3 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y = A ) -> A e. ( X I ( M ` Y ) ) )
31	4	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> G e. TarskiG )
32	6	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> X e. P )
33	13	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> Y e. P )
34	8	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> A e. P )
35	14	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> ( M ` Y ) e. P )
36		simpr 477	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> Y =/= A )
37		simplr 792	. . . . 5 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> Y e. ( A I X ) )
38	1, 2, 3, 31, 34, 33, 32, 37	tgbtwncom 25383	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> Y e. ( X I A ) )
39	1, 2, 3, 4, 14, 8, 13, 18	tgbtwncom 25383	. . . . 5 |- ( ph -> A e. ( Y I ( M ` Y ) ) )
40	39	ad2antrr 762	. . . 4 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> A e. ( Y I ( M ` Y ) ) )
41	1, 2, 3, 31, 32, 33, 34, 35, 36, 38, 40	tgbtwnouttr2 25390	. . 3 |- ( ( ( ph /\ Y e. ( A I X ) ) /\ Y =/= A ) -> A e. ( X I ( M ` Y ) ) )
42	30, 41	pm2.61dane 2881	. 2 |- ( ( ph /\ Y e. ( A I X ) ) -> A e. ( X I ( M ` Y ) ) )
43		mirconn.1	. 2 |- ( ph -> ( X e. ( A I Y ) \/ Y e. ( A I X ) ) )
44	20, 42, 43	mpjaodan 827	1 |- ( ph -> A e. ( X I ( M ` Y ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-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-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  mirbtwnhl  25575