Theorem colmid 25583

Description: Colinearity and equidistance implies midpoint. Theorem 7.20 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 30-Jul-2019.)

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 )
colmid.m	|- M = ( S ` X )
colmid.a	|- ( ph -> A e. P )
colmid.b	|- ( ph -> B e. P )
colmid.x	|- ( ph -> X e. P )
colmid.c	|- ( ph -> ( X e. ( A L B ) \/ A = B ) )
colmid.d	|- ( ph -> ( X .- A ) = ( X .- B ) )

Assertion

Ref	Expression
colmid	|- ( ph -> ( B = ( M ` A ) \/ A = B ) )

Proof of Theorem colmid

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ph /\ A = B ) -> A = B )
2	1	olcd 408	. 2 |- ( ( ph /\ A = B ) -> ( B = ( M ` A ) \/ A = B ) )
3		mirval.p	. . . . 5 |- P = ( Base ` G )
4		mirval.d	. . . . 5 |- .- = ( dist ` G )
5		mirval.i	. . . . 5 |- I = (Itv ` G )
6		mirval.l	. . . . 5 |- L = (LineG ` G )
7		mirval.s	. . . . 5 |- S = (pInvG ` G )
8		mirval.g	. . . . . 6 |- ( ph -> G e. TarskiG )
9	8	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> G e. TarskiG )
10		colmid.x	. . . . . 6 |- ( ph -> X e. P )
11	10	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> X e. P )
12		colmid.m	. . . . 5 |- M = ( S ` X )
13		colmid.a	. . . . . 6 |- ( ph -> A e. P )
14	13	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> A e. P )
15		colmid.b	. . . . . 6 |- ( ph -> B e. P )
16	15	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> B e. P )
17		colmid.d	. . . . . . 7 |- ( ph -> ( X .- A ) = ( X .- B ) )
18	17	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> ( X .- A ) = ( X .- B ) )
19	18	eqcomd 2628	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> ( X .- B ) = ( X .- A ) )
20		simpr 477	. . . . . 6 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> X e. ( A I B ) )
21	3, 4, 5, 9, 14, 11, 16, 20	tgbtwncom 25383	. . . . 5 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> X e. ( B I A ) )
22	3, 4, 5, 6, 7, 9, 11, 12, 14, 16, 19, 21	ismir 25554	. . . 4 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> B = ( M ` A ) )
23	22	orcd 407	. . 3 |- ( ( ( ph /\ A =/= B ) /\ X e. ( A I B ) ) -> ( B = ( M ` A ) \/ A = B ) )
24	8	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( X I B ) ) -> G e. TarskiG )
25	15	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( X I B ) ) -> B e. P )
26	13	adantr 481	. . . . . . 7 |- ( ( ph /\ A e. ( X I B ) ) -> A e. P )
27	10	adantr 481	. . . . . . . 8 |- ( ( ph /\ A e. ( X I B ) ) -> X e. P )
28		simpr 477	. . . . . . . . 9 |- ( ( ph /\ A e. ( X I B ) ) -> A e. ( X I B ) )
29	3, 4, 5, 24, 27, 26, 25, 28	tgbtwncom 25383	. . . . . . . 8 |- ( ( ph /\ A e. ( X I B ) ) -> A e. ( B I X ) )
30	3, 4, 5, 24, 26, 27	tgbtwntriv1 25386	. . . . . . . 8 |- ( ( ph /\ A e. ( X I B ) ) -> A e. ( A I X ) )
31	3, 4, 5, 8, 10, 13, 10, 15, 17	tgcgrcomlr 25375	. . . . . . . . . 10 |- ( ph -> ( A .- X ) = ( B .- X ) )
32	31	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A e. ( X I B ) ) -> ( A .- X ) = ( B .- X ) )
33	32	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ A e. ( X I B ) ) -> ( B .- X ) = ( A .- X ) )
34		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ A e. ( X I B ) ) -> ( A .- X ) = ( A .- X ) )
35	3, 4, 5, 24, 25, 26, 27, 26, 26, 27, 29, 30, 33, 34	tgcgrsub 25404	. . . . . . 7 |- ( ( ph /\ A e. ( X I B ) ) -> ( B .- A ) = ( A .- A ) )
36	3, 4, 5, 24, 25, 26, 26, 35	axtgcgrid 25362	. . . . . 6 |- ( ( ph /\ A e. ( X I B ) ) -> B = A )
37	36	eqcomd 2628	. . . . 5 |- ( ( ph /\ A e. ( X I B ) ) -> A = B )
38	37	adantlr 751	. . . 4 |- ( ( ( ph /\ A =/= B ) /\ A e. ( X I B ) ) -> A = B )
39	38	olcd 408	. . 3 |- ( ( ( ph /\ A =/= B ) /\ A e. ( X I B ) ) -> ( B = ( M ` A ) \/ A = B ) )
40	8	adantr 481	. . . . . 6 |- ( ( ph /\ B e. ( A I X ) ) -> G e. TarskiG )
41	13	adantr 481	. . . . . 6 |- ( ( ph /\ B e. ( A I X ) ) -> A e. P )
42	15	adantr 481	. . . . . 6 |- ( ( ph /\ B e. ( A I X ) ) -> B e. P )
43	10	adantr 481	. . . . . . 7 |- ( ( ph /\ B e. ( A I X ) ) -> X e. P )
44		simpr 477	. . . . . . 7 |- ( ( ph /\ B e. ( A I X ) ) -> B e. ( A I X ) )
45	3, 4, 5, 40, 42, 43	tgbtwntriv1 25386	. . . . . . 7 |- ( ( ph /\ B e. ( A I X ) ) -> B e. ( B I X ) )
46	31	adantr 481	. . . . . . 7 |- ( ( ph /\ B e. ( A I X ) ) -> ( A .- X ) = ( B .- X ) )
47		eqidd 2623	. . . . . . 7 |- ( ( ph /\ B e. ( A I X ) ) -> ( B .- X ) = ( B .- X ) )
48	3, 4, 5, 40, 41, 42, 43, 42, 42, 43, 44, 45, 46, 47	tgcgrsub 25404	. . . . . 6 |- ( ( ph /\ B e. ( A I X ) ) -> ( A .- B ) = ( B .- B ) )
49	3, 4, 5, 40, 41, 42, 42, 48	axtgcgrid 25362	. . . . 5 |- ( ( ph /\ B e. ( A I X ) ) -> A = B )
50	49	adantlr 751	. . . 4 |- ( ( ( ph /\ A =/= B ) /\ B e. ( A I X ) ) -> A = B )
51	50	olcd 408	. . 3 |- ( ( ( ph /\ A =/= B ) /\ B e. ( A I X ) ) -> ( B = ( M ` A ) \/ A = B ) )
52		df-ne 2795	. . . . 5 |- ( A =/= B <-> -. A = B )
53		colmid.c	. . . . . . 7 |- ( ph -> ( X e. ( A L B ) \/ A = B ) )
54	53	orcomd 403	. . . . . 6 |- ( ph -> ( A = B \/ X e. ( A L B ) ) )
55	54	orcanai 952	. . . . 5 |- ( ( ph /\ -. A = B ) -> X e. ( A L B ) )
56	52, 55	sylan2b 492	. . . 4 |- ( ( ph /\ A =/= B ) -> X e. ( A L B ) )
57	8	adantr 481	. . . . 5 |- ( ( ph /\ A =/= B ) -> G e. TarskiG )
58	13	adantr 481	. . . . 5 |- ( ( ph /\ A =/= B ) -> A e. P )
59	15	adantr 481	. . . . 5 |- ( ( ph /\ A =/= B ) -> B e. P )
60		simpr 477	. . . . 5 |- ( ( ph /\ A =/= B ) -> A =/= B )
61	10	adantr 481	. . . . 5 |- ( ( ph /\ A =/= B ) -> X e. P )
62	3, 6, 5, 57, 58, 59, 60, 61	tgellng 25448	. . . 4 |- ( ( ph /\ A =/= B ) -> ( X e. ( A L B ) <-> ( X e. ( A I B ) \/ A e. ( X I B ) \/ B e. ( A I X ) ) ) )
63	56, 62	mpbid 222	. . 3 |- ( ( ph /\ A =/= B ) -> ( X e. ( A I B ) \/ A e. ( X I B ) \/ B e. ( A I X ) ) )
64	23, 39, 51, 63	mpjao3dan 1395	. 2 |- ( ( ph /\ A =/= B ) -> ( B = ( M ` A ) \/ A = B ) )
65	2, 64	pm2.61dane 2881	1 |- ( ph -> ( B = ( M ` A ) \/ A = B ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   = 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-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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  symquadlem  25584  midexlem  25587