Theorem symquadlem 25584

Description: Lemma of the symetrial quadrilateral. The diagonals of quadrilaterals with congruent opposing sides intersect at their middle point. In Euclidean geometry, such quadrilaterals are called parallelograms, as opposing sides are parallel. However, this is not necessarily true in the case of absolute geometry. Lemma 7.21 of [Schwabhauser] p. 52. (Contributed by Thierry Arnoux, 6-Aug-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 )
symquadlem.m	|- M = ( S ` X )
symquadlem.a	|- ( ph -> A e. P )
symquadlem.b	|- ( ph -> B e. P )
symquadlem.c	|- ( ph -> C e. P )
symquadlem.d	|- ( ph -> D e. P )
symquadlem.x	|- ( ph -> X e. P )
symquadlem.1	|- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
symquadlem.2	|- ( ph -> B =/= D )
symquadlem.3	|- ( ph -> ( A .- B ) = ( C .- D ) )
symquadlem.4	|- ( ph -> ( B .- C ) = ( D .- A ) )
symquadlem.5	|- ( ph -> ( X e. ( A L C ) \/ A = C ) )
symquadlem.6	|- ( ph -> ( X e. ( B L D ) \/ B = D ) )

Assertion

Ref	Expression
symquadlem	|- ( ph -> A = ( M ` C ) )

Proof of Theorem symquadlem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		symquadlem.1	. . . . . . . 8 |- ( ph -> -. ( A e. ( B L C ) \/ B = C ) )
2		mirval.p	. . . . . . . . . . 11 |- P = ( Base ` G )
3		mirval.l	. . . . . . . . . . 11 |- L = (LineG ` G )
4		mirval.i	. . . . . . . . . . 11 |- I = (Itv ` G )
5		mirval.g	. . . . . . . . . . 11 |- ( ph -> G e. TarskiG )
6		symquadlem.b	. . . . . . . . . . 11 |- ( ph -> B e. P )
7		symquadlem.a	. . . . . . . . . . 11 |- ( ph -> A e. P )
8		mirval.d	. . . . . . . . . . . 12 |- .- = ( dist ` G )
9	2, 8, 4, 5, 6, 7	tgbtwntriv2 25382	. . . . . . . . . . 11 |- ( ph -> A e. ( B I A ) )
10	2, 3, 4, 5, 6, 7, 7, 9	btwncolg1 25450	. . . . . . . . . 10 |- ( ph -> ( A e. ( B L A ) \/ B = A ) )
11	10	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A = C ) -> ( A e. ( B L A ) \/ B = A ) )
12		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ A = C ) -> A = C )
13	12	oveq2d 6666	. . . . . . . . . . 11 |- ( ( ph /\ A = C ) -> ( B L A ) = ( B L C ) )
14	13	eleq2d 2687	. . . . . . . . . 10 |- ( ( ph /\ A = C ) -> ( A e. ( B L A ) <-> A e. ( B L C ) ) )
15	12	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ph /\ A = C ) -> ( B = A <-> B = C ) )
16	14, 15	orbi12d 746	. . . . . . . . 9 |- ( ( ph /\ A = C ) -> ( ( A e. ( B L A ) \/ B = A ) <-> ( A e. ( B L C ) \/ B = C ) ) )
17	11, 16	mpbid 222	. . . . . . . 8 |- ( ( ph /\ A = C ) -> ( A e. ( B L C ) \/ B = C ) )
18	1, 17	mtand 691	. . . . . . 7 |- ( ph -> -. A = C )
19	18	neqned 2801	. . . . . 6 |- ( ph -> A =/= C )
20	19	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> A =/= C )
21	20	necomd 2849	. . . 4 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> C =/= A )
22	21	neneqd 2799	. . 3 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> -. C = A )
23		mirval.s	. . . . . 6 |- S = (pInvG ` G )
24	5	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> G e. TarskiG )
25		symquadlem.m	. . . . . 6 |- M = ( S ` X )
26		symquadlem.c	. . . . . . 7 |- ( ph -> C e. P )
27	26	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> C e. P )
28	7	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> A e. P )
29		symquadlem.x	. . . . . . 7 |- ( ph -> X e. P )
30	29	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> X e. P )
31		symquadlem.5	. . . . . . . 8 |- ( ph -> ( X e. ( A L C ) \/ A = C ) )
32	31	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X e. ( A L C ) \/ A = C ) )
33	2, 3, 4, 24, 28, 27, 30, 32	colcom 25453	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X e. ( C L A ) \/ C = A ) )
34	6	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> B e. P )
35		symquadlem.d	. . . . . . . . 9 |- ( ph -> D e. P )
36	35	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> D e. P )
37		eqid 2622	. . . . . . . 8 |- (cgrG ` G ) = (cgrG ` G )
38		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> x e. P )
39		symquadlem.6	. . . . . . . . . . 11 |- ( ph -> ( X e. ( B L D ) \/ B = D ) )
40	2, 3, 4, 5, 6, 35, 29, 39	colrot2 25455	. . . . . . . . . 10 |- ( ph -> ( D e. ( X L B ) \/ X = B ) )
41	2, 3, 4, 5, 29, 6, 35, 40	colcom 25453	. . . . . . . . 9 |- ( ph -> ( D e. ( B L X ) \/ B = X ) )
42	41	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( D e. ( B L X ) \/ B = X ) )
43		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> <" B D X "> (cgrG ` G ) <" D B x "> )
44		symquadlem.4	. . . . . . . . 9 |- ( ph -> ( B .- C ) = ( D .- A ) )
45	44	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( B .- C ) = ( D .- A ) )
46		symquadlem.3	. . . . . . . . . . 11 |- ( ph -> ( A .- B ) = ( C .- D ) )
47	2, 8, 4, 5, 7, 6, 26, 35, 46	tgcgrcomlr 25375	. . . . . . . . . 10 |- ( ph -> ( B .- A ) = ( D .- C ) )
48	47	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( B .- A ) = ( D .- C ) )
49	48	eqcomd 2628	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( D .- C ) = ( B .- A ) )
50		symquadlem.2	. . . . . . . . 9 |- ( ph -> B =/= D )
51	50	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> B =/= D )
52	2, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 27, 38, 28, 42, 43, 45, 49, 51	tgfscgr 25463	. . . . . . 7 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X .- C ) = ( x .- A ) )
53	2, 3, 4, 5, 6, 26, 7, 1	ncolcom 25456	. . . . . . . . . 10 |- ( ph -> -. ( A e. ( C L B ) \/ C = B ) )
54	53	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> -. ( A e. ( C L B ) \/ C = B ) )
55	31	orcomd 403	. . . . . . . . . . . 12 |- ( ph -> ( A = C \/ X e. ( A L C ) ) )
56	55	ord 392	. . . . . . . . . . 11 |- ( ph -> ( -. A = C -> X e. ( A L C ) ) )
57	18, 56	mpd 15	. . . . . . . . . 10 |- ( ph -> X e. ( A L C ) )
58	57	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> X e. ( A L C ) )
59	18	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> -. A = C )
60	45	eqcomd 2628	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( D .- A ) = ( B .- C ) )
61	2, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 28, 38, 27, 42, 43, 48, 60, 51	tgfscgr 25463	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X .- A ) = ( x .- C ) )
62	2, 8, 4, 24, 30, 28, 38, 27, 61	tgcgrcomlr 25375	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( A .- X ) = ( C .- x ) )
63	2, 8, 4, 24, 27, 28	axtgcgrrflx 25361	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( C .- A ) = ( A .- C ) )
64	2, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63	trgcgr 25411	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> <" A X C "> (cgrG ` G ) <" C x A "> )
65	2, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64	lnxfr 25461	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( x e. ( C L A ) \/ C = A ) )
66	2, 3, 4, 24, 27, 28, 38, 65	colcom 25453	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( x e. ( A L C ) \/ A = C ) )
67	66	orcomd 403	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( A = C \/ x e. ( A L C ) ) )
68	67	ord 392	. . . . . . . . . 10 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( -. A = C -> x e. ( A L C ) ) )
69	59, 68	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> x e. ( A L C ) )
70	50	neneqd 2799	. . . . . . . . . . 11 |- ( ph -> -. B = D )
71	39	orcomd 403	. . . . . . . . . . . 12 |- ( ph -> ( B = D \/ X e. ( B L D ) ) )
72	71	ord 392	. . . . . . . . . . 11 |- ( ph -> ( -. B = D -> X e. ( B L D ) ) )
73	70, 72	mpd 15	. . . . . . . . . 10 |- ( ph -> X e. ( B L D ) )
74	73	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> X e. ( B L D ) )
75	70	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> -. B = D )
76	39	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X e. ( B L D ) \/ B = D ) )
77	2, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43	cgr3swap23 25419	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> <" B X D "> (cgrG ` G ) <" D x B "> )
78	2, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77	lnxfr 25461	. . . . . . . . . . . . 13 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( x e. ( D L B ) \/ D = B ) )
79	2, 3, 4, 24, 36, 34, 38, 78	colcom 25453	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( x e. ( B L D ) \/ B = D ) )
80	79	orcomd 403	. . . . . . . . . . 11 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( B = D \/ x e. ( B L D ) ) )
81	80	ord 392	. . . . . . . . . 10 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( -. B = D -> x e. ( B L D ) ) )
82	75, 81	mpd 15	. . . . . . . . 9 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> x e. ( B L D ) )
83	2, 4, 3, 24, 28, 27, 34, 36, 54, 58, 69, 74, 82	tglineinteq 25540	. . . . . . . 8 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> X = x )
84	83	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X .- A ) = ( x .- A ) )
85	52, 84	eqtr4d 2659	. . . . . 6 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( X .- C ) = ( X .- A ) )
86	2, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85	colmid 25583	. . . . 5 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( A = ( M ` C ) \/ C = A ) )
87	86	orcomd 403	. . . 4 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( C = A \/ A = ( M ` C ) ) )
88	87	ord 392	. . 3 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> ( -. C = A -> A = ( M ` C ) ) )
89	22, 88	mpd 15	. 2 |- ( ( ( ph /\ x e. P ) /\ <" B D X "> (cgrG ` G ) <" D B x "> ) -> A = ( M ` C ) )
90	2, 8, 4, 5, 6, 35	axtgcgrrflx 25361	. . 3 |- ( ph -> ( B .- D ) = ( D .- B ) )
91	2, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90	lnext 25462	. 2 |- ( ph -> E. x e. P <" B D X "> (cgrG ` G ) <" D B x "> )
92	89, 91	r19.29a 3078	1 |- ( ph -> A = ( M ` C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  cgrGccgrg 25405  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-mir 25548

This theorem is referenced by:  opphllem  25627