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Theorem symquadlem 24192
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 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.
StepHypRef 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 )
92, 8, 4, 5, 6, 7tgbtwntriv2 24004 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  ( B I A ) )
102, 3, 4, 5, 6, 7, 7, 9btwncolg1 24068 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( B L A )  \/  B  =  A ) )
1110adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L A )  \/  B  =  A ) )
12 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  A  =  C )  ->  A  =  C )
1312oveq2d 6312 . . . . . . . . . . 11  |-  ( (
ph  /\  A  =  C )  ->  ( B L A )  =  ( B L C ) )
1413eleq2d 2527 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L A )  <->  A  e.  ( B L C ) ) )
1512eqeq2d 2471 . . . . . . . . . 10  |-  ( (
ph  /\  A  =  C )  ->  ( B  =  A  <->  B  =  C ) )
1614, 15orbi12d 709 . . . . . . . . 9  |-  ( (
ph  /\  A  =  C )  ->  (
( A  e.  ( B L A )  \/  B  =  A )  <->  ( A  e.  ( B L C )  \/  B  =  C ) ) )
1711, 16mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  A  =  C )  ->  ( A  e.  ( B L C )  \/  B  =  C ) )
181, 17mtand 659 . . . . . . 7  |-  ( ph  ->  -.  A  =  C )
1918neqned 2660 . . . . . 6  |-  ( ph  ->  A  =/=  C )
2019ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  =/=  C )
2120necomd 2728 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  C  =/=  A )
2221neneqd 2659 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  C  =  A
)
23 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
245ad2antrr 725 . . . . . 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 )
2726ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  C  e.  P )
287ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  e.  P )
29 symquadlem.x . . . . . . 7  |-  ( ph  ->  X  e.  P )
3029ad2antrr 725 . . . . . 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 ) )
3231ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( A L C )  \/  A  =  C ) )
332, 3, 4, 24, 28, 27, 30, 32colcom 24071 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( C L A )  \/  C  =  A ) )
346ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  B  e.  P )
35 symquadlem.d . . . . . . . . 9  |-  ( ph  ->  D  e.  P )
3635ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  D  e.  P )
37 eqid 2457 . . . . . . . 8  |-  (cgrG `  G )  =  (cgrG `  G )
38 simplr 755 . . . . . . . 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 ) )
402, 3, 4, 5, 6, 35, 29, 39colrot2 24073 . . . . . . . . . 10  |-  ( ph  ->  ( D  e.  ( X L B )  \/  X  =  B ) )
412, 3, 4, 5, 29, 6, 35, 40colcom 24071 . . . . . . . . 9  |-  ( ph  ->  ( D  e.  ( B L X )  \/  B  =  X ) )
4241ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  e.  ( B L X )  \/  B  =  X ) )
43 simpr 461 . . . . . . . 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 ) )
4544ad2antrr 725 . . . . . . . 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 ) )
472, 8, 4, 5, 7, 6, 26, 35, 46tgcgrcomlr 23997 . . . . . . . . . 10  |-  ( ph  ->  ( B  .-  A
)  =  ( D 
.-  C ) )
4847ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( B  .-  A
)  =  ( D 
.-  C ) )
4948eqcomd 2465 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  .-  C
)  =  ( B 
.-  A ) )
50 symquadlem.2 . . . . . . . . 9  |-  ( ph  ->  B  =/=  D )
5150ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  B  =/=  D )
522, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 27, 38, 28, 42, 43, 45, 49, 51tgfscgr 24081 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  C
)  =  ( x 
.-  A ) )
532, 3, 4, 5, 6, 26, 7, 1ncolcom 24074 . . . . . . . . . 10  |-  ( ph  ->  -.  ( A  e.  ( C L B )  \/  C  =  B ) )
5453ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  ( A  e.  ( C L B )  \/  C  =  B ) )
5531orcomd 388 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  =  C  \/  X  e.  ( A L C ) ) )
5655ord 377 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  A  =  C  ->  X  e.  ( A L C ) ) )
5718, 56mpd 15 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( A L C ) )
5857ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  e.  ( A L C ) )
5918ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  A  =  C
)
6045eqcomd 2465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( D  .-  A
)  =  ( B 
.-  C ) )
612, 3, 4, 24, 34, 36, 30, 37, 36, 34, 8, 28, 38, 27, 42, 43, 48, 60, 51tgfscgr 24081 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( x 
.-  C ) )
622, 8, 4, 24, 30, 28, 38, 27, 61tgcgrcomlr 23997 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  .-  X
)  =  ( C 
.-  x ) )
632, 8, 4, 24, 27, 28axtgcgrrflx 23985 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( C  .-  A
)  =  ( A 
.-  C ) )
642, 8, 37, 24, 28, 30, 27, 27, 38, 28, 62, 52, 63trgcgr 24033 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  <" A X C "> (cgrG `  G ) <" C x A "> )
652, 3, 4, 24, 28, 30, 27, 37, 27, 38, 28, 32, 64lnxfr 24079 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( C L A )  \/  C  =  A ) )
662, 3, 4, 24, 27, 28, 38, 65colcom 24071 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( A L C )  \/  A  =  C ) )
6766orcomd 388 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  =  C  \/  x  e.  ( A L C ) ) )
6867ord 377 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  A  =  C  ->  x  e.  ( A L C ) ) )
6959, 68mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  x  e.  ( A L C ) )
7050neneqd 2659 . . . . . . . . . . 11  |-  ( ph  ->  -.  B  =  D )
7139orcomd 388 . . . . . . . . . . . 12  |-  ( ph  ->  ( B  =  D  \/  X  e.  ( B L D ) ) )
7271ord 377 . . . . . . . . . . 11  |-  ( ph  ->  ( -.  B  =  D  ->  X  e.  ( B L D ) ) )
7370, 72mpd 15 . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( B L D ) )
7473ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  e.  ( B L D ) )
7570ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  -.  B  =  D
)
7639ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  e.  ( B L D )  \/  B  =  D ) )
772, 8, 4, 37, 24, 34, 36, 30, 36, 34, 38, 43cgr3swap23 24041 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  <" B X D "> (cgrG `  G ) <" D x B "> )
782, 3, 4, 24, 34, 30, 36, 37, 36, 38, 34, 76, 77lnxfr 24079 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( D L B )  \/  D  =  B ) )
792, 3, 4, 24, 36, 34, 38, 78colcom 24071 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( x  e.  ( B L D )  \/  B  =  D ) )
8079orcomd 388 . . . . . . . . . . 11  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( B  =  D  \/  x  e.  ( B L D ) ) )
8180ord 377 . . . . . . . . . 10  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  B  =  D  ->  x  e.  ( B L D ) ) )
8275, 81mpd 15 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  x  e.  ( B L D ) )
832, 4, 3, 24, 28, 27, 34, 36, 54, 58, 69, 74, 82tglineinteq 24151 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  X  =  x )
8483oveq1d 6311 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  A
)  =  ( x 
.-  A ) )
8552, 84eqtr4d 2501 . . . . . 6  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( X  .-  C
)  =  ( X 
.-  A ) )
862, 8, 4, 3, 23, 24, 25, 27, 28, 30, 33, 85colmid 24191 . . . . 5  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( A  =  ( M `  C )  \/  C  =  A ) )
8786orcomd 388 . . . 4  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( C  =  A  \/  A  =  ( M `  C ) ) )
8887ord 377 . . 3  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  -> 
( -.  C  =  A  ->  A  =  ( M `  C ) ) )
8922, 88mpd 15 . 2  |-  ( ( ( ph  /\  x  e.  P )  /\  <" B D X "> (cgrG `  G ) <" D B x "> )  ->  A  =  ( M `  C ) )
902, 8, 4, 5, 6, 35axtgcgrrflx 23985 . . 3  |-  ( ph  ->  ( B  .-  D
)  =  ( D 
.-  B ) )
912, 3, 4, 5, 6, 35, 29, 37, 35, 6, 8, 41, 90lnext 24080 . 2  |-  ( ph  ->  E. x  e.  P  <" B D X "> (cgrG `  G ) <" D B x "> )
9289, 91r19.29a 2999 1  |-  ( ph  ->  A  =  ( M `
 C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   <"cs3 12819   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959  cgrGccgrg 24028  pInvGcmir 24159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029  df-mir 24160
This theorem is referenced by:  opphllem  24235
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