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Theorem miduniq 23798
Description: Unicity of the middle point, expressed with point inversion. Theorem 7.17 of [Schwabhauser] p. 51. (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 )
miduniq.a  |-  ( ph  ->  A  e.  P )
miduniq.b  |-  ( ph  ->  B  e.  P )
miduniq.x  |-  ( ph  ->  X  e.  P )
miduniq.y  |-  ( ph  ->  Y  e.  P )
miduniq.e  |-  ( ph  ->  ( ( S `  A ) `  X
)  =  Y )
miduniq.f  |-  ( ph  ->  ( ( S `  B ) `  X
)  =  Y )
Assertion
Ref Expression
miduniq  |-  ( ph  ->  A  =  B )

Proof of Theorem miduniq
StepHypRef Expression
1 mirval.p . . . 4  |-  P  =  ( Base `  G
)
2 mirval.l . . . 4  |-  L  =  (LineG `  G )
3 mirval.i . . . 4  |-  I  =  (Itv `  G )
4 mirval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
5 miduniq.x . . . 4  |-  ( ph  ->  X  e.  P )
6 miduniq.y . . . 4  |-  ( ph  ->  Y  e.  P )
7 miduniq.b . . . 4  |-  ( ph  ->  B  e.  P )
8 eqid 2467 . . . 4  |-  (cgrG `  G )  =  (cgrG `  G )
9 mirval.d . . . . 5  |-  .-  =  ( dist `  G )
10 mirval.s . . . . 5  |-  S  =  (pInvG `  G )
11 miduniq.a . . . . 5  |-  ( ph  ->  A  e.  P )
12 eqid 2467 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
131, 9, 3, 2, 10, 4, 11, 12, 7mircl 23783 . . . 4  |-  ( ph  ->  ( ( S `  A ) `  B
)  e.  P )
14 eqid 2467 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
151, 9, 3, 2, 10, 4, 7, 14, 5mirbtwn 23780 . . . . . 6  |-  ( ph  ->  B  e.  ( ( ( S `  B
) `  X )
I X ) )
16 miduniq.f . . . . . . 7  |-  ( ph  ->  ( ( S `  B ) `  X
)  =  Y )
1716oveq1d 6299 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 B ) `  X ) I X )  =  ( Y I X ) )
1815, 17eleqtrd 2557 . . . . 5  |-  ( ph  ->  B  e.  ( Y I X ) )
191, 9, 3, 4, 6, 7, 5, 18tgbtwncom 23635 . . . 4  |-  ( ph  ->  B  e.  ( X I Y ) )
201, 9, 3, 2, 10, 4, 7, 14, 5mircgr 23779 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  X ) )
2116oveq2d 6300 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  Y ) )
2220, 21eqtr3d 2510 . . . . . . 7  |-  ( ph  ->  ( B  .-  X
)  =  ( B 
.-  Y ) )
2322eqcomd 2475 . . . . . 6  |-  ( ph  ->  ( B  .-  Y
)  =  ( B 
.-  X ) )
241, 9, 3, 4, 7, 6, 7, 5, 23tgcgrcomlr 23627 . . . . 5  |-  ( ph  ->  ( Y  .-  B
)  =  ( X 
.-  B ) )
251, 9, 3, 2, 10, 4, 11, 12, 6, 7miriso 23791 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  B ) )
26 miduniq.e . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A ) `  X
)  =  Y )
2726fveq2d 5870 . . . . . . . 8  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  A
) `  X )
)  =  ( ( S `  A ) `
 Y ) )
281, 9, 3, 2, 10, 4, 11, 12, 5mirmir 23784 . . . . . . . 8  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  A
) `  X )
)  =  X )
2927, 28eqtr3d 2510 . . . . . . 7  |-  ( ph  ->  ( ( S `  A ) `  Y
)  =  X )
3029oveq1d 6299 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
3125, 30eqtr3d 2510 . . . . 5  |-  ( ph  ->  ( Y  .-  B
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
3224, 31eqtr3d 2510 . . . 4  |-  ( ph  ->  ( X  .-  B
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
331, 9, 3, 4, 7, 5, 7, 6, 22tgcgrcomlr 23627 . . . . 5  |-  ( ph  ->  ( X  .-  B
)  =  ( Y 
.-  B ) )
341, 9, 3, 2, 10, 4, 11, 12, 5, 7miriso 23791 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  B ) )
3526oveq1d 6299 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
3634, 35eqtr3d 2510 . . . . 5  |-  ( ph  ->  ( X  .-  B
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
3733, 36eqtr3d 2510 . . . 4  |-  ( ph  ->  ( Y  .-  B
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
381, 2, 3, 4, 5, 6, 7, 8, 13, 11, 9, 19, 32, 37tgidinside 23713 . . 3  |-  ( ph  ->  B  =  ( ( S `  A ) `
 B ) )
3938eqcomd 2475 . 2  |-  ( ph  ->  ( ( S `  A ) `  B
)  =  B )
401, 9, 3, 2, 10, 4, 11, 12, 7mirinv 23788 . 2  |-  ( ph  ->  ( ( ( S `
 A ) `  B )  =  B  <-> 
A  =  B ) )
4139, 40mpbid 210 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   Basecbs 14490   distcds 14564  TarskiGcstrkg 23581  Itvcitv 23588  LineGclng 23589  cgrGccgrg 23658  pInvGcmir 23774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-s2 12776  df-s3 12777  df-trkgc 23600  df-trkgb 23601  df-trkgcb 23602  df-trkg 23606  df-cgrg 23659  df-mir 23775
This theorem is referenced by:  miduniq1  23799  miduniq2  23800  krippenlem  23803  mideu  23842
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