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Theorem miduniq 24587
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 2429 . . . 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 2429 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
131, 9, 3, 2, 10, 4, 11, 12, 7mircl 24566 . . . 4  |-  ( ph  ->  ( ( S `  A ) `  B
)  e.  P )
14 eqid 2429 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
151, 9, 3, 2, 10, 4, 7, 14, 5mirbtwn 24563 . . . . . 6  |-  ( ph  ->  B  e.  ( ( ( S `  B
) `  X )
I X ) )
16 miduniq.f . . . . . . 7  |-  ( ph  ->  ( ( S `  B ) `  X
)  =  Y )
1716oveq1d 6320 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 B ) `  X ) I X )  =  ( Y I X ) )
1815, 17eleqtrd 2519 . . . . 5  |-  ( ph  ->  B  e.  ( Y I X ) )
191, 9, 3, 4, 6, 7, 5, 18tgbtwncom 24395 . . . 4  |-  ( ph  ->  B  e.  ( X I Y ) )
201, 9, 3, 2, 10, 4, 11, 12, 6, 7miriso 24574 . . . . 5  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  B ) )
21 miduniq.e . . . . . . 7  |-  ( ph  ->  ( ( S `  A ) `  X
)  =  Y )
221, 9, 3, 2, 10, 4, 11, 12, 5, 21mircom 24568 . . . . . 6  |-  ( ph  ->  ( ( S `  A ) `  Y
)  =  X )
2322oveq1d 6320 . . . . 5  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
241, 9, 3, 2, 10, 4, 7, 14, 5mircgr 24562 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  X ) )
2516oveq2d 6321 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  Y ) )
2624, 25eqtr3d 2472 . . . . . . 7  |-  ( ph  ->  ( B  .-  X
)  =  ( B 
.-  Y ) )
2726eqcomd 2437 . . . . . 6  |-  ( ph  ->  ( B  .-  Y
)  =  ( B 
.-  X ) )
281, 9, 3, 4, 7, 6, 7, 5, 27tgcgrcomlr 24387 . . . . 5  |-  ( ph  ->  ( Y  .-  B
)  =  ( X 
.-  B ) )
2920, 23, 283eqtr3rd 2479 . . . 4  |-  ( ph  ->  ( X  .-  B
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
301, 9, 3, 2, 10, 4, 11, 12, 5, 7miriso 24574 . . . . 5  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  B ) )
3121oveq1d 6320 . . . . 5  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
321, 9, 3, 4, 7, 5, 7, 6, 26tgcgrcomlr 24387 . . . . 5  |-  ( ph  ->  ( X  .-  B
)  =  ( Y 
.-  B ) )
3330, 31, 323eqtr3rd 2479 . . . 4  |-  ( ph  ->  ( Y  .-  B
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
341, 2, 3, 4, 5, 6, 7, 8, 13, 11, 9, 19, 29, 33tgidinside 24476 . . 3  |-  ( ph  ->  B  =  ( ( S `  A ) `
 B ) )
3534eqcomd 2437 . 2  |-  ( ph  ->  ( ( S `  A ) `  B
)  =  B )
361, 9, 3, 2, 10, 4, 11, 12, 7mirinv 24571 . 2  |-  ( ph  ->  ( ( ( S `
 A ) `  B )  =  B  <-> 
A  =  B ) )
3735, 36mpbid 213 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   Basecbs 15084   distcds 15161  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348  cgrGccgrg 24418  pInvGcmir 24557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-hash 12513  df-word 12651  df-concat 12653  df-s1 12654  df-s2 12929  df-s3 12930  df-trkgc 24359  df-trkgb 24360  df-trkgcb 24361  df-trkg 24364  df-cgrg 24419  df-mir 24558
This theorem is referenced by:  miduniq1  24588  krippenlem  24592  mideu  24637  opphllem3  24648
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