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Theorem miduniq 23215
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 2451 . . . 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 2451 . . . . 5  |-  ( S `
 A )  =  ( S `  A
)
131, 9, 3, 2, 10, 4, 11, 12, 7mircl 23201 . . . 4  |-  ( ph  ->  ( ( S `  A ) `  B
)  e.  P )
14 eqid 2451 . . . . . . 7  |-  ( S `
 B )  =  ( S `  B
)
151, 9, 3, 2, 10, 4, 7, 14, 5mirbtwn 23198 . . . . . 6  |-  ( ph  ->  B  e.  ( ( ( S `  B
) `  X )
I X ) )
16 miduniq.f . . . . . . 7  |-  ( ph  ->  ( ( S `  B ) `  X
)  =  Y )
1716oveq1d 6208 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 B ) `  X ) I X )  =  ( Y I X ) )
1815, 17eleqtrd 2541 . . . . 5  |-  ( ph  ->  B  e.  ( Y I X ) )
191, 9, 3, 4, 6, 7, 5, 18tgbtwncom 23069 . . . 4  |-  ( ph  ->  B  e.  ( X I Y ) )
201, 9, 3, 2, 10, 4, 7, 14, 5mircgr 23197 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  X ) )
2116oveq2d 6209 . . . . . . . 8  |-  ( ph  ->  ( B  .-  (
( S `  B
) `  X )
)  =  ( B 
.-  Y ) )
2220, 21eqtr3d 2494 . . . . . . 7  |-  ( ph  ->  ( B  .-  X
)  =  ( B 
.-  Y ) )
2322eqcomd 2459 . . . . . 6  |-  ( ph  ->  ( B  .-  Y
)  =  ( B 
.-  X ) )
241, 9, 3, 4, 7, 6, 7, 5, 23tgcgrcomlr 23061 . . . . 5  |-  ( ph  ->  ( Y  .-  B
)  =  ( X 
.-  B ) )
251, 9, 3, 2, 10, 4, 11, 12, 6, 7miriso 23209 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  B ) )
26 miduniq.e . . . . . . . . 9  |-  ( ph  ->  ( ( S `  A ) `  X
)  =  Y )
2726fveq2d 5796 . . . . . . . 8  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  A
) `  X )
)  =  ( ( S `  A ) `
 Y ) )
281, 9, 3, 2, 10, 4, 11, 12, 5mirmir 23202 . . . . . . . 8  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  A
) `  X )
)  =  X )
2927, 28eqtr3d 2494 . . . . . . 7  |-  ( ph  ->  ( ( S `  A ) `  Y
)  =  X )
3029oveq1d 6208 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  Y )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
3125, 30eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( Y  .-  B
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
3224, 31eqtr3d 2494 . . . 4  |-  ( ph  ->  ( X  .-  B
)  =  ( X 
.-  ( ( S `
 A ) `  B ) ) )
331, 9, 3, 4, 7, 5, 7, 6, 22tgcgrcomlr 23061 . . . . 5  |-  ( ph  ->  ( X  .-  B
)  =  ( Y 
.-  B ) )
341, 9, 3, 2, 10, 4, 11, 12, 5, 7miriso 23209 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( X 
.-  B ) )
3526oveq1d 6208 . . . . . 6  |-  ( ph  ->  ( ( ( S `
 A ) `  X )  .-  (
( S `  A
) `  B )
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
3634, 35eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( X  .-  B
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
3733, 36eqtr3d 2494 . . . 4  |-  ( ph  ->  ( Y  .-  B
)  =  ( Y 
.-  ( ( S `
 A ) `  B ) ) )
381, 2, 3, 4, 5, 6, 7, 8, 13, 11, 9, 19, 32, 37tgidinside 23133 . . 3  |-  ( ph  ->  B  =  ( ( S `  A ) `
 B ) )
3938eqcomd 2459 . 2  |-  ( ph  ->  ( ( S `  A ) `  B
)  =  B )
401, 9, 3, 2, 10, 4, 11, 12, 7mirinv 23206 . 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 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193   Basecbs 14285   distcds 14358  TarskiGcstrkg 23015  Itvcitv 23022  LineGclng 23023  cgrGccgrg 23092  pInvGcmir 23191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-pm 7320  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-s2 12586  df-s3 12587  df-trkgc 23034  df-trkgb 23035  df-trkgcb 23036  df-trkg 23040  df-cgrg 23093  df-mir 23192
This theorem is referenced by:  miduniq1  23216  miduniq2  23217  krippenlem  23220  mideu  23255
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