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Theorem miduniq2 24184
Description: If two point inversions commute, they are identical. Theorem 7.19 of [Schwabhauser] p. 52. (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 )
miduniq2.a  |-  ( ph  ->  A  e.  P )
miduniq2.b  |-  ( ph  ->  B  e.  P )
miduniq2.x  |-  ( ph  ->  X  e.  P )
miduniq2.e  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  B
) `  X )
)  =  ( ( S `  B ) `
 ( ( S `
 A ) `  X ) ) )
Assertion
Ref Expression
miduniq2  |-  ( ph  ->  A  =  B )

Proof of Theorem miduniq2
StepHypRef Expression
1 mirval.p . . . 4  |-  P  =  ( Base `  G
)
2 mirval.d . . . 4  |-  .-  =  ( dist `  G )
3 mirval.i . . . 4  |-  I  =  (Itv `  G )
4 mirval.l . . . 4  |-  L  =  (LineG `  G )
5 mirval.s . . . 4  |-  S  =  (pInvG `  G )
6 mirval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
7 miduniq2.b . . . . . 6  |-  ( ph  ->  B  e.  P )
8 eqid 2382 . . . . . 6  |-  ( S `
 B )  =  ( S `  B
)
91, 2, 3, 4, 5, 6, 7, 8mirf 24161 . . . . 5  |-  ( ph  ->  ( S `  B
) : P --> P )
10 miduniq2.a . . . . 5  |-  ( ph  ->  A  e.  P )
119, 10ffvelrnd 5934 . . . 4  |-  ( ph  ->  ( ( S `  B ) `  A
)  e.  P )
12 miduniq2.x . . . 4  |-  ( ph  ->  X  e.  P )
13 eqid 2382 . . . . . 6  |-  ( ( S `  B ) `
 A )  =  ( ( S `  B ) `  A
)
14 eqid 2382 . . . . . 6  |-  ( ( S `  B ) `
 ( ( S `
 B ) `  X ) )  =  ( ( S `  B ) `  (
( S `  B
) `  X )
)
15 eqid 2382 . . . . . 6  |-  ( ( S `  B ) `
 ( ( S `
 B ) `  ( ( S `  A ) `  X
) ) )  =  ( ( S `  B ) `  (
( S `  B
) `  ( ( S `  A ) `  X ) ) )
169, 12ffvelrnd 5934 . . . . . 6  |-  ( ph  ->  ( ( S `  B ) `  X
)  e.  P )
17 eqid 2382 . . . . . . . 8  |-  ( S `
 A )  =  ( S `  A
)
181, 2, 3, 4, 5, 6, 10, 17, 12mircl 24162 . . . . . . 7  |-  ( ph  ->  ( ( S `  A ) `  X
)  e.  P )
199, 18ffvelrnd 5934 . . . . . 6  |-  ( ph  ->  ( ( S `  B ) `  (
( S `  A
) `  X )
)  e.  P )
20 miduniq2.e . . . . . 6  |-  ( ph  ->  ( ( S `  A ) `  (
( S `  B
) `  X )
)  =  ( ( S `  B ) `
 ( ( S `
 A ) `  X ) ) )
211, 2, 3, 4, 5, 6, 8, 13, 14, 15, 7, 10, 16, 19, 20mirauto 24181 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( S `  B ) `  A
) ) `  (
( S `  B
) `  ( ( S `  B ) `  X ) ) )  =  ( ( S `
 B ) `  ( ( S `  B ) `  (
( S `  A
) `  X )
) ) )
221, 2, 3, 4, 5, 6, 7, 8, 12mirmir 24163 . . . . . 6  |-  ( ph  ->  ( ( S `  B ) `  (
( S `  B
) `  X )
)  =  X )
2322fveq2d 5778 . . . . 5  |-  ( ph  ->  ( ( S `  ( ( S `  B ) `  A
) ) `  (
( S `  B
) `  ( ( S `  B ) `  X ) ) )  =  ( ( S `
 ( ( S `
 B ) `  A ) ) `  X ) )
241, 2, 3, 4, 5, 6, 7, 8, 18mirmir 24163 . . . . 5  |-  ( ph  ->  ( ( S `  B ) `  (
( S `  B
) `  ( ( S `  A ) `  X ) ) )  =  ( ( S `
 A ) `  X ) )
2521, 23, 243eqtr3d 2431 . . . 4  |-  ( ph  ->  ( ( S `  ( ( S `  B ) `  A
) ) `  X
)  =  ( ( S `  A ) `
 X ) )
261, 2, 3, 4, 5, 6, 11, 10, 12, 25miduniq1 24183 . . 3  |-  ( ph  ->  ( ( S `  B ) `  A
)  =  A )
271, 2, 3, 4, 5, 6, 7, 8, 10mirinv 24167 . . 3  |-  ( ph  ->  ( ( ( S `
 B ) `  A )  =  A  <-> 
B  =  A ) )
2826, 27mpbid 210 . 2  |-  ( ph  ->  B  =  A )
2928eqcomd 2390 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1826   ` cfv 5496   Basecbs 14634   distcds 14711  TarskiGcstrkg 23942  Itvcitv 23949  LineGclng 23950  pInvGcmir 24153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-pm 7341  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-concat 12448  df-s1 12449  df-s2 12724  df-s3 12725  df-trkgc 23961  df-trkgb 23962  df-trkgcb 23963  df-trkg 23967  df-cgrg 24023  df-mir 24154
This theorem is referenced by: (None)
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