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Theorem lmicom 24026
Description: The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)
Hypotheses
Ref Expression
ismid.p  |-  P  =  ( Base `  G
)
ismid.d  |-  .-  =  ( dist `  G )
ismid.i  |-  I  =  (Itv `  G )
ismid.g  |-  ( ph  ->  G  e. TarskiG )
ismid.1  |-  ( ph  ->  GDimTarskiG 2 )
lmif.m  |-  M  =  ( (lInvG `  G
) `  D )
lmif.l  |-  L  =  (LineG `  G )
lmif.d  |-  ( ph  ->  D  e.  ran  L
)
lmicl.1  |-  ( ph  ->  A  e.  P )
islmib.b  |-  ( ph  ->  B  e.  P )
lmicom.1  |-  ( ph  ->  ( M `  A
)  =  B )
Assertion
Ref Expression
lmicom  |-  ( ph  ->  ( M `  B
)  =  A )

Proof of Theorem lmicom
StepHypRef Expression
1 ismid.p . . . . 5  |-  P  =  ( Base `  G
)
2 ismid.d . . . . 5  |-  .-  =  ( dist `  G )
3 ismid.i . . . . 5  |-  I  =  (Itv `  G )
4 ismid.g . . . . 5  |-  ( ph  ->  G  e. TarskiG )
5 ismid.1 . . . . 5  |-  ( ph  ->  GDimTarskiG 2 )
6 lmicl.1 . . . . 5  |-  ( ph  ->  A  e.  P )
7 islmib.b . . . . 5  |-  ( ph  ->  B  e.  P )
81, 2, 3, 4, 5, 6, 7midcom 24020 . . . 4  |-  ( ph  ->  ( A (midG `  G ) B )  =  ( B (midG `  G ) A ) )
9 lmicom.1 . . . . . . 7  |-  ( ph  ->  ( M `  A
)  =  B )
109eqcomd 2451 . . . . . 6  |-  ( ph  ->  B  =  ( M `
 A ) )
11 lmif.m . . . . . . 7  |-  M  =  ( (lInvG `  G
) `  D )
12 lmif.l . . . . . . 7  |-  L  =  (LineG `  G )
13 lmif.d . . . . . . 7  |-  ( ph  ->  D  e.  ran  L
)
141, 2, 3, 4, 5, 11, 12, 13, 6, 7islmib 24025 . . . . . 6  |-  ( ph  ->  ( B  =  ( M `  A )  <-> 
( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G )
( A L B )  \/  A  =  B ) ) ) )
1510, 14mpbid 210 . . . . 5  |-  ( ph  ->  ( ( A (midG `  G ) B )  e.  D  /\  ( D (⟂G `  G )
( A L B )  \/  A  =  B ) ) )
1615simpld 459 . . . 4  |-  ( ph  ->  ( A (midG `  G ) B )  e.  D )
178, 16eqeltrrd 2532 . . 3  |-  ( ph  ->  ( B (midG `  G ) A )  e.  D )
1815simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( D (⟂G `  G
) ( A L B )  \/  A  =  B ) )
1918orcomd 388 . . . . . . . 8  |-  ( ph  ->  ( A  =  B  \/  D (⟂G `  G
) ( A L B ) ) )
2019ord 377 . . . . . . 7  |-  ( ph  ->  ( -.  A  =  B  ->  D (⟂G `  G ) ( A L B ) ) )
214adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  B )  ->  G  e. TarskiG )
226adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  B )  ->  A  e.  P )
237adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  B )  ->  B  e.  P )
24 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  A  =  B )  ->  -.  A  =  B )
2524neqned 2646 . . . . . . . . . 10  |-  ( (
ph  /\  -.  A  =  B )  ->  A  =/=  B )
261, 3, 12, 21, 22, 23, 25tglinecom 23887 . . . . . . . . 9  |-  ( (
ph  /\  -.  A  =  B )  ->  ( A L B )  =  ( B L A ) )
2726breq2d 4449 . . . . . . . 8  |-  ( (
ph  /\  -.  A  =  B )  ->  ( D (⟂G `  G )
( A L B )  <->  D (⟂G `  G
) ( B L A ) ) )
2827pm5.74da 687 . . . . . . 7  |-  ( ph  ->  ( ( -.  A  =  B  ->  D (⟂G `  G ) ( A L B ) )  <-> 
( -.  A  =  B  ->  D (⟂G `  G ) ( B L A ) ) ) )
2920, 28mpbid 210 . . . . . 6  |-  ( ph  ->  ( -.  A  =  B  ->  D (⟂G `  G ) ( B L A ) ) )
3029orrd 378 . . . . 5  |-  ( ph  ->  ( A  =  B  \/  D (⟂G `  G
) ( B L A ) ) )
3130orcomd 388 . . . 4  |-  ( ph  ->  ( D (⟂G `  G
) ( B L A )  \/  A  =  B ) )
32 eqcom 2452 . . . . 5  |-  ( A  =  B  <->  B  =  A )
3332orbi2i 519 . . . 4  |-  ( ( D (⟂G `  G
) ( B L A )  \/  A  =  B )  <->  ( D
(⟂G `  G ) ( B L A )  \/  B  =  A ) )
3431, 33sylib 196 . . 3  |-  ( ph  ->  ( D (⟂G `  G
) ( B L A )  \/  B  =  A ) )
351, 2, 3, 4, 5, 11, 12, 13, 7, 6islmib 24025 . . 3  |-  ( ph  ->  ( A  =  ( M `  B )  <-> 
( ( B (midG `  G ) A )  e.  D  /\  ( D (⟂G `  G )
( B L A )  \/  B  =  A ) ) ) )
3617, 34, 35mpbir2and 922 . 2  |-  ( ph  ->  A  =  ( M `
 B ) )
3736eqcomd 2451 1  |-  ( ph  ->  ( M `  B
)  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804   class class class wbr 4437   ran crn 4990   ` cfv 5578  (class class class)co 6281   2c2 10591   Basecbs 14509   distcds 14583  TarskiGcstrkg 23697  DimTarskiGcstrkgld 23701  Itvcitv 23704  LineGclng 23705  ⟂Gcperpg 23944  midGcmid 24010  lInvGclmi 24011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-s2 12792  df-s3 12793  df-trkgc 23716  df-trkgb 23717  df-trkgcb 23718  df-trkgld 23720  df-trkg 23722  df-cgrg 23775  df-leg 23842  df-mir 23906  df-rag 23943  df-perpg 23945  df-mid 24012  df-lmi 24013
This theorem is referenced by:  lmilmi  24027
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