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Theorem mirhl 24724
Description: If two points  X and  Y are on the same half-line from  Z, the same applies to the mirror points. (Contributed by Thierry Arnoux, 21-Feb-2020.)
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 )
mirhl.m  |-  M  =  ( S `  A
)
mirhl.k  |-  K  =  (hlG `  G )
mirhl.a  |-  ( ph  ->  A  e.  P )
mirhl.x  |-  ( ph  ->  X  e.  P )
mirhl.y  |-  ( ph  ->  Y  e.  P )
mirhl.z  |-  ( ph  ->  Z  e.  P )
mirhl.1  |-  ( ph  ->  X ( K `  Z ) Y )
Assertion
Ref Expression
mirhl  |-  ( ph  ->  ( M `  X
) ( K `  ( M `  Z ) ) ( M `  Y ) )

Proof of Theorem mirhl
StepHypRef Expression
1 mirval.p . . . . . 6  |-  P  =  ( Base `  G
)
2 mirval.d . . . . . 6  |-  .-  =  ( dist `  G )
3 mirval.i . . . . . 6  |-  I  =  (Itv `  G )
4 mirval.l . . . . . 6  |-  L  =  (LineG `  G )
5 mirval.s . . . . . 6  |-  S  =  (pInvG `  G )
6 mirval.g . . . . . . 7  |-  ( ph  ->  G  e. TarskiG )
76adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  G  e. TarskiG )
8 mirhl.a . . . . . . 7  |-  ( ph  ->  A  e.  P )
98adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  A  e.  P )
10 mirhl.m . . . . . 6  |-  M  =  ( S `  A
)
11 mirhl.x . . . . . . 7  |-  ( ph  ->  X  e.  P )
1211adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  X  e.  P )
13 mirhl.z . . . . . . 7  |-  ( ph  ->  Z  e.  P )
1413adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  Z  e.  P )
15 simpr 463 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  ( M `  X )  =  ( M `  Z ) )
161, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15mireq 24710 . . . . 5  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  X  =  Z )
17 mirhl.1 . . . . . . . . 9  |-  ( ph  ->  X ( K `  Z ) Y )
18 mirhl.k . . . . . . . . . 10  |-  K  =  (hlG `  G )
19 mirhl.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  P )
201, 3, 18, 11, 19, 13, 6ishlg 24647 . . . . . . . . 9  |-  ( ph  ->  ( X ( K `
 Z ) Y  <-> 
( X  =/=  Z  /\  Y  =/=  Z  /\  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) ) ) )
2117, 20mpbid 214 . . . . . . . 8  |-  ( ph  ->  ( X  =/=  Z  /\  Y  =/=  Z  /\  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) ) )
2221simp1d 1020 . . . . . . 7  |-  ( ph  ->  X  =/=  Z )
2322adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  X  =/=  Z )
2423neneqd 2629 . . . . 5  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  -.  X  =  Z )
2516, 24pm2.65da 580 . . . 4  |-  ( ph  ->  -.  ( M `  X )  =  ( M `  Z ) )
2625neqned 2631 . . 3  |-  ( ph  ->  ( M `  X
)  =/=  ( M `
 Z ) )
276adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  G  e. TarskiG )
288adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  A  e.  P )
2919adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  e.  P )
3013adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Z  e.  P )
31 simpr 463 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  ( M `  Y )  =  ( M `  Z ) )
321, 2, 3, 4, 5, 27, 28, 10, 29, 30, 31mireq 24710 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  =  Z )
3321simp2d 1021 . . . . . . 7  |-  ( ph  ->  Y  =/=  Z )
3433adantr 467 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  =/=  Z )
3534neneqd 2629 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  -.  Y  =  Z )
3632, 35pm2.65da 580 . . . 4  |-  ( ph  ->  -.  ( M `  Y )  =  ( M `  Z ) )
3736neqned 2631 . . 3  |-  ( ph  ->  ( M `  Y
)  =/=  ( M `
 Z ) )
3821simp3d 1022 . . . 4  |-  ( ph  ->  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) )
396adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  G  e. TarskiG )
408adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  A  e.  P )
4113adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  Z  e.  P )
4211adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  X  e.  P )
4319adantr 467 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  Y  e.  P )
44 simpr 463 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  X  e.  ( Z I Y ) )
451, 2, 3, 4, 5, 39, 40, 10, 41, 42, 43, 44mirbtwni 24716 . . . . . 6  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) ) )
4645ex 436 . . . . 5  |-  ( ph  ->  ( X  e.  ( Z I Y )  ->  ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) ) ) )
476adantr 467 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  G  e. TarskiG )
488adantr 467 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  A  e.  P )
4913adantr 467 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Z  e.  P )
5019adantr 467 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Y  e.  P )
5111adantr 467 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  X  e.  P )
52 simpr 463 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Y  e.  ( Z I X ) )
531, 2, 3, 4, 5, 47, 48, 10, 49, 50, 51, 52mirbtwni 24716 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) )
5453ex 436 . . . . 5  |-  ( ph  ->  ( Y  e.  ( Z I X )  ->  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) ) )
5546, 54orim12d 849 . . . 4  |-  ( ph  ->  ( ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) )  ->  (
( M `  X
)  e.  ( ( M `  Z ) I ( M `  Y ) )  \/  ( M `  Y
)  e.  ( ( M `  Z ) I ( M `  X ) ) ) ) )
5638, 55mpd 15 . . 3  |-  ( ph  ->  ( ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) )  \/  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) ) )
5726, 37, 563jca 1188 . 2  |-  ( ph  ->  ( ( M `  X )  =/=  ( M `  Z )  /\  ( M `  Y
)  =/=  ( M `
 Z )  /\  ( ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) )  \/  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) ) ) )
581, 2, 3, 4, 5, 6, 8, 10, 11mircl 24706 . . 3  |-  ( ph  ->  ( M `  X
)  e.  P )
591, 2, 3, 4, 5, 6, 8, 10, 19mircl 24706 . . 3  |-  ( ph  ->  ( M `  Y
)  e.  P )
601, 2, 3, 4, 5, 6, 8, 10, 13mircl 24706 . . 3  |-  ( ph  ->  ( M `  Z
)  e.  P )
611, 3, 18, 58, 59, 60, 6ishlg 24647 . 2  |-  ( ph  ->  ( ( M `  X ) ( K `
 ( M `  Z ) ) ( M `  Y )  <-> 
( ( M `  X )  =/=  ( M `  Z )  /\  ( M `  Y
)  =/=  ( M `
 Z )  /\  ( ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) )  \/  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) ) ) ) )
6257, 61mpbird 236 1  |-  ( ph  ->  ( M `  X
) ( K `  ( M `  Z ) ) ( M `  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   distcds 15199  TarskiGcstrkg 24478  Itvcitv 24484  LineGclng 24485  hlGchlg 24645  pInvGcmir 24697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-concat 12666  df-s1 12667  df-s2 12944  df-s3 12945  df-trkgc 24496  df-trkgb 24497  df-trkgcb 24498  df-trkg 24501  df-cgrg 24556  df-hlg 24646  df-mir 24698
This theorem is referenced by:  opphllem3  24791
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