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Theorem mirhl 24803
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 472 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  G  e. TarskiG )
8 mirhl.a . . . . . . 7  |-  ( ph  ->  A  e.  P )
98adantr 472 . . . . . 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 472 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  X  e.  P )
13 mirhl.z . . . . . . 7  |-  ( ph  ->  Z  e.  P )
1413adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  Z  e.  P )
15 simpr 468 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  ( M `  X )  =  ( M `  Z ) )
161, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15mireq 24789 . . . . 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 24726 . . . . . . . . 9  |-  ( ph  ->  ( X ( K `
 Z ) Y  <-> 
( X  =/=  Z  /\  Y  =/=  Z  /\  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) ) ) )
2117, 20mpbid 215 . . . . . . . 8  |-  ( ph  ->  ( X  =/=  Z  /\  Y  =/=  Z  /\  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) ) )
2221simp1d 1042 . . . . . . 7  |-  ( ph  ->  X  =/=  Z )
2322adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  X  =/=  Z )
2423neneqd 2648 . . . . 5  |-  ( (
ph  /\  ( M `  X )  =  ( M `  Z ) )  ->  -.  X  =  Z )
2516, 24pm2.65da 586 . . . 4  |-  ( ph  ->  -.  ( M `  X )  =  ( M `  Z ) )
2625neqned 2650 . . 3  |-  ( ph  ->  ( M `  X
)  =/=  ( M `
 Z ) )
276adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  G  e. TarskiG )
288adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  A  e.  P )
2919adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  e.  P )
3013adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Z  e.  P )
31 simpr 468 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  ( M `  Y )  =  ( M `  Z ) )
321, 2, 3, 4, 5, 27, 28, 10, 29, 30, 31mireq 24789 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  =  Z )
3321simp2d 1043 . . . . . . 7  |-  ( ph  ->  Y  =/=  Z )
3433adantr 472 . . . . . 6  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  Y  =/=  Z )
3534neneqd 2648 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  =  ( M `  Z ) )  ->  -.  Y  =  Z )
3632, 35pm2.65da 586 . . . 4  |-  ( ph  ->  -.  ( M `  Y )  =  ( M `  Z ) )
3736neqned 2650 . . 3  |-  ( ph  ->  ( M `  Y
)  =/=  ( M `
 Z ) )
3821simp3d 1044 . . . 4  |-  ( ph  ->  ( X  e.  ( Z I Y )  \/  Y  e.  ( Z I X ) ) )
396adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  G  e. TarskiG )
408adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  A  e.  P )
4113adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  Z  e.  P )
4211adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  X  e.  P )
4319adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  Y  e.  P )
44 simpr 468 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  X  e.  ( Z I Y ) )
451, 2, 3, 4, 5, 39, 40, 10, 41, 42, 43, 44mirbtwni 24795 . . . . . 6  |-  ( (
ph  /\  X  e.  ( Z I Y ) )  ->  ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) ) )
4645ex 441 . . . . 5  |-  ( ph  ->  ( X  e.  ( Z I Y )  ->  ( M `  X )  e.  ( ( M `  Z
) I ( M `
 Y ) ) ) )
476adantr 472 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  G  e. TarskiG )
488adantr 472 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  A  e.  P )
4913adantr 472 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Z  e.  P )
5019adantr 472 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Y  e.  P )
5111adantr 472 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  X  e.  P )
52 simpr 468 . . . . . . 7  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  Y  e.  ( Z I X ) )
531, 2, 3, 4, 5, 47, 48, 10, 49, 50, 51, 52mirbtwni 24795 . . . . . 6  |-  ( (
ph  /\  Y  e.  ( Z I X ) )  ->  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) )
5453ex 441 . . . . 5  |-  ( ph  ->  ( Y  e.  ( Z I X )  ->  ( M `  Y )  e.  ( ( M `  Z
) I ( M `
 X ) ) ) )
5546, 54orim12d 856 . . . 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 1210 . 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 24785 . . 3  |-  ( ph  ->  ( M `  X
)  e.  P )
591, 2, 3, 4, 5, 6, 8, 10, 19mircl 24785 . . 3  |-  ( ph  ->  ( M `  Y
)  e.  P )
601, 2, 3, 4, 5, 6, 8, 10, 13mircl 24785 . . 3  |-  ( ph  ->  ( M `  Z
)  e.  P )
611, 3, 18, 58, 59, 60, 6ishlg 24726 . 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 240 1  |-  ( ph  ->  ( M `  X
) ( K `  ( M `  Z ) ) ( M `  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   distcds 15277  TarskiGcstrkg 24557  Itvcitv 24563  LineGclng 24564  hlGchlg 24724  pInvGcmir 24776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003  df-s3 13004  df-trkgc 24575  df-trkgb 24576  df-trkgcb 24577  df-trkg 24580  df-cgrg 24635  df-hlg 24725  df-mir 24777
This theorem is referenced by:  opphllem3  24870
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