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Theorem mirbtwnhl 24725
Description: If the center of the point inversion  A is between two points  X and  Y, then the half lines are mirrored. (Contributed by Thierry Arnoux, 3-Mar-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 )
mirbtwnhl.1  |-  ( ph  ->  X  =/=  A )
mirbtwnhl.2  |-  ( ph  ->  Y  =/=  A )
mirbtwnhl.3  |-  ( ph  ->  A  e.  ( X I Y ) )
Assertion
Ref Expression
mirbtwnhl  |-  ( ph  ->  ( Z ( K `
 A ) X  <-> 
( M `  Z
) ( K `  A ) Y ) )

Proof of Theorem mirbtwnhl
StepHypRef Expression
1 mirval.p . . . . . 6  |-  P  =  ( Base `  G
)
2 mirval.i . . . . . 6  |-  I  =  (Itv `  G )
3 mirhl.k . . . . . 6  |-  K  =  (hlG `  G )
4 mirhl.a . . . . . 6  |-  ( ph  ->  A  e.  P )
5 mirhl.x . . . . . 6  |-  ( ph  ->  X  e.  P )
6 mirval.g . . . . . 6  |-  ( ph  ->  G  e. TarskiG )
71, 2, 3, 4, 5, 4, 6hleqnid 24653 . . . . 5  |-  ( ph  ->  -.  A ( K `
 A ) X )
87adantr 467 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  -.  A ( K `  A ) X )
9 simpr 463 . . . . 5  |-  ( (
ph  /\  Z  =  A )  ->  Z  =  A )
109breq1d 4412 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  ( Z ( K `  A ) X  <->  A ( K `  A ) X ) )
118, 10mtbird 303 . . 3  |-  ( (
ph  /\  Z  =  A )  ->  -.  Z ( K `  A ) X )
12 mirhl.y . . . . . 6  |-  ( ph  ->  Y  e.  P )
131, 2, 3, 4, 12, 4, 6hleqnid 24653 . . . . 5  |-  ( ph  ->  -.  A ( K `
 A ) Y )
1413adantr 467 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  -.  A ( K `  A ) Y )
159fveq2d 5869 . . . . . 6  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  Z )  =  ( M `  A ) )
16 mirval.d . . . . . . . 8  |-  .-  =  ( dist `  G )
17 mirval.l . . . . . . . 8  |-  L  =  (LineG `  G )
18 mirval.s . . . . . . . 8  |-  S  =  (pInvG `  G )
19 mirhl.m . . . . . . . 8  |-  M  =  ( S `  A
)
201, 16, 2, 17, 18, 6, 4, 19mircinv 24713 . . . . . . 7  |-  ( ph  ->  ( M `  A
)  =  A )
2120adantr 467 . . . . . 6  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  A )  =  A )
2215, 21eqtrd 2485 . . . . 5  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  Z )  =  A )
2322breq1d 4412 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  (
( M `  Z
) ( K `  A ) Y  <->  A ( K `  A ) Y ) )
2414, 23mtbird 303 . . 3  |-  ( (
ph  /\  Z  =  A )  ->  -.  ( M `  Z ) ( K `  A
) Y )
2511, 242falsed 353 . 2  |-  ( (
ph  /\  Z  =  A )  ->  ( Z ( K `  A ) X  <->  ( M `  Z ) ( K `
 A ) Y ) )
26 simplr 762 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Z  =/=  A )
2726neneqd 2629 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  -.  Z  =  A
)
286ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  G  e. TarskiG )
294ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  A  e.  P )
30 mirhl.z . . . . . . . . 9  |-  ( ph  ->  Z  e.  P )
3130ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  Z  e.  P )
32 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  Z )  =  A )
3320ad3antrrr 736 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  A )  =  A )
3432, 33eqtr4d 2488 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  Z )  =  ( M `  A ) )
351, 16, 2, 17, 18, 28, 29, 19, 31, 29, 34mireq 24710 . . . . . . 7  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  Z  =  A )
3627, 35mtand 665 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  -.  ( M `  Z
)  =  A )
3736neqned 2631 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
)  =/=  A )
38 mirbtwnhl.2 . . . . . 6  |-  ( ph  ->  Y  =/=  A )
3938ad2antrr 732 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Y  =/=  A )
406ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  G  e. TarskiG )
415ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  X  e.  P )
424ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  A  e.  P )
431, 16, 2, 17, 18, 6, 4, 19, 30mircl 24706 . . . . . . 7  |-  ( ph  ->  ( M `  Z
)  e.  P )
4443ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
)  e.  P )
4512ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Y  e.  P )
46 mirbtwnhl.1 . . . . . . 7  |-  ( ph  ->  X  =/=  A )
4746ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  X  =/=  A )
4830ad2antrr 732 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Z  e.  P )
491, 2, 3, 30, 5, 4, 6ishlg 24647 . . . . . . . . . . 11  |-  ( ph  ->  ( Z ( K `
 A ) X  <-> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) ) )
5049adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  Z  =/=  A )  ->  ( Z
( K `  A
) X  <->  ( Z  =/=  A  /\  X  =/= 
A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) ) )
5150biimpa 487 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) )
5251simp3d 1022 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) )
5352orcomd 390 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( X  e.  ( A I Z )  \/  Z  e.  ( A I X ) ) )
541, 16, 2, 17, 18, 40, 19, 42, 41, 48, 53mirconn 24723 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  A  e.  ( X I ( M `  Z ) ) )
55 mirbtwnhl.3 . . . . . . 7  |-  ( ph  ->  A  e.  ( X I Y ) )
5655ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  A  e.  ( X I Y ) )
571, 2, 40, 41, 42, 44, 45, 47, 54, 56tgbtwnconn2 24621 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) )
5837, 39, 573jca 1188 . . . 4  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( ( M `  Z )  =/=  A  /\  Y  =/=  A  /\  ( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) ) )
591, 2, 3, 43, 12, 4, 6ishlg 24647 . . . . . 6  |-  ( ph  ->  ( ( M `  Z ) ( K `
 A ) Y  <-> 
( ( M `  Z )  =/=  A  /\  Y  =/=  A  /\  ( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) ) ) )
6059adantr 467 . . . . 5  |-  ( (
ph  /\  Z  =/=  A )  ->  ( ( M `  Z )
( K `  A
) Y  <->  ( ( M `  Z )  =/=  A  /\  Y  =/= 
A  /\  ( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `  Z ) ) ) ) ) )
6160adantr 467 . . . 4  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( ( M `  Z ) ( K `
 A ) Y  <-> 
( ( M `  Z )  =/=  A  /\  Y  =/=  A  /\  ( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) ) ) )
6258, 61mpbird 236 . . 3  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
) ( K `  A ) Y )
63 simplr 762 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z  =/=  A )
6446ad2antrr 732 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  X  =/=  A )
656ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  G  e. TarskiG )
6612ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Y  e.  P )
674ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  P )
6830ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z  e.  P )
695ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  X  e.  P )
7038ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Y  =/=  A )
7120ad2antrr 732 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  A
)  =  A )
7243ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  Z
)  e.  P )
731, 16, 2, 17, 18, 65, 67, 19, 66mircl 24706 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  Y
)  e.  P )
7460biimpa 487 . . . . . . . . . . 11  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( ( M `  Z )  =/=  A  /\  Y  =/=  A  /\  ( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) ) )
7574simp3d 1022 . . . . . . . . . 10  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) )
761, 16, 2, 17, 18, 65, 19, 67, 72, 66, 75mirconn 24723 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( ( M `  Z )
I ( M `  Y ) ) )
771, 16, 2, 65, 72, 67, 73, 76tgbtwncom 24532 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( ( M `  Y )
I ( M `  Z ) ) )
7871, 77eqeltrd 2529 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  A
)  e.  ( ( M `  Y ) I ( M `  Z ) ) )
791, 16, 2, 17, 18, 65, 67, 19, 66, 67, 68mirbtwnb 24717 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( A  e.  ( Y I Z )  <-> 
( M `  A
)  e.  ( ( M `  Y ) I ( M `  Z ) ) ) )
8078, 79mpbird 236 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( Y I Z ) )
811, 16, 2, 6, 5, 4, 12, 55tgbtwncom 24532 . . . . . . 7  |-  ( ph  ->  A  e.  ( Y I X ) )
8281ad2antrr 732 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( Y I X ) )
831, 2, 65, 66, 67, 68, 69, 70, 80, 82tgbtwnconn2 24621 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) )
8463, 64, 833jca 1188 . . . 4  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) )
8550adantr 467 . . . 4  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( Z ( K `
 A ) X  <-> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) ) )
8684, 85mpbird 236 . . 3  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z ( K `  A ) X )
8762, 86impbida 843 . 2  |-  ( (
ph  /\  Z  =/=  A )  ->  ( Z
( K `  A
) X  <->  ( M `  Z ) ( K `
 A ) Y ) )
8825, 87pm2.61dane 2711 1  |-  ( ph  ->  ( Z ( K `
 A ) X  <-> 
( M `  Z
) ( K `  A ) Y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ 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:  opphllem6  24794
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