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Theorem mirbtwnhl 24186
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  =  ( c  e.  P  |->  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )
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 ) )
Distinct variable groups:    A, a,
b, c    I, a,
b, c    M, a,
b, c    P, a,
b, c    X, a,
b    Y, a, b    Z, a, b, c
Allowed substitution hints:    ph( a, b, c)    S( a, b, c)    G( a, b, c)    K( a, b, c)    L( a, b, c)    .- ( a, b, c)    X( c)    Y( c)

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  =  ( c  e.  P  |->  { <. a ,  b
>.  |  ( (
a  e.  P  /\  b  e.  P )  /\  ( a  =/=  c  /\  b  =/=  c  /\  ( a  e.  ( c I b )  \/  b  e.  ( c I a ) ) ) ) } )
4 mirhl.a . . . . . 6  |-  ( ph  ->  A  e.  P )
5 mirhl.x . . . . . 6  |-  ( ph  ->  X  e.  P )
61, 2, 3, 4, 5, 4hleqnid 24118 . . . . 5  |-  ( ph  ->  -.  A ( K `
 A ) X )
76adantr 465 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  -.  A ( K `  A ) X )
8 simpr 461 . . . . 5  |-  ( (
ph  /\  Z  =  A )  ->  Z  =  A )
98breq1d 4466 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  ( Z ( K `  A ) X  <->  A ( K `  A ) X ) )
107, 9mtbird 301 . . 3  |-  ( (
ph  /\  Z  =  A )  ->  -.  Z ( K `  A ) X )
11 mirhl.y . . . . . 6  |-  ( ph  ->  Y  e.  P )
121, 2, 3, 4, 11, 4hleqnid 24118 . . . . 5  |-  ( ph  ->  -.  A ( K `
 A ) Y )
1312adantr 465 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  -.  A ( K `  A ) Y )
148fveq2d 5876 . . . . . 6  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  Z )  =  ( M `  A ) )
15 mirval.d . . . . . . . 8  |-  .-  =  ( dist `  G )
16 mirval.l . . . . . . . 8  |-  L  =  (LineG `  G )
17 mirval.s . . . . . . . 8  |-  S  =  (pInvG `  G )
18 mirval.g . . . . . . . 8  |-  ( ph  ->  G  e. TarskiG )
19 mirhl.m . . . . . . . 8  |-  M  =  ( S `  A
)
201, 15, 2, 16, 17, 18, 4, 19mircinv 24174 . . . . . . 7  |-  ( ph  ->  ( M `  A
)  =  A )
2120adantr 465 . . . . . 6  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  A )  =  A )
2214, 21eqtrd 2498 . . . . 5  |-  ( (
ph  /\  Z  =  A )  ->  ( M `  Z )  =  A )
2322breq1d 4466 . . . 4  |-  ( (
ph  /\  Z  =  A )  ->  (
( M `  Z
) ( K `  A ) Y  <->  A ( K `  A ) Y ) )
2413, 23mtbird 301 . . 3  |-  ( (
ph  /\  Z  =  A )  ->  -.  ( M `  Z ) ( K `  A
) Y )
2510, 242falsed 351 . 2  |-  ( (
ph  /\  Z  =  A )  ->  ( Z ( K `  A ) X  <->  ( M `  Z ) ( K `
 A ) Y ) )
26 simplr 755 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Z  =/=  A )
2726neneqd 2659 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  -.  Z  =  A
)
2818ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  G  e. TarskiG )
294ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  A  e.  P )
30 mirhl.z . . . . . . . . 9  |-  ( ph  ->  Z  e.  P )
3130ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  Z  e.  P )
32 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  Z )  =  A )
3320ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  A )  =  A )
3432, 33eqtr4d 2501 . . . . . . . 8  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  ( M `  Z )  =  ( M `  A ) )
351, 15, 2, 16, 17, 28, 29, 19, 31, 29, 34mireq 24172 . . . . . . 7  |-  ( ( ( ( ph  /\  Z  =/=  A )  /\  Z ( K `  A ) X )  /\  ( M `  Z )  =  A )  ->  Z  =  A )
3627, 35mtand 659 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  -.  ( M `  Z
)  =  A )
3736neqned 2660 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
)  =/=  A )
38 mirbtwnhl.2 . . . . . 6  |-  ( ph  ->  Y  =/=  A )
3938ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Y  =/=  A )
4018ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  G  e. TarskiG )
415ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  X  e.  P )
424ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  A  e.  P )
431, 15, 2, 16, 17, 18, 4, 19, 30mircl 24168 . . . . . . 7  |-  ( ph  ->  ( M `  Z
)  e.  P )
4443ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
)  e.  P )
4511ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Y  e.  P )
46 mirbtwnhl.1 . . . . . . 7  |-  ( ph  ->  X  =/=  A )
4746ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  X  =/=  A )
4830ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  Z  e.  P )
491, 2, 3, 30, 5, 4ishlg 24112 . . . . . . . . . . 11  |-  ( ph  ->  ( Z ( K `
 A ) X  <-> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) ) )
5049adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  Z  =/=  A )  ->  ( Z
( K `  A
) X  <->  ( Z  =/=  A  /\  X  =/= 
A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) ) )
5150biimpa 484 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) )
5251simp3d 1010 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) )
5352orcomd 388 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( X  e.  ( A I Z )  \/  Z  e.  ( A I X ) ) )
541, 15, 2, 16, 17, 40, 19, 42, 41, 48, 53mirconn 24184 . . . . . 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 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  ->  A  e.  ( X I Y ) )
571, 2, 40, 41, 42, 44, 45, 47, 54, 56tgbtwnconn2 24089 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) )
5837, 39, 573jca 1176 . . . 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, 11, 4ishlg 24112 . . . . . 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 465 . . . . 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 465 . . . 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 232 . . 3  |-  ( ( ( ph  /\  Z  =/=  A )  /\  Z
( K `  A
) X )  -> 
( M `  Z
) ( K `  A ) Y )
63 simplr 755 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z  =/=  A )
6446ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  X  =/=  A )
6518ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  G  e. TarskiG )
6611ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Y  e.  P )
674ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  P )
6830ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z  e.  P )
695ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  X  e.  P )
7038ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Y  =/=  A )
7120ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  A
)  =  A )
7243ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  Z
)  e.  P )
731, 15, 2, 16, 17, 65, 67, 19, 66mircl 24168 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  Y
)  e.  P )
7460biimpa 484 . . . . . . . . . . 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 1010 . . . . . . . . . 10  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( ( M `  Z )  e.  ( A I Y )  \/  Y  e.  ( A I ( M `
 Z ) ) ) )
761, 15, 2, 16, 17, 65, 19, 67, 72, 66, 75mirconn 24184 . . . . . . . . 9  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( ( M `  Z )
I ( M `  Y ) ) )
771, 15, 2, 65, 72, 67, 73, 76tgbtwncom 24005 . . . . . . . 8  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( ( M `  Y )
I ( M `  Z ) ) )
7871, 77eqeltrd 2545 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( M `  A
)  e.  ( ( M `  Y ) I ( M `  Z ) ) )
791, 15, 2, 16, 17, 65, 67, 19, 66, 67, 68mirbtwnb 24178 . . . . . . 7  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( A  e.  ( Y I Z )  <-> 
( M `  A
)  e.  ( ( M `  Y ) I ( M `  Z ) ) ) )
8078, 79mpbird 232 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( Y I Z ) )
811, 15, 2, 18, 5, 4, 11, 55tgbtwncom 24005 . . . . . . 7  |-  ( ph  ->  A  e.  ( Y I X ) )
8281ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  A  e.  ( Y I X ) )
831, 2, 65, 66, 67, 68, 69, 70, 80, 82tgbtwnconn2 24089 . . . . 5  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) )
8463, 64, 833jca 1176 . . . 4  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  -> 
( Z  =/=  A  /\  X  =/=  A  /\  ( Z  e.  ( A I X )  \/  X  e.  ( A I Z ) ) ) )
8550adantr 465 . . . 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 232 . . 3  |-  ( ( ( ph  /\  Z  =/=  A )  /\  ( M `  Z )
( K `  A
) Y )  ->  Z ( K `  A ) X )
8762, 86impbida 832 . 2  |-  ( (
ph  /\  Z  =/=  A )  ->  ( Z
( K `  A
) X  <->  ( M `  Z ) ( K `
 A ) Y ) )
8825, 87pm2.61dane 2775 1  |-  ( ph  ->  ( Z ( K `
 A ) X  <-> 
( M `  Z
) ( K `  A ) Y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456   {copab 4514    |-> cmpt 4515   ` cfv 5594  (class class class)co 6296   Basecbs 14644   distcds 14721  TarskiGcstrkg 23951  Itvcitv 23958  LineGclng 23959  pInvGcmir 24159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409  df-word 12546  df-concat 12548  df-s1 12549  df-s2 12825  df-s3 12826  df-trkgc 23970  df-trkgb 23971  df-trkgcb 23972  df-trkg 23976  df-cgrg 24029  df-mir 24160
This theorem is referenced by:  opphllem6  24250
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