MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mirbtwnb Structured version   Unicode version

Theorem mirbtwnb 24253
Description: Point inversion preserves betweenness. Theorem 7.15 of [Schwabhauser] p. 51. (Contributed by Thierry Arnoux, 9-Jun-2019.)
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 )
mirval.a  |-  ( ph  ->  A  e.  P )
mirfv.m  |-  M  =  ( S `  A
)
miriso.1  |-  ( ph  ->  X  e.  P )
miriso.2  |-  ( ph  ->  Y  e.  P )
mirbtwnb.z  |-  ( ph  ->  Z  e.  P )
Assertion
Ref Expression
mirbtwnb  |-  ( ph  ->  ( Y  e.  ( X I Z )  <-> 
( M `  Y
)  e.  ( ( M `  X ) I ( M `  Z ) ) ) )

Proof of Theorem mirbtwnb
StepHypRef Expression
1 mirval.p . . 3  |-  P  =  ( Base `  G
)
2 mirval.d . . 3  |-  .-  =  ( dist `  G )
3 mirval.i . . 3  |-  I  =  (Itv `  G )
4 mirval.l . . 3  |-  L  =  (LineG `  G )
5 mirval.s . . 3  |-  S  =  (pInvG `  G )
6 mirval.g . . . 4  |-  ( ph  ->  G  e. TarskiG )
76adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  G  e. TarskiG )
8 mirval.a . . . 4  |-  ( ph  ->  A  e.  P )
98adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  A  e.  P )
10 mirfv.m . . 3  |-  M  =  ( S `  A
)
11 miriso.1 . . . 4  |-  ( ph  ->  X  e.  P )
1211adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  X  e.  P )
13 miriso.2 . . . 4  |-  ( ph  ->  Y  e.  P )
1413adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  P )
15 mirbtwnb.z . . . 4  |-  ( ph  ->  Z  e.  P )
1615adantr 463 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Z  e.  P )
17 simpr 459 . . 3  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  Y  e.  ( X I Z ) )
181, 2, 3, 4, 5, 7, 9, 10, 12, 14, 16, 17mirbtwni 24252 . 2  |-  ( (
ph  /\  Y  e.  ( X I Z ) )  ->  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )
196adantr 463 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  G  e. TarskiG )
208adantr 463 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  A  e.  P )
211, 2, 3, 4, 5, 19, 20, 10mirf 24242 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  M : P
--> P )
2211adantr 463 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  X  e.  P )
2321, 22ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( M `  X )  e.  P
)
2413adantr 463 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  Y  e.  P )
2521, 24ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( M `  Y )  e.  P
)
2615adantr 463 . . . . 5  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  Z  e.  P )
2721, 26ffvelrnd 6008 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( M `  Z )  e.  P
)
28 simpr 459 . . . 4  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )
291, 2, 3, 4, 5, 19, 20, 10, 23, 25, 27, 28mirbtwni 24252 . . 3  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( M `  ( M `  Y
) )  e.  ( ( M `  ( M `  X )
) I ( M `
 ( M `  Z ) ) ) )
301, 2, 3, 4, 5, 6, 8, 10, 13mirmir 24244 . . . . 5  |-  ( ph  ->  ( M `  ( M `  Y )
)  =  Y )
311, 2, 3, 4, 5, 6, 8, 10, 11mirmir 24244 . . . . . 6  |-  ( ph  ->  ( M `  ( M `  X )
)  =  X )
321, 2, 3, 4, 5, 6, 8, 10, 15mirmir 24244 . . . . . 6  |-  ( ph  ->  ( M `  ( M `  Z )
)  =  Z )
3331, 32oveq12d 6288 . . . . 5  |-  ( ph  ->  ( ( M `  ( M `  X ) ) I ( M `
 ( M `  Z ) ) )  =  ( X I Z ) )
3430, 33eleq12d 2536 . . . 4  |-  ( ph  ->  ( ( M `  ( M `  Y ) )  e.  ( ( M `  ( M `
 X ) ) I ( M `  ( M `  Z ) ) )  <->  Y  e.  ( X I Z ) ) )
3534adantr 463 . . 3  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  ( ( M `  ( M `  Y ) )  e.  ( ( M `  ( M `  X ) ) I ( M `
 ( M `  Z ) ) )  <-> 
Y  e.  ( X I Z ) ) )
3629, 35mpbid 210 . 2  |-  ( (
ph  /\  ( M `  Y )  e.  ( ( M `  X
) I ( M `
 Z ) ) )  ->  Y  e.  ( X I Z ) )
3718, 36impbida 830 1  |-  ( ph  ->  ( Y  e.  ( X I Z )  <-> 
( M `  Y
)  e.  ( ( M `  X ) I ( M `  Z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14716   distcds 14793  TarskiGcstrkg 24023  Itvcitv 24030  LineGclng 24031  pInvGcmir 24234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-concat 12528  df-s1 12529  df-s2 12804  df-s3 12805  df-trkgc 24042  df-trkgb 24043  df-trkgcb 24044  df-trkg 24048  df-cgrg 24104  df-mir 24235
This theorem is referenced by:  mirbtwnhl  24261
  Copyright terms: Public domain W3C validator