Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isibg1a6 Unicode version

Theorem isibg1a6 26125
Description: If  Y is between  X and 
Z, it belongs to the line XZ (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
isibg1a3a.1  |-  P  =  (PPoints `  G )
isibg1a3a.2  |-  B  =  (btw `  G )
isibg1a3a.3  |-  ( ph  ->  G  e. Ibg )
isibg1a3a.4  |-  ( ph  ->  X  e.  P )
isibg1a3a.5  |-  ( ph  ->  Y  e.  P )
isibg1a3a.6  |-  ( ph  ->  Z  e.  P )
isibg1a3a.7  |-  ( ph  ->  Y  e.  ( X B Z ) )
isibg1a3a.8  |-  M  =  ( line `  G
)
Assertion
Ref Expression
isibg1a6  |-  ( ph  ->  Y  e.  ( X M Z ) )

Proof of Theorem isibg1a6
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 isibg1a3a.1 . . 3  |-  P  =  (PPoints `  G )
2 isibg1a3a.2 . . 3  |-  B  =  (btw `  G )
3 isibg1a3a.3 . . 3  |-  ( ph  ->  G  e. Ibg )
4 isibg1a3a.4 . . 3  |-  ( ph  ->  X  e.  P )
5 isibg1a3a.5 . . 3  |-  ( ph  ->  Y  e.  P )
6 eqid 2283 . . 3  |-  (coln `  G )  =  (coln `  G )
7 isibg1a3a.7 . . 3  |-  ( ph  ->  Y  e.  ( X B Z ) )
8 isibg1a3a.6 . . 3  |-  ( ph  ->  Z  e.  P )
91, 2, 3, 4, 5, 6, 7, 8isibg1a2 26117 . 2  |-  ( ph  ->  { X ,  Y ,  Z }  e.  (coln `  G ) )
103isibg1a 26111 . . . . 5  |-  ( ph  ->  G  e. Ig )
11 tpex 4519 . . . . . 6  |-  { X ,  Y ,  Z }  e.  _V
1211a1i 10 . . . . 5  |-  ( ph  ->  { X ,  Y ,  Z }  e.  _V )
1310, 12iscol2 26093 . . . 4  |-  ( ph  ->  ( { X ,  Y ,  Z }  e.  (coln `  G )  <->  E. l  e.  (PLines `  G ) { X ,  Y ,  Z }  C_  l ) )
144, 5, 8tpssg 24932 . . . . . . . . 9  |-  ( ph  ->  ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  <->  { X ,  Y ,  Z }  C_  l ) )
15 simp12 986 . . . . . . . . . . 11  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  Y  e.  l )
16 eqid 2283 . . . . . . . . . . . 12  |-  (PLines `  G )  =  (PLines `  G )
17 isibg1a3a.8 . . . . . . . . . . . 12  |-  M  =  ( line `  G
)
18103ad2ant3 978 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  G  e. Ig )
1943ad2ant3 978 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  X  e.  P )
2083ad2ant3 978 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  Z  e.  P )
211, 2, 3, 4, 5, 8, 7isibg1spa 26123 . . . . . . . . . . . . 13  |-  ( ph  ->  X  =/=  Z )
22213ad2ant3 978 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  X  =/=  Z )
23 simp11 985 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  X  e.  l )
24 simp13 987 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  Z  e.  l )
25 simp2 956 . . . . . . . . . . . 12  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  l  e.  (PLines `  G )
)
261, 16, 17, 18, 19, 20, 22, 23, 24, 25lineval42 26080 . . . . . . . . . . 11  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  l  =  ( X M Z ) )
2715, 26eleqtrd 2359 . . . . . . . . . 10  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  Y  e.  ( X M Z ) )
28273exp 1150 . . . . . . . . 9  |-  ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  ->  ( l  e.  (PLines `  G )  ->  ( ph  ->  Y  e.  ( X M Z ) ) ) )
2914, 28syl6bir 220 . . . . . . . 8  |-  ( ph  ->  ( { X ,  Y ,  Z }  C_  l  ->  ( l  e.  (PLines `  G )  ->  ( ph  ->  Y  e.  ( X M Z ) ) ) ) )
3029com24 81 . . . . . . 7  |-  ( ph  ->  ( ph  ->  (
l  e.  (PLines `  G )  ->  ( { X ,  Y ,  Z }  C_  l  ->  Y  e.  ( X M Z ) ) ) ) )
3130pm2.43i 43 . . . . . 6  |-  ( ph  ->  ( l  e.  (PLines `  G )  ->  ( { X ,  Y ,  Z }  C_  l  ->  Y  e.  ( X M Z ) ) ) )
3231com3l 75 . . . . 5  |-  ( l  e.  (PLines `  G
)  ->  ( { X ,  Y ,  Z }  C_  l  -> 
( ph  ->  Y  e.  ( X M Z ) ) ) )
3332rexlimiv 2661 . . . 4  |-  ( E. l  e.  (PLines `  G ) { X ,  Y ,  Z }  C_  l  ->  ( ph  ->  Y  e.  ( X M Z ) ) )
3413, 33syl6bi 219 . . 3  |-  ( ph  ->  ( { X ,  Y ,  Z }  e.  (coln `  G )  ->  ( ph  ->  Y  e.  ( X M Z ) ) ) )
3534pm2.43a 45 . 2  |-  ( ph  ->  ( { X ,  Y ,  Z }  e.  (coln `  G )  ->  Y  e.  ( X M Z ) ) )
369, 35mpd 14 1  |-  ( ph  ->  Y  e.  ( X M Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788    C_ wss 3152   {ctp 3642   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076  colnccol 26090  btwcbtw 26106  Ibgcibg 26107
This theorem is referenced by:  isibg1a7  26126  isibg1a8  26127  segline  26141  lppotos  26144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-ig2 26061  df-li 26077  df-col 26091  df-ibg2 26109
  Copyright terms: Public domain W3C validator