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Theorem isibg1a6 26228
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 2296 . . 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 26220 . 2  |-  ( ph  ->  { X ,  Y ,  Z }  e.  (coln `  G ) )
103isibg1a 26214 . . . . 5  |-  ( ph  ->  G  e. Ig )
11 tpex 4535 . . . . . 6  |-  { X ,  Y ,  Z }  e.  _V
1211a1i 10 . . . . 5  |-  ( ph  ->  { X ,  Y ,  Z }  e.  _V )
1310, 12iscol2 26196 . . . 4  |-  ( ph  ->  ( { X ,  Y ,  Z }  e.  (coln `  G )  <->  E. l  e.  (PLines `  G ) { X ,  Y ,  Z }  C_  l ) )
144, 5, 8tpssg 25035 . . . . . . . . 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 2296 . . . . . . . . . . . 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 26226 . . . . . . . . . . . . 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 26183 . . . . . . . . . . 11  |-  ( ( ( X  e.  l  /\  Y  e.  l  /\  Z  e.  l )  /\  l  e.  (PLines `  G )  /\  ph )  ->  l  =  ( X M Z ) )
2715, 26eleqtrd 2372 . . . . . . . . . 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 2674 . . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {ctp 3655   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179  colnccol 26193  btwcbtw 26209  Ibgcibg 26210
This theorem is referenced by:  isibg1a7  26229  isibg1a8  26230  segline  26244  lppotos  26247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-ig2 26164  df-li 26180  df-col 26194  df-ibg2 26212
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