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Theorem rayline 26259
Description: A ray is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-2016.)
Hypotheses
Ref Expression
rayline.1  |-  P  =  (PPoints `  G )
rayline.2  |-  L  =  ( line `  G
)
rayline.3  |-  R  =  (ray `  G )
rayline.4  |-  ( ph  ->  G  e. Ibg )
rayline.5  |-  ( ph  ->  X  e.  P )
rayline.6  |-  ( ph  ->  Y  e.  P )
Assertion
Ref Expression
rayline  |-  ( ph  ->  ( X R Y )  C_  ( X L Y ) )

Proof of Theorem rayline
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 ssid 3210 . . . 4  |-  { Y }  C_  { Y }
2 rayline.1 . . . . . 6  |-  P  =  (PPoints `  G )
3 rayline.3 . . . . . 6  |-  R  =  (ray `  G )
4 rayline.4 . . . . . 6  |-  ( ph  ->  G  e. Ibg )
5 rayline.6 . . . . . 6  |-  ( ph  ->  Y  e.  P )
62, 3, 4, 5isray2 26256 . . . . 5  |-  ( ph  ->  ( Y R Y )  =  { Y } )
7 rayline.2 . . . . . 6  |-  L  =  ( line `  G
)
84isibg1a 26214 . . . . . 6  |-  ( ph  ->  G  e. Ig )
92, 7, 8, 5lineval3a 26186 . . . . 5  |-  ( ph  ->  ( Y L Y )  =  { Y } )
106, 9sseq12d 3220 . . . 4  |-  ( ph  ->  ( ( Y R Y )  C_  ( Y L Y )  <->  { Y }  C_  { Y }
) )
111, 10mpbiri 224 . . 3  |-  ( ph  ->  ( Y R Y )  C_  ( Y L Y ) )
12 oveq1 5881 . . . 4  |-  ( X  =  Y  ->  ( X R Y )  =  ( Y R Y ) )
13 oveq1 5881 . . . 4  |-  ( X  =  Y  ->  ( X L Y )  =  ( Y L Y ) )
1412, 13sseq12d 3220 . . 3  |-  ( X  =  Y  ->  (
( X R Y )  C_  ( X L Y )  <->  ( Y R Y )  C_  ( Y L Y ) ) )
1511, 14syl5ibr 212 . 2  |-  ( X  =  Y  ->  ( ph  ->  ( X R Y )  C_  ( X L Y ) ) )
16 eqid 2296 . . . . 5  |-  ( seg `  G )  =  ( seg `  G )
174adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  G  e. Ibg )
18 rayline.5 . . . . . 6  |-  ( ph  ->  X  e.  P )
1918adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  X  e.  P )
205adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  P )
21 eqid 2296 . . . . 5  |-  (btw `  G )  =  (btw
`  G )
22 simpl 443 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  X  =/=  Y )
232, 16, 3, 17, 19, 20, 21, 22isray 26257 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  ( X R Y )  =  ( ( X ( seg `  G ) Y )  u.  { p  e.  P  |  Y  e.  ( X (btw `  G ) p ) } ) )
242, 16, 4, 18, 7, 5segline 26244 . . . . . 6  |-  ( ph  ->  ( X ( seg `  G ) Y ) 
C_  ( X L Y ) )
2524adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  ( X
( seg `  G
) Y )  C_  ( X L Y ) )
26173ad2ant1 976 . . . . . . 7  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  G  e. Ibg )
27193ad2ant1 976 . . . . . . 7  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  X  e.  P )
28203ad2ant1 976 . . . . . . 7  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  Y  e.  P )
29 simp2 956 . . . . . . 7  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  p  e.  P )
30 simp3 957 . . . . . . 7  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  Y  e.  ( X (btw `  G
) p ) )
312, 21, 26, 27, 28, 29, 30, 7isibg1a8 26230 . . . . . 6  |-  ( ( ( X  =/=  Y  /\  ph )  /\  p  e.  P  /\  Y  e.  ( X (btw `  G ) p ) )  ->  p  e.  ( X L Y ) )
3231rabssdv 3266 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  { p  e.  P  |  Y  e.  ( X (btw `  G ) p ) }  C_  ( X L Y ) )
3325, 32unssd 3364 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  ( ( X ( seg `  G
) Y )  u. 
{ p  e.  P  |  Y  e.  ( X (btw `  G )
p ) } ) 
C_  ( X L Y ) )
3423, 33eqsstrd 3225 . . 3  |-  ( ( X  =/=  Y  /\  ph )  ->  ( X R Y )  C_  ( X L Y ) )
3534ex 423 . 2  |-  ( X  =/=  Y  ->  ( ph  ->  ( X R Y )  C_  ( X L Y ) ) )
3615, 35pm2.61ine 2535 1  |-  ( ph  ->  ( X R Y )  C_  ( X L Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560    u. cun 3163    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159   linecline 26179  btwcbtw 26209  Ibgcibg 26210   segcseg 26233  raycray2 26254
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  df-seg2 26234  df-ray2 26255
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