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Theorem sgplpte21d1 26139
Description: The extremities belong to a segment. (For my private use only. Don't use.) (Contributed by FL, 29-Jul-2016.)
Hypotheses
Ref Expression
sgplpte.1  |-  P  =  (PPoints `  G )
sgplpte.3  |-  S  =  ( seg `  G
)
sgplpte.4  |-  ( ph  ->  G  e. Ibg )
sgplpte.5  |-  ( ph  ->  X  e.  P )
sgplpte21d1.6  |-  ( ph  ->  Y  e.  P )
Assertion
Ref Expression
sgplpte21d1  |-  ( ph  ->  X  e.  ( X S Y ) )

Proof of Theorem sgplpte21d1
StepHypRef Expression
1 sgplpte.5 . . . . . 6  |-  ( ph  ->  X  e.  P )
2 snidg 3665 . . . . . 6  |-  ( X  e.  P  ->  X  e.  { X } )
31, 2syl 15 . . . . 5  |-  ( ph  ->  X  e.  { X } )
4 sgplpte.1 . . . . . 6  |-  P  =  (PPoints `  G )
5 sgplpte.3 . . . . . 6  |-  S  =  ( seg `  G
)
6 sgplpte.4 . . . . . 6  |-  ( ph  ->  G  e. Ibg )
74, 5, 6, 1sgplpte22 26138 . . . . 5  |-  ( ph  ->  ( X S X )  =  { X } )
83, 7eleqtrrd 2360 . . . 4  |-  ( ph  ->  X  e.  ( X S X ) )
9 oveq2 5866 . . . . 5  |-  ( Y  =  X  ->  ( X S Y )  =  ( X S X ) )
109eleq2d 2350 . . . 4  |-  ( Y  =  X  ->  ( X  e.  ( X S Y )  <->  X  e.  ( X S X ) ) )
118, 10syl5ibr 212 . . 3  |-  ( Y  =  X  ->  ( ph  ->  X  e.  ( X S Y ) ) )
1211eqcoms 2286 . 2  |-  ( X  =  Y  ->  ( ph  ->  X  e.  ( X S Y ) ) )
131adantl 452 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  X  e.  P )
146adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  G  e. Ibg )
15 eqid 2283 . . . . 5  |-  (btw `  G )  =  (btw
`  G )
16 sgplpte21d1.6 . . . . . 6  |-  ( ph  ->  Y  e.  P )
1716adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  P )
18 simpl 443 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  X  =/=  Y )
194, 5, 14, 13, 15, 17, 18sgplpte21d 26136 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  ( ( X  e.  P  /\  ( X  e.  ( X (btw `  G ) Y )  \/  X  =  X  \/  X  =  Y ) )  ->  X  e.  ( X S Y ) ) )
20 eqid 2283 . . . . 5  |-  X  =  X
21 simpr 447 . . . . . 6  |-  ( ( X  =  X  /\  X  e.  P )  ->  X  e.  P )
22 3mix2 1125 . . . . . . 7  |-  ( X  =  X  ->  ( X  e.  ( X
(btw `  G ) Y )  \/  X  =  X  \/  X  =  Y ) )
2322adantr 451 . . . . . 6  |-  ( ( X  =  X  /\  X  e.  P )  ->  ( X  e.  ( X (btw `  G
) Y )  \/  X  =  X  \/  X  =  Y )
)
24 pm2.27 35 . . . . . 6  |-  ( ( X  e.  P  /\  ( X  e.  ( X (btw `  G ) Y )  \/  X  =  X  \/  X  =  Y ) )  -> 
( ( ( X  e.  P  /\  ( X  e.  ( X
(btw `  G ) Y )  \/  X  =  X  \/  X  =  Y ) )  ->  X  e.  ( X S Y ) )  ->  X  e.  ( X S Y ) ) )
2521, 23, 24syl2anc 642 . . . . 5  |-  ( ( X  =  X  /\  X  e.  P )  ->  ( ( ( X  e.  P  /\  ( X  e.  ( X
(btw `  G ) Y )  \/  X  =  X  \/  X  =  Y ) )  ->  X  e.  ( X S Y ) )  ->  X  e.  ( X S Y ) ) )
2620, 25mpan 651 . . . 4  |-  ( X  e.  P  ->  (
( ( X  e.  P  /\  ( X  e.  ( X (btw
`  G ) Y )  \/  X  =  X  \/  X  =  Y ) )  ->  X  e.  ( X S Y ) )  ->  X  e.  ( X S Y ) ) )
2713, 19, 26sylc 56 . . 3  |-  ( ( X  =/=  Y  /\  ph )  ->  X  e.  ( X S Y ) )
2827ex 423 . 2  |-  ( X  =/=  Y  ->  ( ph  ->  X  e.  ( X S Y ) ) )
2912, 28pm2.61ine 2522 1  |-  ( ph  ->  X  e.  ( X S Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684    =/= wne 2446   {csn 3640   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107   segcseg 26130
This theorem is referenced by:  sgplpte21d2  26140
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-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-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-seg2 26131
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