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Theorem sgplpte21d2 26140
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
sgplpte21d2  |-  ( ph  ->  Y  e.  ( X S Y ) )

Proof of Theorem sgplpte21d2
StepHypRef Expression
1 sgplpte.1 . . . 4  |-  P  =  (PPoints `  G )
2 sgplpte.3 . . . 4  |-  S  =  ( seg `  G
)
3 sgplpte.4 . . . 4  |-  ( ph  ->  G  e. Ibg )
4 sgplpte21d1.6 . . . 4  |-  ( ph  ->  Y  e.  P )
51, 2, 3, 4, 4sgplpte21d1 26139 . . 3  |-  ( ph  ->  Y  e.  ( Y S Y ) )
6 oveq1 5865 . . . 4  |-  ( X  =  Y  ->  ( X S Y )  =  ( Y S Y ) )
76eleq2d 2350 . . 3  |-  ( X  =  Y  ->  ( Y  e.  ( X S Y )  <->  Y  e.  ( Y S Y ) ) )
85, 7syl5ibr 212 . 2  |-  ( X  =  Y  ->  ( ph  ->  Y  e.  ( X S Y ) ) )
94adantl 452 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  P )
103adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  G  e. Ibg )
11 sgplpte.5 . . . . . 6  |-  ( ph  ->  X  e.  P )
1211adantl 452 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  X  e.  P )
13 eqid 2283 . . . . 5  |-  (btw `  G )  =  (btw
`  G )
14 simpl 443 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  X  =/=  Y )
151, 2, 10, 12, 13, 9, 14sgplpte21d 26136 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  ( ( Y  e.  P  /\  ( Y  e.  ( X (btw `  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )  ->  Y  e.  ( X S Y ) ) )
16 eqid 2283 . . . . 5  |-  Y  =  Y
17 simpr 447 . . . . . 6  |-  ( ( Y  =  Y  /\  Y  e.  P )  ->  Y  e.  P )
18 3mix3 1126 . . . . . . 7  |-  ( Y  =  Y  ->  ( Y  e.  ( X
(btw `  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )
1918adantr 451 . . . . . 6  |-  ( ( Y  =  Y  /\  Y  e.  P )  ->  ( Y  e.  ( X (btw `  G
) Y )  \/  Y  =  X  \/  Y  =  Y )
)
20 pm2.27 35 . . . . . 6  |-  ( ( Y  e.  P  /\  ( Y  e.  ( X (btw `  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )  -> 
( ( ( Y  e.  P  /\  ( Y  e.  ( X
(btw `  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )  ->  Y  e.  ( X S Y ) )  ->  Y  e.  ( X S Y ) ) )
2117, 19, 20syl2anc 642 . . . . 5  |-  ( ( Y  =  Y  /\  Y  e.  P )  ->  ( ( ( Y  e.  P  /\  ( Y  e.  ( X
(btw `  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )  ->  Y  e.  ( X S Y ) )  ->  Y  e.  ( X S Y ) ) )
2216, 21mpan 651 . . . 4  |-  ( Y  e.  P  ->  (
( ( Y  e.  P  /\  ( Y  e.  ( X (btw
`  G ) Y )  \/  Y  =  X  \/  Y  =  Y ) )  ->  Y  e.  ( X S Y ) )  ->  Y  e.  ( X S Y ) ) )
239, 15, 22sylc 56 . . 3  |-  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  ( X S Y ) )
2423ex 423 . 2  |-  ( X  =/=  Y  ->  ( ph  ->  Y  e.  ( X S Y ) ) )
258, 24pm2.61ine 2522 1  |-  ( ph  ->  Y  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   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  btwcbtw 26106  Ibgcibg 26107   segcseg 26130
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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