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Theorem segline 26141
Description: A segment is a part of a line. (For my private use only. Don't use.) (Contributed by FL, 20-May-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 )
segline.1  |-  M  =  ( line `  G
)
segline.2  |-  ( ph  ->  Y  e.  P )
Assertion
Ref Expression
segline  |-  ( ph  ->  ( X S Y )  C_  ( X M Y ) )

Proof of Theorem segline
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssid 3197 . . . 4  |-  { X }  C_  { X }
2 sgplpte.1 . . . . . 6  |-  P  =  (PPoints `  G )
3 sgplpte.3 . . . . . 6  |-  S  =  ( seg `  G
)
4 sgplpte.4 . . . . . 6  |-  ( ph  ->  G  e. Ibg )
5 sgplpte.5 . . . . . 6  |-  ( ph  ->  X  e.  P )
62, 3, 4, 5sgplpte22 26138 . . . . 5  |-  ( ph  ->  ( X S X )  =  { X } )
7 segline.1 . . . . . 6  |-  M  =  ( line `  G
)
84isibg1a 26111 . . . . . 6  |-  ( ph  ->  G  e. Ig )
92, 7, 8, 5lineval3a 26083 . . . . 5  |-  ( ph  ->  ( X M X )  =  { X } )
106, 9sseq12d 3207 . . . 4  |-  ( ph  ->  ( ( X S X )  C_  ( X M X )  <->  { X }  C_  { X }
) )
111, 10mpbiri 224 . . 3  |-  ( ph  ->  ( X S X )  C_  ( X M X ) )
12 id 19 . . . . . 6  |-  ( X  =  Y  ->  X  =  Y )
1312eqcomd 2288 . . . . 5  |-  ( X  =  Y  ->  Y  =  X )
1413oveq2d 5874 . . . 4  |-  ( X  =  Y  ->  ( X S Y )  =  ( X S X ) )
1513oveq2d 5874 . . . 4  |-  ( X  =  Y  ->  ( X M Y )  =  ( X M X ) )
1614, 15sseq12d 3207 . . 3  |-  ( X  =  Y  ->  (
( X S Y )  C_  ( X M Y )  <->  ( X S X )  C_  ( X M X ) ) )
1711, 16syl5ibr 212 . 2  |-  ( X  =  Y  ->  ( ph  ->  ( X S Y )  C_  ( X M Y ) ) )
184adantl 452 . . . . . 6  |-  ( ( X  =/=  Y  /\  ph )  ->  G  e. Ibg )
195adantl 452 . . . . . 6  |-  ( ( X  =/=  Y  /\  ph )  ->  X  e.  P )
20 eqid 2283 . . . . . 6  |-  (btw `  G )  =  (btw
`  G )
21 segline.2 . . . . . . 7  |-  ( ph  ->  Y  e.  P )
2221adantl 452 . . . . . 6  |-  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  P )
23 simpl 443 . . . . . 6  |-  ( ( X  =/=  Y  /\  ph )  ->  X  =/=  Y )
242, 3, 18, 19, 20, 22, 23sgplpte21e 26137 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  ( x  e.  ( X S Y )  <->  ( x  e.  P  /\  ( x  e.  ( X (btw
`  G ) Y )  \/  x  =  X  \/  x  =  Y ) ) ) )
25183ad2ant3 978 . . . . . . . . . 10  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  G  e. Ibg )
26193ad2ant3 978 . . . . . . . . . 10  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  X  e.  P )
27 simp2 956 . . . . . . . . . 10  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  x  e.  P )
28223ad2ant3 978 . . . . . . . . . 10  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  Y  e.  P )
29 simp1 955 . . . . . . . . . 10  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  x  e.  ( X (btw `  G ) Y ) )
302, 20, 25, 26, 27, 28, 29, 7isibg1a6 26125 . . . . . . . . 9  |-  ( ( x  e.  ( X (btw `  G ) Y )  /\  x  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  x  e.  ( X M Y ) )
31303exp 1150 . . . . . . . 8  |-  ( x  e.  ( X (btw
`  G ) Y )  ->  ( x  e.  P  ->  ( ( X  =/=  Y  /\  ph )  ->  x  e.  ( X M Y ) ) ) )
328adantl 452 . . . . . . . . . . . 12  |-  ( ( X  =/=  Y  /\  ph )  ->  G  e. Ig )
3332adantl 452 . . . . . . . . . . 11  |-  ( ( X  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  G  e. Ig )
34 simpl 443 . . . . . . . . . . 11  |-  ( ( X  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  X  e.  P )
3522adantl 452 . . . . . . . . . . 11  |-  ( ( X  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  Y  e.  P )
362, 7, 33, 34, 35lineval2a 26085 . . . . . . . . . 10  |-  ( ( X  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  X  e.  ( X M Y ) )
3736ex 423 . . . . . . . . 9  |-  ( X  e.  P  ->  (
( X  =/=  Y  /\  ph )  ->  X  e.  ( X M Y ) ) )
38 eleq1 2343 . . . . . . . . . 10  |-  ( x  =  X  ->  (
x  e.  P  <->  X  e.  P ) )
39 eleq1 2343 . . . . . . . . . . 11  |-  ( x  =  X  ->  (
x  e.  ( X M Y )  <->  X  e.  ( X M Y ) ) )
4039imbi2d 307 . . . . . . . . . 10  |-  ( x  =  X  ->  (
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) )  <->  ( ( X  =/=  Y  /\  ph )  ->  X  e.  ( X M Y ) ) ) )
4138, 40imbi12d 311 . . . . . . . . 9  |-  ( x  =  X  ->  (
( x  e.  P  ->  ( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) )  <-> 
( X  e.  P  ->  ( ( X  =/= 
Y  /\  ph )  ->  X  e.  ( X M Y ) ) ) ) )
4237, 41mpbiri 224 . . . . . . . 8  |-  ( x  =  X  ->  (
x  e.  P  -> 
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) ) )
4332adantl 452 . . . . . . . . . . 11  |-  ( ( Y  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  G  e. Ig )
4419adantl 452 . . . . . . . . . . 11  |-  ( ( Y  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  X  e.  P )
45 simpl 443 . . . . . . . . . . 11  |-  ( ( Y  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  Y  e.  P )
462, 7, 43, 44, 45lineval2b 26086 . . . . . . . . . 10  |-  ( ( Y  e.  P  /\  ( X  =/=  Y  /\  ph ) )  ->  Y  e.  ( X M Y ) )
4746ex 423 . . . . . . . . 9  |-  ( Y  e.  P  ->  (
( X  =/=  Y  /\  ph )  ->  Y  e.  ( X M Y ) ) )
48 eleq1 2343 . . . . . . . . . 10  |-  ( x  =  Y  ->  (
x  e.  P  <->  Y  e.  P ) )
49 eleq1 2343 . . . . . . . . . . 11  |-  ( x  =  Y  ->  (
x  e.  ( X M Y )  <->  Y  e.  ( X M Y ) ) )
5049imbi2d 307 . . . . . . . . . 10  |-  ( x  =  Y  ->  (
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) )  <->  ( ( X  =/=  Y  /\  ph )  ->  Y  e.  ( X M Y ) ) ) )
5148, 50imbi12d 311 . . . . . . . . 9  |-  ( x  =  Y  ->  (
( x  e.  P  ->  ( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) )  <-> 
( Y  e.  P  ->  ( ( X  =/= 
Y  /\  ph )  ->  Y  e.  ( X M Y ) ) ) ) )
5247, 51mpbiri 224 . . . . . . . 8  |-  ( x  =  Y  ->  (
x  e.  P  -> 
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) ) )
5331, 42, 523jaoi 1245 . . . . . . 7  |-  ( ( x  e.  ( X (btw `  G ) Y )  \/  x  =  X  \/  x  =  Y )  ->  (
x  e.  P  -> 
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) ) )
5453impcom 419 . . . . . 6  |-  ( ( x  e.  P  /\  ( x  e.  ( X (btw `  G ) Y )  \/  x  =  X  \/  x  =  Y ) )  -> 
( ( X  =/= 
Y  /\  ph )  ->  x  e.  ( X M Y ) ) )
5554com12 27 . . . . 5  |-  ( ( X  =/=  Y  /\  ph )  ->  ( (
x  e.  P  /\  ( x  e.  ( X (btw `  G ) Y )  \/  x  =  X  \/  x  =  Y ) )  ->  x  e.  ( X M Y ) ) )
5624, 55sylbid 206 . . . 4  |-  ( ( X  =/=  Y  /\  ph )  ->  ( x  e.  ( X S Y )  ->  x  e.  ( X M Y ) ) )
5756ssrdv 3185 . . 3  |-  ( ( X  =/=  Y  /\  ph )  ->  ( X S Y )  C_  ( X M Y ) )
5857ex 423 . 2  |-  ( X  =/=  Y  ->  ( ph  ->  ( X S Y )  C_  ( X M Y ) ) )
5917, 58pm2.61ine 2522 1  |-  ( ph  ->  ( X S Y )  C_  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858  PPointscpoints 26056  Igcig 26060   linecline 26076  btwcbtw 26106  Ibgcibg 26107   segcseg 26130
This theorem is referenced by:  rayline  26156
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  df-seg2 26131
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