Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isconcl5a Unicode version

Theorem isconcl5a 25267
Description: Two distinct lines intersect in at most one point. Proposition 4 of [AitkenIBG] p. 3. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
isconcl5a.1  |-  L  =  (PLines `  F )
isconcl5a.2  |-  P  =  (PPoints `  F )
isconcl5a.3  |-  ( ph  ->  F  e. Ig )
isconcl5a.4  |-  ( ph  -> 
L1  e.  L )
isconcl5a.5  |-  ( ph  ->  L 2  e.  L
)
isconcl5a.6  |-  ( ph  -> 
L1  =/=  L 2
)
Assertion
Ref Expression
isconcl5a  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Distinct variable groups:    p, L1    p, L 2    P, p    ph, p
Allowed substitution hints:    F( p)    L( p)

Proof of Theorem isconcl5a
StepHypRef Expression
1 vex 2730 . . . . . . 7  |-  q  e. 
_V
2 eleq1 2313 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  P  <->  q  e.  P ) )
3 eleq1 2313 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L1  <->  q  e.  L1 ) )
4 eleq1 2313 . . . . . . . 8  |-  ( p  =  q  ->  (
p  e.  L 2  <->  q  e.  L 2 )
)
52, 3, 43anbi123d 1257 . . . . . . 7  |-  ( p  =  q  ->  (
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 ) ) )
61, 5sbcie 2955 . . . . . 6  |-  ( [. q  /  p ]. (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 ) )
7 idd 23 . . . . . . . . 9  |-  ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) )
87a1ii 26 . . . . . . . 8  |-  ( ph  ->  ( ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  ->  (
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( p  =  q  ->  p  =  q ) ) ) )
98com14 84 . . . . . . 7  |-  ( p  =  q  ->  (
( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) ) )
10 isconcl5a.2 . . . . . . . . . . . . 13  |-  P  =  (PPoints `  F )
11 isconcl5a.1 . . . . . . . . . . . . 13  |-  L  =  (PLines `  F )
12 eqid 2253 . . . . . . . . . . . . 13  |-  ( line `  F )  =  (
line `  F )
13 isconcl5a.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e. Ig )
1413adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  F  e. Ig )
15 simpl31 1041 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  P )
16 simpl21 1038 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  P )
17 simpl1 963 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  =/=  q )
18 simpl32 1042 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L1 )
19 simpl22 1039 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L1 )
20 isconcl5a.4 . . . . . . . . . . . . . 14  |-  ( ph  -> 
L1  e.  L )
2120adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  e.  L )
2210, 11, 12, 14, 15, 16, 17, 18, 19, 21lineval42 25246 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  ( p (
line `  F )
q ) )
23 simpl33 1043 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  p  e.  L 2
)
24 simpl23 1040 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
q  e.  L 2
)
25 isconcl5a.5 . . . . . . . . . . . . . 14  |-  ( ph  ->  L 2  e.  L
)
2625adantl 454 . . . . . . . . . . . . 13  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  e.  L
)
2710, 11, 12, 14, 15, 16, 17, 23, 24, 26lineval42 25246 . . . . . . . . . . . 12  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L 2  =  (
p ( line `  F
) q ) )
2822, 27eqtr4d 2288 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  L1  =  L 2 )
29 isconcl5a.6 . . . . . . . . . . . . 13  |-  ( ph  -> 
L1  =/=  L 2
)
3029neneqd 2428 . . . . . . . . . . . 12  |-  ( ph  ->  -.  L1  =  L 2 )
3130adantl 454 . . . . . . . . . . 11  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  ->  -.  L1  =  L 2
)
3228, 31jca 520 . . . . . . . . . 10  |-  ( ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  /\  ph )  -> 
( L1  =  L 2  /\  -.  L1  =  L 2 ) )
3332ex 425 . . . . . . . . 9  |-  ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  (
L1  =  L 2  /\  -.  L1  =  L 2 ) ) )
34 pm2.24 103 . . . . . . . . . 10  |-  ( L1  =  L 2  ->  ( -.  L1  =  L 2  ->  p  =  q ) )
3534imp 420 . . . . . . . . 9  |-  ( (
L1  =  L 2  /\  -.  L1  =  L 2 )  ->  p  =  q )
3633, 35syl6 31 . . . . . . . 8  |-  ( ( p  =/=  q  /\  ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  /\  ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  =  q ) )
37363exp 1155 . . . . . . 7  |-  ( p  =/=  q  ->  (
( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) ) )
389, 37pm2.61ine 2488 . . . . . 6  |-  ( ( q  e.  P  /\  q  e.  L1  /\  q  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
396, 38sylbi 189 . . . . 5  |-  ( [. q  /  p ]. (
p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  -> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  ->  ( ph  ->  p  =  q ) ) )
4039impcom 421 . . . 4  |-  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  ( ph  ->  p  =  q ) )
4140com12 29 . . 3  |-  ( ph  ->  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
4241alrimivv 2013 . 2  |-  ( ph  ->  A. p A. q
( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
43 nfv 1629 . . . 4  |-  F/ q ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )
4443mo3 2144 . . 3  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
45 sbsbc 2925 . . . . . 6  |-  ( [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
4645anbi2i 678 . . . . 5  |-  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  <-> 
( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) ) )
4746imbi1i 317 . . . 4  |-  ( ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [
q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )  <->  ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
48472albii 1555 . . 3  |-  ( A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [ q  /  p ]
( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q )  <->  A. p A. q
( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
4944, 48bitri 242 . 2  |-  ( E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  <->  A. p A. q ( ( ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 )  /\  [. q  /  p ]. ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )  ->  p  =  q ) )
5042, 49sylibr 205 1  |-  ( ph  ->  E* p ( p  e.  P  /\  p  e.  L1  /\  p  e.  L 2 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939   A.wal 1532    = wceq 1619    e. wcel 1621   [wsb 1882   E*wmo 2115    =/= wne 2412   [.wsbc 2921   ` cfv 4592  (class class class)co 5710  PPointscpoints 25222  PLinescplines 25224  Igcig 25226   linecline 25242
This theorem is referenced by:  isconcl6a  25269
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-ig2 25227  df-li 25243
  Copyright terms: Public domain W3C validator