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Theorem lineval22 26082
Description: The points  A and  B belong to the line passing through two distinct points  A and  B. (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval2.1  |-  P  =  (PPoints `  G )
lineval2.2  |-  L  =  (PLines `  G )
lineval2.3  |-  M  =  ( line `  G
)
lineval2.4  |-  ( ph  ->  G  e. Ig )
lineval22.5  |-  ( ph  ->  A  e.  P )
lineval22.6  |-  ( ph  ->  B  e.  P )
lineval22.7  |-  ( ph  ->  A  =/=  B )
Assertion
Ref Expression
lineval22  |-  ( ph  ->  ( A  e.  ( A M B )  /\  B  e.  ( A M B ) ) )

Proof of Theorem lineval22
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 lineval2.1 . . . . 5  |-  P  =  (PPoints `  G )
2 lineval2.2 . . . . 5  |-  L  =  (PLines `  G )
3 lineval2.4 . . . . 5  |-  ( ph  ->  G  e. Ig )
4 lineval22.5 . . . . 5  |-  ( ph  ->  A  e.  P )
5 lineval22.6 . . . . 5  |-  ( ph  ->  B  e.  P )
6 lineval22.7 . . . . 5  |-  ( ph  ->  A  =/=  B )
71, 2, 3, 4, 5, 6isig2a2 26066 . . . 4  |-  ( ph  ->  E! l  e.  L  ( A  e.  l  /\  B  e.  l
) )
8 nfriota1 6312 . . . . . . 7  |-  F/_ l
( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )
98nfel2 2431 . . . . . 6  |-  F/ l  A  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )
108nfel2 2431 . . . . . 6  |-  F/ l  B  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )
119, 10nfan 1771 . . . . 5  |-  F/ l ( A  e.  (
iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) )
12 eqid 2283 . . . . 5  |-  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  =  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )
13 eleq2 2344 . . . . . 6  |-  ( l  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  -> 
( A  e.  l  <-> 
A  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) )
14 eleq2 2344 . . . . . 6  |-  ( l  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  -> 
( B  e.  l  <-> 
B  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) )
1513, 14anbi12d 691 . . . . 5  |-  ( l  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  -> 
( ( A  e.  l  /\  B  e.  l )  <->  ( A  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) ) ) )
1611, 12, 15riotaprop 6328 . . . 4  |-  ( E! l  e.  L  ( A  e.  l  /\  B  e.  l )  ->  ( ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  e.  L  /\  ( A  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) ) )
177, 16syl 15 . . 3  |-  ( ph  ->  ( ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  e.  L  /\  ( A  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) ) )
1817simprd 449 . 2  |-  ( ph  ->  ( A  e.  (
iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) ) )
19 lineval2.3 . . . . 5  |-  M  =  ( line `  G
)
201, 2, 19, 3, 4, 5, 6lineval222 26079 . . . 4  |-  ( ph  ->  ( A M B )  =  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) )
2120eleq2d 2350 . . 3  |-  ( ph  ->  ( A  e.  ( A M B )  <-> 
A  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) )
2220eleq2d 2350 . . 3  |-  ( ph  ->  ( B  e.  ( A M B )  <-> 
B  e.  ( iota_ l  e.  L ( A  e.  l  /\  B  e.  l ) ) ) )
2321, 22anbi12d 691 . 2  |-  ( ph  ->  ( ( A  e.  ( A M B )  /\  B  e.  ( A M B ) )  <->  ( A  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) )  /\  B  e.  ( iota_ l  e.  L
( A  e.  l  /\  B  e.  l ) ) ) ) )
2418, 23mpbird 223 1  |-  ( ph  ->  ( A  e.  ( A M B )  /\  B  e.  ( A M B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E!wreu 2545   ` cfv 5255  (class class class)co 5858   iota_crio 6297  PPointscpoints 26056  PLinescplines 26058  Igcig 26060   linecline 26076
This theorem is referenced by:  lineval2a  26085  lineval2b  26086
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-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-riota 6304  df-ig2 26061  df-li 26077
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