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Theorem linethru 30705
Description: If  A is a line containing two distinct points  P and  Q, then  A is the line through  P and  Q. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
linethru  |-  ( ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) )

Proof of Theorem linethru
Dummy variables  a 
b  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ellines 30704 . . 3  |-  ( A  e. LinesEE 
<->  E. n  e.  NN  E. a  e.  ( EE
`  n ) E. b  e.  ( EE
`  n ) ( a  =/=  b  /\  A  =  ( aLine b ) ) )
2 simpll1 1044 . . . . . . . . . . . 12  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  n  e.  NN )
3 simpll2 1045 . . . . . . . . . . . 12  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  a  e.  ( EE `  n ) )
4 simpll3 1046 . . . . . . . . . . . 12  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  b  e.  ( EE `  n ) )
5 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  a  =/=  b
)
6 liness 30697 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
) )  ->  (
aLine b )  C_  ( EE `  n ) )
72, 3, 4, 5, 6syl13anc 1266 . . . . . . . . . . 11  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  ( aLine b )  C_  ( EE `  n ) )
8 simprll 770 . . . . . . . . . . 11  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  P  e.  ( aLine b ) )
97, 8sseldd 3471 . . . . . . . . . 10  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  P  e.  ( EE `  n ) )
10 simprlr 771 . . . . . . . . . . 11  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  Q  e.  ( aLine b ) )
117, 10sseldd 3471 . . . . . . . . . 10  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  Q  e.  ( EE `  n ) )
12 simplll 766 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  ->  P  e.  ( aLine b ) )
1312adantl 467 . . . . . . . . . . . . . . 15  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  ->  P  e.  ( aLine b ) )
14 simpll1 1044 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  ->  n  e.  NN )
15 simpll2 1045 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
a  e.  ( EE
`  n ) )
16 simpll3 1046 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
b  e.  ( EE
`  n ) )
17 simplr 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
a  =/=  b )
18 simprrl 772 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  ->  P  e.  ( EE `  n ) )
19 simprlr 771 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  ->  P  =/=  a )
2019necomd 2702 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
a  =/=  P )
21 lineelsb2 30700 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  ( P  e.  ( EE `  n
)  /\  a  =/=  P ) )  ->  ( P  e.  ( aLine b )  ->  (
aLine b )  =  ( aLine P ) ) )
2214, 15, 16, 17, 18, 20, 21syl132anc 1282 . . . . . . . . . . . . . . 15  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( P  e.  ( aLine b )  -> 
( aLine b )  =  ( aLine P
) ) )
2313, 22mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( aLine P
) )
24 linecom 30702 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  a  =/=  P
) )  ->  (
aLine P )  =  ( PLine a ) )
2514, 15, 18, 20, 24syl13anc 1266 . . . . . . . . . . . . . 14  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine P )  =  ( PLine a
) )
2623, 25eqtrd 2470 . . . . . . . . . . . . 13  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine a
) )
27 neeq2 2714 . . . . . . . . . . . . . . . . 17  |-  ( Q  =  a  ->  ( P  =/=  Q  <->  P  =/=  a ) )
2827anbi2d 708 . . . . . . . . . . . . . . . 16  |-  ( Q  =  a  ->  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a ) ) )
2928anbi1d 709 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  (
( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q )  /\  ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
) ) )  <->  ( (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a )  /\  ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n ) ) ) ) )
3029anbi2d 708 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  <->  ( (
( n  e.  NN  /\  a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a )  /\  ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
) ) ) ) ) )
31 oveq2 6313 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  ( PLine Q )  =  ( PLine a ) )
3231eqeq2d 2443 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( aLine b )  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine a ) ) )
3330, 32imbi12d 321 . . . . . . . . . . . . 13  |-  ( Q  =  a  ->  (
( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n )  /\  b  e.  ( EE `  n
) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine Q
) )  <->  ( (
( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine a
) ) ) )
3426, 33mpbiri 236 . . . . . . . . . . . 12  |-  ( Q  =  a  ->  (
( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine Q
) ) )
35 simp1 1005 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
( n  e.  NN  /\  a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b ) )
36 simp2l 1031 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )
3735, 36, 10syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  Q  e.  ( aLine b ) )
38 simp1l1 1098 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  n  e.  NN )
39 simp1l2 1099 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  a  e.  ( EE `  n
) )
40 simp1l3 1100 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  b  e.  ( EE `  n
) )
41 simp1r 1030 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  a  =/=  b )
42 simp2rr 1075 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  Q  e.  ( EE `  n
) )
43 simp3 1007 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  Q  =/=  a )
4443necomd 2702 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  a  =/=  Q )
45 lineelsb2 30700 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
)  /\  ( Q  e.  ( EE `  n
)  /\  a  =/=  Q ) )  ->  ( Q  e.  ( aLine b )  ->  (
aLine b )  =  ( aLine Q ) ) )
4638, 39, 40, 41, 42, 44, 45syl132anc 1282 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  ( Q  e.  ( aLine b )  ->  (
aLine b )  =  ( aLine Q ) ) )
4737, 46mpd 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
aLine b )  =  ( aLine Q ) )
48 linecom 30702 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  a  =/=  Q
) )  ->  (
aLine Q )  =  ( QLine a ) )
4938, 39, 42, 44, 48syl13anc 1266 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
aLine Q )  =  ( QLine a ) )
5047, 49eqtrd 2470 . . . . . . . . . . . . . . 15  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
aLine b )  =  ( QLine a ) )
5136simplld 759 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  P  e.  ( aLine b ) )
5251, 50eleqtrd 2519 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  P  e.  ( QLine a ) )
53 simp2rl 1074 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  P  e.  ( EE `  n
) )
54 simp2lr 1073 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  P  =/=  Q )
5554necomd 2702 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  Q  =/=  P )
56 lineelsb2 30700 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  a  e.  ( EE `  n )  /\  Q  =/=  a )  /\  ( P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( P  e.  ( QLine a )  -> 
( QLine a )  =  ( QLine P
) ) )
5738, 42, 39, 43, 53, 55, 56syl132anc 1282 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  ( P  e.  ( QLine a )  ->  ( QLine a )  =  ( QLine P ) ) )
5852, 57mpd 15 . . . . . . . . . . . . . . 15  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  ( QLine a )  =  ( QLine P ) )
59 linecom 30702 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  ( PLine Q
) )
6038, 42, 53, 55, 59syl13anc 1266 . . . . . . . . . . . . . . 15  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  ( QLine P )  =  ( PLine Q ) )
6150, 58, 603eqtrd 2474 . . . . . . . . . . . . . 14  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  /\  Q  =/=  a )  ->  (
aLine b )  =  ( PLine Q ) )
62613expa 1205 . . . . . . . . . . . . 13  |-  ( ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  /\  Q  =/=  a )  -> 
( aLine b )  =  ( PLine Q
) )
6362expcom 436 . . . . . . . . . . . 12  |-  ( Q  =/=  a  ->  (
( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine Q
) ) )
6434, 63pm2.61ine 2744 . . . . . . . . . . 11  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) ) )  -> 
( aLine b )  =  ( PLine Q
) )
6564expr 618 . . . . . . . . . 10  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  ( ( P  e.  ( EE `  n )  /\  Q  e.  ( EE `  n
) )  ->  (
aLine b )  =  ( PLine Q ) ) )
669, 11, 65mp2and 683 . . . . . . . . 9  |-  ( ( ( ( n  e.  NN  /\  a  e.  ( EE `  n
)  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  /\  (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q ) )  ->  ( aLine b )  =  ( PLine Q ) )
6766ex 435 . . . . . . . 8  |-  ( ( ( n  e.  NN  /\  a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  -> 
( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q )  ->  (
aLine b )  =  ( PLine Q ) ) )
68 eleq2 2502 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( P  e.  A  <->  P  e.  (
aLine b ) ) )
69 eleq2 2502 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( Q  e.  A  <->  Q  e.  (
aLine b ) ) )
7068, 69anbi12d 715 . . . . . . . . . 10  |-  ( A  =  ( aLine b )  ->  ( ( P  e.  A  /\  Q  e.  A )  <->  ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) ) ) )
7170anbi1d 709 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q ) ) )
72 eqeq1 2433 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( A  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine Q ) ) )
7371, 72imbi12d 321 . . . . . . . 8  |-  ( A  =  ( aLine b )  ->  ( (
( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  ->  A  =  ( PLine Q
) )  <->  ( (
( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q )  -> 
( aLine b )  =  ( PLine Q
) ) ) )
7467, 73syl5ibrcom 225 . . . . . . 7  |-  ( ( ( n  e.  NN  /\  a  e.  ( EE
`  n )  /\  b  e.  ( EE `  n ) )  /\  a  =/=  b )  -> 
( A  =  ( aLine b )  -> 
( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  ->  A  =  ( PLine Q
) ) ) )
7574expimpd 606 . . . . . 6  |-  ( ( n  e.  NN  /\  a  e.  ( EE `  n )  /\  b  e.  ( EE `  n
) )  ->  (
( a  =/=  b  /\  A  =  (
aLine b ) )  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) ) )
76753expa 1205 . . . . 5  |-  ( ( ( n  e.  NN  /\  a  e.  ( EE
`  n ) )  /\  b  e.  ( EE `  n ) )  ->  ( (
a  =/=  b  /\  A  =  ( aLine b ) )  -> 
( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q )  ->  A  =  ( PLine Q
) ) ) )
7776rexlimdva 2924 . . . 4  |-  ( ( n  e.  NN  /\  a  e.  ( EE `  n ) )  -> 
( E. b  e.  ( EE `  n
) ( a  =/=  b  /\  A  =  ( aLine b ) )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  A  =  ( PLine Q ) ) ) )
7877rexlimivv 2929 . . 3  |-  ( E. n  e.  NN  E. a  e.  ( EE `  n ) E. b  e.  ( EE `  n
) ( a  =/=  b  /\  A  =  ( aLine b ) )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  ->  A  =  ( PLine Q ) ) )
791, 78sylbi 198 . 2  |-  ( A  e. LinesEE  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) )
80793impib 1203 1  |-  ( ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   E.wrex 2783    C_ wss 3442   ` cfv 5601  (class class class)co 6305   NNcn 10609   EEcee 24764  Linecline2 30686  LinesEEclines2 30688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-ec 7373  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-ee 24767  df-btwn 24768  df-cgr 24769  df-ofs 30535  df-colinear 30591  df-ifs 30592  df-cgr3 30593  df-fs 30594  df-line2 30689  df-lines2 30691
This theorem is referenced by:  hilbert1.2  30707  lineintmo  30709
  Copyright terms: Public domain W3C validator