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Theorem linethru 24152
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
StepHypRef Expression
1 ellines 24151 . . 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 999 . . . . . . . . . . . 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 1000 . . . . . . . . . . . 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 1001 . . . . . . . . . . . 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 734 . . . . . . . . . . . 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 24144 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
) )  ->  (
aLine b )  C_  ( EE `  n ) )
72, 3, 4, 5, 6syl13anc 1189 . . . . . . . . . . 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 741 . . . . . . . . . . 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 3156 . . . . . . . . . 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 742 . . . . . . . . . . 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 3156 . . . . . . . . . 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 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  ->  P  e.  ( aLine b ) )
1312adantl 454 . . . . . . . . . . . . . . 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 999 . . . . . . . . . . . . . . . 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 1000 . . . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . 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 742 . . . . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . . . . 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 24147 . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . 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 24149 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  a  =/=  P
) )  ->  (
aLine P )  =  ( PLine a ) )
2514, 15, 18, 20, 24syl13anc 1189 . . . . . . . . . . . . . 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 2290 . . . . . . . . . . . . 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 2430 . . . . . . . . . . . . . . . . 17  |-  ( Q  =  a  ->  ( P  =/=  Q  <->  P  =/=  a ) )
2827anbi2d 687 . . . . . . . . . . . . . . . 16  |-  ( Q  =  a  ->  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a ) ) )
2928anbi1d 688 . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . 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 5800 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  ( PLine Q )  =  ( PLine a ) )
3231eqeq2d 2269 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( aLine b )  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine a ) ) )
3330, 32imbi12d 313 . . . . . . . . . . . . 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 226 . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . . . 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 645 . . . . . . . . . . . . . . . . 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 1053 . . . . . . . . . . . . . . . . . 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 1054 . . . . . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . . . . 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 985 . . . . . . . . . . . . . . . . . 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 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 )  ->  Q  e.  ( EE `  n
) )
43 simp3 962 . . . . . . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . . . . . . 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 24147 . . . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . 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 24149 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  a  =/=  Q
) )  ->  (
aLine Q )  =  ( QLine a ) )
4938, 39, 42, 44, 48syl13anc 1189 . . . . . . . . . . . . . . . 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 2290 . . . . . . . . . . . . . . 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 ) )
51 simpll 733 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q )  ->  P  e.  ( aLine b ) )
5236, 51syl 17 . . . . . . . . . . . . . . . . 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 ) )
5352, 50eleqtrd 2334 . . . . . . . . . . . . . . . 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 ) )
54 simp2rl 1029 . . . . . . . . . . . . . . . . 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
) )
55 simp2lr 1028 . . . . . . . . . . . . . . . . . 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 )
5655necomd 2504 . . . . . . . . . . . . . . . . 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 )
57 lineelsb2 24147 . . . . . . . . . . . . . . . . 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
) ) )
5838, 42, 39, 43, 54, 56, 57syl132anc 1205 . . . . . . . . . . . . . . . 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 ) ) )
5953, 58mpd 16 . . . . . . . . . . . . . . 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 ) )
60 linecom 24149 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  ( PLine Q
) )
6138, 42, 54, 56, 60syl13anc 1189 . . . . . . . . . . . . . . 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 ) )
6250, 59, 613eqtrd 2294 . . . . . . . . . . . . . 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 ) )
63623expa 1156 . . . . . . . . . . . . 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
) )
6463expcom 426 . . . . . . . . . . . 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
) ) )
6534, 64pm2.61ine 2497 . . . . . . . . . . 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
) )
6665expr 601 . . . . . . . . . 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 ) ) )
679, 11, 66mp2and 663 . . . . . . . . 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 ) )
6867ex 425 . . . . . . . 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 ) ) )
69 eleq2 2319 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( P  e.  A  <->  P  e.  (
aLine b ) ) )
70 eleq2 2319 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( Q  e.  A  <->  Q  e.  (
aLine b ) ) )
7169, 70anbi12d 694 . . . . . . . . . 10  |-  ( A  =  ( aLine b )  ->  ( ( P  e.  A  /\  Q  e.  A )  <->  ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) ) ) )
7271anbi1d 688 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q ) ) )
73 eqeq1 2264 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( A  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine Q ) ) )
7472, 73imbi12d 313 . . . . . . . 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
) ) ) )
7568, 74syl5ibrcom 215 . . . . . . 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
) ) ) )
7675expimpd 589 . . . . . 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 ) ) ) )
77763expa 1156 . . . . 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
) ) ) )
7877rexlimdva 2642 . . . 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 ) ) ) )
7978rexlimivv 2647 . . 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 ) ) )
801, 79sylbi 189 . 2  |-  ( A  e. LinesEE  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) )
81803impib 1154 1  |-  ( ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519    C_ wss 3127   ` cfv 4673  (class class class)co 5792   NNcn 9714   EEcee 23892  Linecline2 24133  LinesEEclines2 24135
This theorem is referenced by:  hilbert1.2  24154  lineintmo  24156
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-ec 6630  df-map 6742  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9934  df-z 9993  df-uz 10199  df-rp 10323  df-ico 10629  df-icc 10630  df-fz 10750  df-fzo 10838  df-seq 11014  df-exp 11072  df-hash 11305  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-clim 11928  df-sum 12125  df-ee 23895  df-btwn 23896  df-cgr 23897  df-ofs 23982  df-ifs 24038  df-cgr3 24039  df-colinear 24040  df-fs 24041  df-line2 24136  df-lines2 24138
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