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Theorem linethru 26052
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 26051 . . 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 996 . . . . . . . . . . . 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 997 . . . . . . . . . . . 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 998 . . . . . . . . . . . 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 732 . . . . . . . . . . . 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 26044 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
) )  ->  (
aLine b )  C_  ( EE `  n ) )
72, 3, 4, 5, 6syl13anc 1186 . . . . . . . . . . 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 739 . . . . . . . . . . 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 3341 . . . . . . . . . 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 740 . . . . . . . . . . 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 3341 . . . . . . . . . 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 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  ->  P  e.  ( aLine b ) )
1312adantl 453 . . . . . . . . . . . . . . 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 996 . . . . . . . . . . . . . . . 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 997 . . . . . . . . . . . . . . . 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 998 . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 741 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . . 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 2681 . . . . . . . . . . . . . . . 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 26047 . . . . . . . . . . . . . . . 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 1202 . . . . . . . . . . . . . . 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 26049 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  a  =/=  P
) )  ->  (
aLine P )  =  ( PLine a ) )
2514, 15, 18, 20, 24syl13anc 1186 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 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 2607 . . . . . . . . . . . . . . . . 17  |-  ( Q  =  a  ->  ( P  =/=  Q  <->  P  =/=  a ) )
2827anbi2d 685 . . . . . . . . . . . . . . . 16  |-  ( Q  =  a  ->  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a ) ) )
2928anbi1d 686 . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . 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 6081 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  ( PLine Q )  =  ( PLine a ) )
3231eqeq2d 2446 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( aLine b )  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine a ) ) )
3330, 32imbi12d 312 . . . . . . . . . . . . 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 225 . . . . . . . . . . . 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 957 . . . . . . . . . . . . . . . . . 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 983 . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . 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 1050 . . . . . . . . . . . . . . . . . 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 1051 . . . . . . . . . . . . . . . . . 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 1052 . . . . . . . . . . . . . . . . . 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 982 . . . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . . . 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 959 . . . . . . . . . . . . . . . . . . 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 2681 . . . . . . . . . . . . . . . . . 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 26047 . . . . . . . . . . . . . . . . . 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 1202 . . . . . . . . . . . . . . . . 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 26049 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  a  =/=  Q
) )  ->  (
aLine Q )  =  ( QLine a ) )
4938, 39, 42, 44, 48syl13anc 1186 . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q )  ->  P  e.  ( aLine b ) )
5236, 51syl 16 . . . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . . . . 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 1026 . . . . . . . . . . . . . . . . 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 1025 . . . . . . . . . . . . . . . . . 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 2681 . . . . . . . . . . . . . . . . 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 26047 . . . . . . . . . . . . . . . . 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 1202 . . . . . . . . . . . . . . . 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 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 ) )
60 linecom 26049 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  ( PLine Q
) )
6138, 42, 54, 56, 60syl13anc 1186 . . . . . . . . . . . . . . 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 2471 . . . . . . . . . . . . . 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 1153 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 2674 . . . . . . . . . . 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 599 . . . . . . . . . 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 661 . . . . . . . . 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 424 . . . . . . . 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 2496 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( P  e.  A  <->  P  e.  (
aLine b ) ) )
70 eleq2 2496 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( Q  e.  A  <->  Q  e.  (
aLine b ) ) )
7169, 70anbi12d 692 . . . . . . . . . 10  |-  ( A  =  ( aLine b )  ->  ( ( P  e.  A  /\  Q  e.  A )  <->  ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) ) ) )
7271anbi1d 686 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q ) ) )
73 eqeq1 2441 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( A  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine Q ) ) )
7472, 73imbi12d 312 . . . . . . . 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 214 . . . . . . 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 587 . . . . . 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 1153 . . . . 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 2822 . . . 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 2827 . . 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 188 . 2  |-  ( A  e. LinesEE  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) )
81803impib 1151 1  |-  ( ( A  e. LinesEE  /\  ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    C_ wss 3312   ` cfv 5446  (class class class)co 6073   NNcn 9990   EEcee 25792  Linecline2 26033  LinesEEclines2 26035
This theorem is referenced by:  hilbert1.2  26054  lineintmo  26056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-ico 10912  df-icc 10913  df-fz 11034  df-fzo 11126  df-seq 11314  df-exp 11373  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-ee 25795  df-btwn 25796  df-cgr 25797  df-ofs 25882  df-ifs 25938  df-cgr3 25939  df-colinear 25940  df-fs 25941  df-line2 26036  df-lines2 26038
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