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Theorem linethru 25335
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 25334 . . 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 994 . . . . . . . . . . . 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 995 . . . . . . . . . . . 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 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 ) )  ->  b  e.  ( EE `  n ) )
5 simplr 731 . . . . . . . . . . . 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 25327 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
) )  ->  (
aLine b )  C_  ( EE `  n ) )
72, 3, 4, 5, 6syl13anc 1184 . . . . . . . . . . 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 738 . . . . . . . . . . 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 3257 . . . . . . . . . 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 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 ) )  ->  Q  e.  ( aLine b ) )
117, 10sseldd 3257 . . . . . . . . . 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 734 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  ->  P  e.  ( aLine b ) )
1312adantl 452 . . . . . . . . . . . . . . 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 994 . . . . . . . . . . . . . . . 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 995 . . . . . . . . . . . . . . . 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 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 ) ) ) )  -> 
b  e.  ( EE
`  n ) )
17 simplr 731 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . . 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 25330 . . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . . 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 25332 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  a  =/=  P
) )  ->  (
aLine P )  =  ( PLine a ) )
2514, 15, 18, 20, 24syl13anc 1184 . . . . . . . . . . . . . 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 2390 . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . 17  |-  ( Q  =  a  ->  ( P  =/=  Q  <->  P  =/=  a ) )
2827anbi2d 684 . . . . . . . . . . . . . . . 16  |-  ( Q  =  a  ->  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a ) ) )
2928anbi1d 685 . . . . . . . . . . . . . . 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 684 . . . . . . . . . . . . . 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 5953 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  ( PLine Q )  =  ( PLine a ) )
3231eqeq2d 2369 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( aLine b )  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine a ) ) )
3330, 32imbi12d 311 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 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 955 . . . . . . . . . . . . . . . . . 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 981 . . . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . . . 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 1048 . . . . . . . . . . . . . . . . . 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 1049 . . . . . . . . . . . . . . . . . 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 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 )  ->  b  e.  ( EE `  n
) )
41 simp1r 980 . . . . . . . . . . . . . . . . . 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 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 )  ->  Q  e.  ( EE `  n
) )
43 simp3 957 . . . . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . . . . 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 25330 . . . . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . . . . 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 25332 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  a  =/=  Q
) )  ->  (
aLine Q )  =  ( QLine a ) )
4938, 39, 42, 44, 48syl13anc 1184 . . . . . . . . . . . . . . . 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 2390 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q )  ->  P  e.  ( aLine b ) )
5236, 51syl 15 . . . . . . . . . . . . . . . . 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 2434 . . . . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . . 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 1023 . . . . . . . . . . . . . . . . . 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 2604 . . . . . . . . . . . . . . . . 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 25330 . . . . . . . . . . . . . . . . 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 1200 . . . . . . . . . . . . . . . 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 14 . . . . . . . . . . . . . . 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 25332 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  ( PLine Q
) )
6138, 42, 54, 56, 60syl13anc 1184 . . . . . . . . . . . . . . 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 2394 . . . . . . . . . . . . . 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 1151 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2597 . . . . . . . . . . 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 598 . . . . . . . . . 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 660 . . . . . . . . 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 423 . . . . . . . 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 2419 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( P  e.  A  <->  P  e.  (
aLine b ) ) )
70 eleq2 2419 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( Q  e.  A  <->  Q  e.  (
aLine b ) ) )
7169, 70anbi12d 691 . . . . . . . . . 10  |-  ( A  =  ( aLine b )  ->  ( ( P  e.  A  /\  Q  e.  A )  <->  ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) ) ) )
7271anbi1d 685 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q ) ) )
73 eqeq1 2364 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( A  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine Q ) ) )
7472, 73imbi12d 311 . . . . . . . 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 213 . . . . . . 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 586 . . . . . 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 1151 . . . . 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 2743 . . . 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 2748 . . 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 187 . 2  |-  ( A  e. LinesEE  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) )
81803impib 1149 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   E.wrex 2620    C_ wss 3228   ` cfv 5337  (class class class)co 5945   NNcn 9836   EEcee 25075  Linecline2 25316  LinesEEclines2 25318
This theorem is referenced by:  hilbert1.2  25337  lineintmo  25339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-ec 6749  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-seq 11139  df-exp 11198  df-hash 11431  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-clim 12058  df-sum 12256  df-ee 25078  df-btwn 25079  df-cgr 25080  df-ofs 25165  df-ifs 25221  df-cgr3 25222  df-colinear 25223  df-fs 25224  df-line2 25319  df-lines2 25321
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