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Theorem linethru 30970
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 30969 . . 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 1053 . . . . . . . . . . . 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 1054 . . . . . . . . . . . 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 1055 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 30962 . . . . . . . . . . . 12  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  b  e.  ( EE `  n )  /\  a  =/=  b
) )  ->  (
aLine b )  C_  ( EE `  n ) )
72, 3, 4, 5, 6syl13anc 1278 . . . . . . . . . . 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 777 . . . . . . . . . . 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 3445 . . . . . . . . . 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 778 . . . . . . . . . . 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 3445 . . . . . . . . . 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 773 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a
)  /\  ( P  e.  ( EE `  n
)  /\  Q  e.  ( EE `  n ) ) )  ->  P  e.  ( aLine b ) )
1312adantl 472 . . . . . . . . . . . . . . 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 1053 . . . . . . . . . . . . . . . 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 1054 . . . . . . . . . . . . . . . 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 1055 . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . . . 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 30965 . . . . . . . . . . . . . . . 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 1294 . . . . . . . . . . . . . . 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 30967 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  a  =/=  P
) )  ->  (
aLine P )  =  ( PLine a ) )
2514, 15, 18, 20, 24syl13anc 1278 . . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . 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 2699 . . . . . . . . . . . . . . . . 17  |-  ( Q  =  a  ->  ( P  =/=  Q  <->  P  =/=  a ) )
2827anbi2d 715 . . . . . . . . . . . . . . . 16  |-  ( Q  =  a  ->  (
( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  Q
)  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/=  a ) ) )
2928anbi1d 716 . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . 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 6328 . . . . . . . . . . . . . . 15  |-  ( Q  =  a  ->  ( PLine Q )  =  ( PLine a ) )
3231eqeq2d 2472 . . . . . . . . . . . . . 14  |-  ( Q  =  a  ->  (
( aLine b )  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine a ) ) )
3330, 32imbi12d 326 . . . . . . . . . . . . 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 241 . . . . . . . . . . . 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 1014 . . . . . . . . . . . . . . . . . 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 1040 . . . . . . . . . . . . . . . . . 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 671 . . . . . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . . . . . . 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 1108 . . . . . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . . . . . . 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 1039 . . . . . . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . . . . . 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 30965 . . . . . . . . . . . . . . . . . 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 1294 . . . . . . . . . . . . . . . . 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 30967 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  NN  /\  ( a  e.  ( EE `  n )  /\  Q  e.  ( EE `  n )  /\  a  =/=  Q
) )  ->  (
aLine Q )  =  ( QLine a ) )
4938, 39, 42, 44, 48syl13anc 1278 . . . . . . . . . . . . . . . 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 2496 . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . 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 2542 . . . . . . . . . . . . . . . 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 1083 . . . . . . . . . . . . . . . . 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 1082 . . . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . . . . 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 30965 . . . . . . . . . . . . . . . . 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 1294 . . . . . . . . . . . . . . . 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 30967 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  NN  /\  ( Q  e.  ( EE `  n )  /\  P  e.  ( EE `  n )  /\  Q  =/=  P ) )  -> 
( QLine P )  =  ( PLine Q
) )
6038, 42, 53, 55, 59syl13anc 1278 . . . . . . . . . . . . . . 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 2500 . . . . . . . . . . . . . 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 1215 . . . . . . . . . . . . 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 441 . . . . . . . . . . . 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 2719 . . . . . . . . . . 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 624 . . . . . . . . . 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 690 . . . . . . . . 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 440 . . . . . . . 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 2529 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( P  e.  A  <->  P  e.  (
aLine b ) ) )
69 eleq2 2529 . . . . . . . . . . 11  |-  ( A  =  ( aLine b )  ->  ( Q  e.  A  <->  Q  e.  (
aLine b ) ) )
7068, 69anbi12d 722 . . . . . . . . . 10  |-  ( A  =  ( aLine b )  ->  ( ( P  e.  A  /\  Q  e.  A )  <->  ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) ) ) )
7170anbi1d 716 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( (
( P  e.  A  /\  Q  e.  A
)  /\  P  =/=  Q )  <->  ( ( P  e.  ( aLine b )  /\  Q  e.  ( aLine b ) )  /\  P  =/= 
Q ) ) )
72 eqeq1 2466 . . . . . . . . 9  |-  ( A  =  ( aLine b )  ->  ( A  =  ( PLine Q
)  <->  ( aLine b )  =  ( PLine Q ) ) )
7371, 72imbi12d 326 . . . . . . . 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 230 . . . . . . 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 612 . . . . . 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 1215 . . . . 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 2891 . . . 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 2896 . . 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 200 . 2  |-  ( A  e. LinesEE  ->  ( ( ( P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  A  =  ( PLine Q ) ) )
80793impib 1213 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 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750    C_ wss 3416   ` cfv 5605  (class class class)co 6320   NNcn 10642   EEcee 24974  Linecline2 30951  LinesEEclines2 30953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-inf2 8177  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-se 4816  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-isom 5614  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-ec 7396  df-map 7505  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-oi 8056  df-card 8404  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-ico 11675  df-icc 11676  df-fz 11820  df-fzo 11953  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-clim 13607  df-sum 13808  df-ee 24977  df-btwn 24978  df-cgr 24979  df-ofs 30800  df-colinear 30856  df-ifs 30857  df-cgr3 30858  df-fs 30859  df-line2 30954  df-lines2 30956
This theorem is referenced by:  hilbert1.2  30972  lineintmo  30974
  Copyright terms: Public domain W3C validator