Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cgrextend Structured version   Unicode version

Theorem cgrextend 28175
Description: Link congruence over a pair of line segments. Theorem 2.11 of [Schwabhauser] p. 29. (Contributed by Scott Fenton, 12-Jun-2013.)
Assertion
Ref Expression
cgrextend  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) )

Proof of Theorem cgrextend
StepHypRef Expression
1 opeq1 4159 . . . . . . . . 9  |-  ( A  =  B  ->  <. A ,  B >.  =  <. B ,  B >. )
21breq1d 4402 . . . . . . . 8  |-  ( A  =  B  ->  ( <. A ,  B >.Cgr <. D ,  E >.  <->  <. B ,  B >.Cgr <. D ,  E >. ) )
32adantr 465 . . . . . . 7  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  <->  <. B ,  B >.Cgr <. D ,  E >. ) )
4 simp1 988 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  N  e.  NN )
5 simp22 1022 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
6 simp31 1024 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
7 simp32 1025 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
8 cgrid2 28170 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) ) )  -> 
( <. B ,  B >.Cgr
<. D ,  E >.  ->  D  =  E )
)
94, 5, 6, 7, 8syl13anc 1221 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( <. B ,  B >.Cgr <. D ,  E >.  ->  D  =  E )
)
109adantl 466 . . . . . . 7  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( <. B ,  B >.Cgr
<. D ,  E >.  ->  D  =  E )
)
113, 10sylbid 215 . . . . . 6  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  ->  D  =  E )
)
12 opeq1 4159 . . . . . . . . 9  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
13 opeq1 4159 . . . . . . . . 9  |-  ( D  =  E  ->  <. D ,  F >.  =  <. E ,  F >. )
1412, 13breqan12d 4407 . . . . . . . 8  |-  ( ( A  =  B  /\  D  =  E )  ->  ( <. A ,  C >.Cgr
<. D ,  F >.  <->  <. B ,  C >.Cgr <. E ,  F >. ) )
1514exbiri 622 . . . . . . 7  |-  ( A  =  B  ->  ( D  =  E  ->  (
<. B ,  C >.Cgr <. E ,  F >.  ->  <. A ,  C >.Cgr <. D ,  F >. ) ) )
1615adantr 465 . . . . . 6  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( D  =  E  ->  ( <. B ,  C >.Cgr <. E ,  F >.  ->  <. A ,  C >.Cgr
<. D ,  F >. ) ) )
1711, 16syld 44 . . . . 5  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  -> 
( <. B ,  C >.Cgr
<. E ,  F >.  ->  <. A ,  C >.Cgr <. D ,  F >. ) ) )
1817impd 431 . . . 4  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. )  ->  <. A ,  C >.Cgr
<. D ,  F >. ) )
1918adantld 467 . . 3  |-  ( ( A  =  B  /\  ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) ) )  -> 
( ( ( B 
Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) )
2019ex 434 . 2  |-  ( A  =  B  ->  (
( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) ) )
21 simpl1 991 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  N  e.  NN )
22 simpl21 1066 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  A  e.  ( EE `  N ) )
23 simpl22 1067 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  B  e.  ( EE `  N ) )
2421, 22, 233jca 1168 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) ) )
25 simpl23 1068 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  C  e.  ( EE `  N ) )
26 simpl31 1069 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  D  e.  ( EE `  N ) )
2725, 22, 263jca 1168 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
28 simpl32 1070 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  E  e.  ( EE `  N ) )
29 simpl33 1071 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  F  e.  ( EE `  N ) )
3028, 29, 263jca 1168 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )
3124, 27, 303jca 1168 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( ( N  e.  NN  /\  A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) ) )
32 simprrl 763 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. ) )
33 simprrr 764 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )
34 cgrtriv 28169 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
3521, 22, 26, 34syl3anc 1219 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
3633simpld 459 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. A ,  B >.Cgr <. D ,  E >. )
37 cgrcomlr 28165 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N ) ) )  ->  ( <. A ,  B >.Cgr <. D ,  E >.  <->  <. B ,  A >.Cgr <. E ,  D >. ) )
3821, 22, 23, 26, 28, 37syl122anc 1228 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. A ,  B >.Cgr
<. D ,  E >.  <->  <. B ,  A >.Cgr <. E ,  D >. ) )
3936, 38mpbid 210 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. B ,  A >.Cgr <. E ,  D >. )
4035, 39jca 532 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. A ,  A >.Cgr
<. D ,  D >.  /\ 
<. B ,  A >.Cgr <. E ,  D >. ) )
41 brofs 28172 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  A >. >. 
OuterFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  A >.Cgr <. D ,  D >.  /\  <. B ,  A >.Cgr
<. E ,  D >. ) ) ) )
4221, 22, 23, 25, 22, 26, 28, 29, 26, 41syl333anc 1251 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. <. A ,  B >. ,  <. C ,  A >. >. 
OuterFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. )  /\  ( <. A ,  A >.Cgr <. D ,  D >.  /\  <. B ,  A >.Cgr
<. E ,  D >. ) ) ) )
4332, 33, 40, 42mpbir3and 1171 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. <. A ,  B >. ,  <. C ,  A >. >. 
OuterFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >. )
44 simprl 755 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  A  =/=  B )
4543, 44jca 532 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. <. A ,  B >. ,  <. C ,  A >. >. 
OuterFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >.  /\  A  =/=  B ) )
46 5segofs 28173 . . . . . 6  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  A >. >. 
OuterFiveSeg  <. <. D ,  E >. ,  <. F ,  D >. >.  /\  A  =/=  B )  ->  <. C ,  A >.Cgr <. F ,  D >. ) )
4731, 45, 46sylc 60 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. C ,  A >.Cgr <. F ,  D >. )
48 cgrcomlr 28165 . . . . . 6  |-  ( ( N  e.  NN  /\  ( C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) )  /\  ( F  e.  ( EE `  N )  /\  D  e.  ( EE `  N ) ) )  ->  ( <. C ,  A >.Cgr <. F ,  D >.  <->  <. A ,  C >.Cgr <. D ,  F >. ) )
4921, 25, 22, 29, 26, 48syl122anc 1228 . . . . 5  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  -> 
( <. C ,  A >.Cgr
<. F ,  D >.  <->  <. A ,  C >.Cgr <. D ,  F >. ) )
5047, 49mpbid 210 . . . 4  |-  ( ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  /\  ( A  =/=  B  /\  (
( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) ) )  ->  <. A ,  C >.Cgr <. D ,  F >. )
5150exp32 605 . . 3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  ( A  =/=  B  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) ) )
5251com12 31 . 2  |-  ( A  =/=  B  ->  (
( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N ) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) ) )
5320, 52pm2.61ine 2761 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
)  /\  F  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. A ,  C >.  /\  E  Btwn  <. D ,  F >. )  /\  ( <. A ,  B >.Cgr <. D ,  E >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) )  ->  <. A ,  C >.Cgr <. D ,  F >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   <.cop 3983   class class class wbr 4392   ` cfv 5518   NNcn 10425   EEcee 23271    Btwn cbtwn 23272  Cgrccgr 23273    OuterFiveSeg cofs 28149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-sup 7794  df-oi 7827  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-n0 10683  df-z 10750  df-uz 10965  df-rp 11095  df-ico 11409  df-icc 11410  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-clim 13070  df-sum 13268  df-ee 23274  df-btwn 23275  df-cgr 23276  df-ofs 28150
This theorem is referenced by:  cgrextendand  28176  segconeq  28177  lineext  28243  brofs2  28244
  Copyright terms: Public domain W3C validator