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Theorem cgrextend 29634
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 4202 . . . . . . . . 9  |-  ( A  =  B  ->  <. A ,  B >.  =  <. B ,  B >. )
21breq1d 4447 . . . . . . . 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 997 . . . . . . . . 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 1031 . . . . . . . . 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 1033 . . . . . . . . 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 1034 . . . . . . . . 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 29629 . . . . . . . . 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 1231 . . . . . . . 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 4202 . . . . . . . . 9  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
13 opeq1 4202 . . . . . . . . 9  |-  ( D  =  E  ->  <. D ,  F >.  =  <. E ,  F >. )
1412, 13breqan12d 4452 . . . . . . . 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 1000 . . . . . . . 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 1075 . . . . . . . 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 1076 . . . . . . . 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 1177 . . . . . . 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 1077 . . . . . . . 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 1078 . . . . . . . 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 1177 . . . . . . 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 1079 . . . . . . . 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 1080 . . . . . . . 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 1177 . . . . . . 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 1177 . . . . . 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 765 . . . . . . . 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 766 . . . . . . . 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 29628 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  A  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) )  ->  <. A ,  A >.Cgr <. D ,  D >. )
3521, 22, 26, 34syl3anc 1229 . . . . . . . . 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 29624 . . . . . . . . . . 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 1238 . . . . . . . . . 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 29631 . . . . . . . . 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 1261 . . . . . . . 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 1180 . . . . . . 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 756 . . . . . . 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 29632 . . . . . 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 29624 . . . . . 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 1238 . . . . 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 2756 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   <.cop 4020   class class class wbr 4437   ` cfv 5578   NNcn 10543   EEcee 24169    Btwn cbtwn 24170  Cgrccgr 24171    OuterFiveSeg cofs 29608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-clim 13293  df-sum 13491  df-ee 24172  df-btwn 24173  df-cgr 24174  df-ofs 29609
This theorem is referenced by:  cgrextendand  29635  segconeq  29636  lineext  29702  brofs2  29703
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