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Theorem fscgr 25922
Description: Congruence law for the general five segment configuration. Theorem 4.16 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
fscgr  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >.  /\  A  =/=  B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )

Proof of Theorem fscgr
StepHypRef Expression
1 brfs 25921 . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
21anbi1d 686 . 2  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >.  /\  A  =/=  B )  <->  ( ( A 
Colinear 
<. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B ) ) )
3 simp11 987 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  N  e.  NN )
4 simp12 988 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
5 simp13 989 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
6 simp21 990 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  C  e.  ( EE `  N
) )
7 brcolinear 25901 . . . . . 6  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
83, 4, 5, 6, 7syl13anc 1186 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >.  <-> 
( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
9 simp23 992 . . . . . . . . . . . . 13  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  E  e.  ( EE `  N
) )
10 simp31 993 . . . . . . . . . . . . 13  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
11 simp32 994 . . . . . . . . . . . . 13  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  G  e.  ( EE `  N
) )
12 cgr3permute2 25891 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  G  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  <->  <. B ,  <. A ,  C >. >.Cgr3 <. F ,  <. E ,  G >. >. ) )
133, 4, 5, 6, 9, 10, 11, 12syl133anc 1207 . . . . . . . . . . . 12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  <->  <. B ,  <. A ,  C >. >.Cgr3 <. F ,  <. E ,  G >. >. ) )
14 ancom 438 . . . . . . . . . . . . 13  |-  ( (
<. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  <-> 
( <. B ,  D >.Cgr
<. F ,  H >.  /\ 
<. A ,  D >.Cgr <. E ,  H >. ) )
1514a1i 11 . . . . . . . . . . . 12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  <-> 
( <. B ,  D >.Cgr
<. F ,  H >.  /\ 
<. A ,  D >.Cgr <. E ,  H >. ) ) )
1613, 153anbi23d 1257 . . . . . . . . . . 11  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  ( A  Btwn  <. B ,  C >.  /\  <. B ,  <. A ,  C >. >.Cgr3 <. F ,  <. E ,  G >. >.  /\  ( <. B ,  D >.Cgr
<. F ,  H >.  /\ 
<. A ,  D >.Cgr <. E ,  H >. ) ) ) )
17 simp22 991 . . . . . . . . . . . 12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  D  e.  ( EE `  N
) )
18 simp33 995 . . . . . . . . . . . 12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  H  e.  ( EE `  N
) )
19 brofs2 25919 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. B ,  A >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. F ,  E >. ,  <. G ,  H >. >. 
<->  ( A  Btwn  <. B ,  C >.  /\  <. B ,  <. A ,  C >. >.Cgr3 <. F ,  <. E ,  G >. >.  /\  ( <. B ,  D >.Cgr <. F ,  H >.  /\  <. A ,  D >.Cgr <. E ,  H >. ) ) ) )
203, 5, 4, 6, 17, 10, 9, 11, 18, 19syl333anc 1216 . . . . . . . . . . 11  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. B ,  A >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. F ,  E >. ,  <. G ,  H >. >. 
<->  ( A  Btwn  <. B ,  C >.  /\  <. B ,  <. A ,  C >. >.Cgr3 <. F ,  <. E ,  G >. >.  /\  ( <. B ,  D >.Cgr <. F ,  H >.  /\  <. A ,  D >.Cgr <. E ,  H >. ) ) ) )
2116, 20bitr4d 248 . . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  <. <. B ,  A >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. F ,  E >. ,  <. G ,  H >. >. ) )
22 necom 2652 . . . . . . . . . . 11  |-  ( A  =/=  B  <->  B  =/=  A )
2322a1i 11 . . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( A  =/=  B  <->  B  =/=  A ) )
2421, 23anbi12d 692 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B )  <->  ( <. <. B ,  A >. , 
<. C ,  D >. >.  OuterFiveSeg  <. <. F ,  E >. , 
<. G ,  H >. >.  /\  B  =/=  A
) ) )
25 5segofs 25848 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  B  e.  ( EE
`  N )  /\  A  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  F  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. B ,  A >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. F ,  E >. ,  <. G ,  H >. >.  /\  B  =/=  A )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
263, 5, 4, 6, 17, 10, 9, 11, 18, 25syl333anc 1216 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. B ,  A >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. F ,  E >. ,  <. G ,  H >. >.  /\  B  =/=  A )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
2724, 26sylbid 207 . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( A  Btwn  <. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
2827exp3a 426 . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  ->  ( A  =/=  B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) )
29283expd 1170 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( A  Btwn  <. B ,  C >.  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  ->  ( ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  ->  ( A  =/= 
B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) ) ) )
30 btwncom 25856 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( B  Btwn  <. C ,  A >. 
<->  B  Btwn  <. A ,  C >. ) )
313, 5, 6, 4, 30syl13anc 1186 . . . . . . . . . . . 12  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( B  Btwn  <. C ,  A >.  <-> 
B  Btwn  <. A ,  C >. ) )
32313anbi1d 1258 . . . . . . . . . . 11  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. C ,  A >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) ) ) )
33 brofs2 25919 . . . . . . . . . . 11  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( B  Btwn  <. A ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
3432, 33bitr4d 248 . . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. C ,  A >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. ) )
3534anbi1d 686 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. C ,  A >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B )  <->  ( <. <. A ,  B >. , 
<. C ,  D >. >.  OuterFiveSeg  <. <. E ,  F >. , 
<. G ,  H >. >.  /\  A  =/=  B
) ) )
36 5segofs 25848 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  D >. >. 
OuterFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >.  /\  A  =/=  B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
3735, 36sylbid 207 . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( B  Btwn  <. C ,  A >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
3837exp3a 426 . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( B  Btwn  <. C ,  A >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  ->  ( A  =/=  B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) )
39383expd 1170 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( B  Btwn  <. C ,  A >.  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  ->  ( ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  ->  ( A  =/= 
B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) ) ) )
40 cgr3permute1 25890 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  /\  ( E  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  G  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. E ,  <. G ,  F >. >. ) )
413, 4, 5, 6, 9, 10, 11, 40syl133anc 1207 . . . . . . . . . . 11  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  <->  <. A ,  <. C ,  B >. >.Cgr3 <. E ,  <. G ,  F >. >. ) )
42413anbi2d 1259 . . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. A ,  B >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  ( C  Btwn  <. A ,  B >.  /\  <. A ,  <. C ,  B >. >.Cgr3 <. E ,  <. G ,  F >. >.  /\  ( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) ) ) )
43 brifs2 25920 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  /\  ( G  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  C >. ,  <. B ,  D >. >. 
InnerFiveSeg  <. <. E ,  G >. ,  <. F ,  H >. >. 
<->  ( C  Btwn  <. A ,  B >.  /\  <. A ,  <. C ,  B >. >.Cgr3 <. E ,  <. G ,  F >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
443, 4, 6, 5, 17, 9, 11, 10, 18, 43syl333anc 1216 . . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  C >. ,  <. B ,  D >. >. 
InnerFiveSeg  <. <. E ,  G >. ,  <. F ,  H >. >. 
<->  ( C  Btwn  <. A ,  B >.  /\  <. A ,  <. C ,  B >. >.Cgr3 <. E ,  <. G ,  F >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
4542, 44bitr4d 248 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. A ,  B >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  <->  <. <. A ,  C >. ,  <. B ,  D >. >. 
InnerFiveSeg  <. <. E ,  G >. ,  <. F ,  H >. >. ) )
46 ifscgr 25886 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  A  e.  ( EE
`  N )  /\  C  e.  ( EE `  N ) )  /\  ( B  e.  ( EE `  N )  /\  D  e.  ( EE `  N )  /\  E  e.  ( EE `  N
) )  /\  ( G  e.  ( EE `  N )  /\  F  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  C >. ,  <. B ,  D >. >. 
InnerFiveSeg  <. <. E ,  G >. ,  <. F ,  H >. >.  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
473, 4, 6, 5, 17, 9, 11, 10, 18, 46syl333anc 1216 . . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( <. <. A ,  C >. ,  <. B ,  D >. >. 
InnerFiveSeg  <. <. E ,  G >. ,  <. F ,  H >. >.  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
4845, 47sylbid 207 . . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. A ,  B >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
4948a1dd 44 . . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( C  Btwn  <. A ,  B >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  ->  ( A  =/=  B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) )
50493expd 1170 . . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( C  Btwn  <. A ,  B >.  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  ->  ( ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  ->  ( A  =/= 
B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) ) ) )
5129, 39, 503jaod 1248 . . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  ->  (
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  ->  ( A  =/= 
B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) ) ) )
528, 51sylbid 207 . . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  ( A  Colinear  <. B ,  C >.  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  ->  ( ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. )  ->  ( A  =/= 
B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) ) ) )
53523impd 1167 . . 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) )  ->  ( A  =/=  B  ->  <. C ,  D >.Cgr <. G ,  H >. ) ) )
5453imp3a 421 . 2  |-  ( ( ( 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) )  /\  A  =/= 
B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
552, 54sylbid 207 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 )  /\  G  e.  ( EE `  N
)  /\  H  e.  ( EE `  N ) ) )  ->  (
( <. <. A ,  B >. ,  <. C ,  D >. >.  FiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >.  /\  A  =/=  B )  ->  <. C ,  D >.Cgr <. G ,  H >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    e. wcel 1721    =/= wne 2571   <.cop 3781   class class class wbr 4176   ` cfv 5417   NNcn 9960   EEcee 25735    Btwn cbtwn 25736  Cgrccgr 25737    OuterFiveSeg cofs 25824    InnerFiveSeg cifs 25877  Cgr3ccgr3 25878    Colinear ccolin 25879    FiveSeg cfs 25880
This theorem is referenced by:  linecgr  25923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-ee 25738  df-btwn 25739  df-cgr 25740  df-ofs 25825  df-ifs 25881  df-cgr3 25882  df-colinear 25883  df-fs 25884
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