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Theorem fscgr 30391
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 30390 . . 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 703 . 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 1027 . . . . . 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 1028 . . . . . 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 1029 . . . . . 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 1030 . . . . . 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 30370 . . . . . 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 1232 . . . . 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 1032 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . 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 1034 . . . . . . . . . . . . 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 30360 . . . . . . . . . . . . 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 1253 . . . . . . . . . . . 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 448 . . . . . . . . . . . . 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 1304 . . . . . . . . . . 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 1031 . . . . . . . . . . . 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 1035 . . . . . . . . . . . 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 30388 . . . . . . . . . . . 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 1262 . . . . . . . . . . 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 256 . . . . . . . . . 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 2672 . . . . . . . . . . 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 709 . . . . . . . . 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 30317 . . . . . . . . . 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 1262 . . . . . . . . 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 215 . . . . . . . 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 >. ) )
2827expd 434 . . . . . . 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 1214 . . . . . 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 30325 . . . . . . . . . . . . 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 1232 . . . . . . . . . . . 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 1305 . . . . . . . . . . 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 30388 . . . . . . . . . . 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 256 . . . . . . . . . 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 703 . . . . . . . . 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 30317 . . . . . . . . 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 215 . . . . . . . 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 >. ) )
3837expd 434 . . . . . . 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 1214 . . . . . 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 30359 . . . . . . . . . . . 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 1253 . . . . . . . . . . 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 1306 . . . . . . . . . 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 30389 . . . . . . . . . . 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 1262 . . . . . . . . . 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 256 . . . . . . . . 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 30355 . . . . . . . . . 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 1262 . . . . . . . . 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 215 . . . . . . . 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 1214 . . . . . 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 1294 . . . . 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 215 . . . 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 1211 . . 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 >. ) ) )
5453impd 429 . 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 215 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 973    /\ w3a 974    e. wcel 1842    =/= wne 2598   <.cop 3977   class class class wbr 4394   ` cfv 5525   NNcn 10496   EEcee 24489    Btwn cbtwn 24490  Cgrccgr 24491    OuterFiveSeg cofs 30293    InnerFiveSeg cifs 30346  Cgr3ccgr3 30347    Colinear ccolin 30348    FiveSeg cfs 30349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-inf2 8011  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-map 7379  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-oi 7889  df-card 8272  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-div 10168  df-nn 10497  df-2 10555  df-3 10556  df-n0 10757  df-z 10826  df-uz 11046  df-rp 11184  df-ico 11506  df-icc 11507  df-fz 11644  df-fzo 11768  df-seq 12062  df-exp 12121  df-hash 12360  df-cj 12988  df-re 12989  df-im 12990  df-sqrt 13124  df-abs 13125  df-clim 13367  df-sum 13565  df-ee 24492  df-btwn 24493  df-cgr 24494  df-ofs 30294  df-colinear 30350  df-ifs 30351  df-cgr3 30352  df-fs 30353
This theorem is referenced by:  linecgr  30392
  Copyright terms: Public domain W3C validator