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Theorem brifs 25979
Description: Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
Assertion
Ref Expression
brifs  |-  ( ( ( 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 >. >. 
InnerFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr
<. G ,  H >. ) ) ) )

Proof of Theorem brifs
Dummy variables  a 
b  c  d  e  f  g  h  p  q  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3986 . . . . 5  |-  ( a  =  A  ->  <. a ,  c >.  =  <. A ,  c >. )
21breq2d 4226 . . . 4  |-  ( a  =  A  ->  (
b  Btwn  <. a ,  c >.  <->  b  Btwn  <. A , 
c >. ) )
32anbi1d 687 . . 3  |-  ( a  =  A  ->  (
( b  Btwn  <. a ,  c >.  /\  f  Btwn  <. e ,  g
>. )  <->  ( b  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. ) ) )
41breq1d 4224 . . . 4  |-  ( a  =  A  ->  ( <. a ,  c >.Cgr <. e ,  g >.  <->  <. A ,  c >.Cgr <.
e ,  g >.
) )
54anbi1d 687 . . 3  |-  ( a  =  A  ->  (
( <. a ,  c
>.Cgr <. e ,  g
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A , 
c >.Cgr <. e ,  g
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. ) ) )
6 opeq1 3986 . . . . 5  |-  ( a  =  A  ->  <. a ,  d >.  =  <. A ,  d >. )
76breq1d 4224 . . . 4  |-  ( a  =  A  ->  ( <. a ,  d >.Cgr <. e ,  h >.  <->  <. A ,  d >.Cgr <. e ,  h >. ) )
87anbi1d 687 . . 3  |-  ( a  =  A  ->  (
( <. a ,  d
>.Cgr <. e ,  h >.  /\  <. c ,  d
>.Cgr <. g ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. c ,  d
>.Cgr <. g ,  h >. ) ) )
93, 5, 83anbi123d 1255 . 2  |-  ( a  =  A  ->  (
( ( b  Btwn  <.
a ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. a ,  c
>.Cgr <. e ,  g
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) )  <->  ( ( b 
Btwn  <. A ,  c
>.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  c
>.Cgr <. e ,  g
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) ) ) )
10 breq1 4217 . . . 4  |-  ( b  =  B  ->  (
b  Btwn  <. A , 
c >. 
<->  B  Btwn  <. A , 
c >. ) )
1110anbi1d 687 . . 3  |-  ( b  =  B  ->  (
( b  Btwn  <. A , 
c >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. ) ) )
12 opeq1 3986 . . . . 5  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
1312breq1d 4224 . . . 4  |-  ( b  =  B  ->  ( <. b ,  c >.Cgr <. f ,  g >.  <->  <. B ,  c >.Cgr <.
f ,  g >.
) )
1413anbi2d 686 . . 3  |-  ( b  =  B  ->  (
( <. A ,  c
>.Cgr <. e ,  g
>.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A , 
c >.Cgr <. e ,  g
>.  /\  <. B ,  c
>.Cgr <. f ,  g
>. ) ) )
1511, 143anbi12d 1256 . 2  |-  ( b  =  B  ->  (
( ( b  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  c >.Cgr <. e ,  g >.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  c
>.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  c
>.Cgr <. e ,  g
>.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) ) ) )
16 opeq2 3987 . . . . 5  |-  ( c  =  C  ->  <. A , 
c >.  =  <. A ,  C >. )
1716breq2d 4226 . . . 4  |-  ( c  =  C  ->  ( B  Btwn  <. A ,  c
>. 
<->  B  Btwn  <. A ,  C >. ) )
1817anbi1d 687 . . 3  |-  ( c  =  C  ->  (
( B  Btwn  <. A , 
c >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. ) ) )
1916breq1d 4224 . . . 4  |-  ( c  =  C  ->  ( <. A ,  c >.Cgr <. e ,  g >.  <->  <. A ,  C >.Cgr <.
e ,  g >.
) )
20 opeq2 3987 . . . . 5  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
2120breq1d 4224 . . . 4  |-  ( c  =  C  ->  ( <. B ,  c >.Cgr <. f ,  g >.  <->  <. B ,  C >.Cgr <.
f ,  g >.
) )
2219, 21anbi12d 693 . . 3  |-  ( c  =  C  ->  (
( <. A ,  c
>.Cgr <. e ,  g
>.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  <->  ( <. A ,  C >.Cgr <. e ,  g
>.  /\  <. B ,  C >.Cgr
<. f ,  g >.
) ) )
23 opeq1 3986 . . . . 5  |-  ( c  =  C  ->  <. c ,  d >.  =  <. C ,  d >. )
2423breq1d 4224 . . . 4  |-  ( c  =  C  ->  ( <. c ,  d >.Cgr <. g ,  h >.  <->  <. C ,  d >.Cgr <. g ,  h >. ) )
2524anbi2d 686 . . 3  |-  ( c  =  C  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. c ,  d
>.Cgr <. g ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. C ,  d
>.Cgr <. g ,  h >. ) ) )
2618, 22, 253anbi123d 1255 . 2  |-  ( c  =  C  ->  (
( ( B  Btwn  <. A ,  c >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  c >.Cgr <. e ,  g >.  /\  <. B ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  C >.Cgr
<. e ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\  <. C , 
d >.Cgr <. g ,  h >. ) ) ) )
27 opeq2 3987 . . . . 5  |-  ( d  =  D  ->  <. A , 
d >.  =  <. A ,  D >. )
2827breq1d 4224 . . . 4  |-  ( d  =  D  ->  ( <. A ,  d >.Cgr <. e ,  h >.  <->  <. A ,  D >.Cgr <. e ,  h >. ) )
29 opeq2 3987 . . . . 5  |-  ( d  =  D  ->  <. C , 
d >.  =  <. C ,  D >. )
3029breq1d 4224 . . . 4  |-  ( d  =  D  ->  ( <. C ,  d >.Cgr <. g ,  h >.  <->  <. C ,  D >.Cgr <. g ,  h >. ) )
3128, 30anbi12d 693 . . 3  |-  ( d  =  D  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. C ,  d
>.Cgr <. g ,  h >. )  <->  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. C ,  D >.Cgr
<. g ,  h >. ) ) )
32313anbi3d 1261 . 2  |-  ( d  =  D  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  C >.Cgr <.
e ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\  <. C , 
d >.Cgr <. g ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  C >.Cgr
<. e ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. C ,  D >.Cgr <. g ,  h >. ) ) ) )
33 opeq1 3986 . . . . 5  |-  ( e  =  E  ->  <. e ,  g >.  =  <. E ,  g >. )
3433breq2d 4226 . . . 4  |-  ( e  =  E  ->  (
f  Btwn  <. e ,  g >.  <->  f  Btwn  <. E , 
g >. ) )
3534anbi2d 686 . . 3  |-  ( e  =  E  ->  (
( B  Btwn  <. A ,  C >.  /\  f  Btwn  <.
e ,  g >.
)  <->  ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. ) ) )
3633breq2d 4226 . . . 4  |-  ( e  =  E  ->  ( <. A ,  C >.Cgr <.
e ,  g >.  <->  <. A ,  C >.Cgr <. E ,  g >. ) )
3736anbi1d 687 . . 3  |-  ( e  =  E  ->  (
( <. A ,  C >.Cgr
<. e ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  <->  ( <. A ,  C >.Cgr <. E ,  g
>.  /\  <. B ,  C >.Cgr
<. f ,  g >.
) ) )
38 opeq1 3986 . . . . 5  |-  ( e  =  E  ->  <. e ,  h >.  =  <. E ,  h >. )
3938breq2d 4226 . . . 4  |-  ( e  =  E  ->  ( <. A ,  D >.Cgr <.
e ,  h >.  <->  <. A ,  D >.Cgr <. E ,  h >. ) )
4039anbi1d 687 . . 3  |-  ( e  =  E  ->  (
( <. A ,  D >.Cgr
<. e ,  h >.  /\ 
<. C ,  D >.Cgr <.
g ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. C ,  D >.Cgr <.
g ,  h >. ) ) )
4135, 37, 403anbi123d 1255 . 2  |-  ( e  =  E  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. e ,  g >. )  /\  ( <. A ,  C >.Cgr <.
e ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. C ,  D >.Cgr <. g ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. )  /\  ( <. A ,  C >.Cgr <. E ,  g >.  /\ 
<. B ,  C >.Cgr <.
f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr <. g ,  h >. ) ) ) )
42 breq1 4217 . . . 4  |-  ( f  =  F  ->  (
f  Btwn  <. E , 
g >. 
<->  F  Btwn  <. E , 
g >. ) )
4342anbi2d 686 . . 3  |-  ( f  =  F  ->  (
( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E ,  g >. )  <-> 
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  g >. ) ) )
44 opeq1 3986 . . . . 5  |-  ( f  =  F  ->  <. f ,  g >.  =  <. F ,  g >. )
4544breq2d 4226 . . . 4  |-  ( f  =  F  ->  ( <. B ,  C >.Cgr <.
f ,  g >.  <->  <. B ,  C >.Cgr <. F ,  g >. ) )
4645anbi2d 686 . . 3  |-  ( f  =  F  ->  (
( <. A ,  C >.Cgr
<. E ,  g >.  /\  <. B ,  C >.Cgr
<. f ,  g >.
)  <->  ( <. A ,  C >.Cgr <. E ,  g
>.  /\  <. B ,  C >.Cgr
<. F ,  g >.
) ) )
4743, 463anbi12d 1256 . 2  |-  ( f  =  F  ->  (
( ( B  Btwn  <. A ,  C >.  /\  f  Btwn  <. E , 
g >. )  /\  ( <. A ,  C >.Cgr <. E ,  g >.  /\ 
<. B ,  C >.Cgr <.
f ,  g >.
)  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr <. g ,  h >. ) )  <->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E , 
g >. )  /\  ( <. A ,  C >.Cgr <. E ,  g >.  /\ 
<. B ,  C >.Cgr <. F ,  g >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr
<. g ,  h >. ) ) ) )
48 opeq2 3987 . . . . 5  |-  ( g  =  G  ->  <. E , 
g >.  =  <. E ,  G >. )
4948breq2d 4226 . . . 4  |-  ( g  =  G  ->  ( F  Btwn  <. E ,  g
>. 
<->  F  Btwn  <. E ,  G >. ) )
5049anbi2d 686 . . 3  |-  ( g  =  G  ->  (
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  g >. )  <-> 
( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. ) ) )
5148breq2d 4226 . . . 4  |-  ( g  =  G  ->  ( <. A ,  C >.Cgr <. E ,  g >.  <->  <. A ,  C >.Cgr <. E ,  G >. ) )
52 opeq2 3987 . . . . 5  |-  ( g  =  G  ->  <. F , 
g >.  =  <. F ,  G >. )
5352breq2d 4226 . . . 4  |-  ( g  =  G  ->  ( <. B ,  C >.Cgr <. F ,  g >.  <->  <. B ,  C >.Cgr <. F ,  G >. ) )
5451, 53anbi12d 693 . . 3  |-  ( g  =  G  ->  (
( <. A ,  C >.Cgr
<. E ,  g >.  /\  <. B ,  C >.Cgr
<. F ,  g >.
)  <->  ( <. A ,  C >.Cgr <. E ,  G >.  /\  <. B ,  C >.Cgr
<. F ,  G >. ) ) )
55 opeq1 3986 . . . . 5  |-  ( g  =  G  ->  <. g ,  h >.  =  <. G ,  h >. )
5655breq2d 4226 . . . 4  |-  ( g  =  G  ->  ( <. C ,  D >.Cgr <.
g ,  h >.  <->  <. C ,  D >.Cgr <. G ,  h >. ) )
5756anbi2d 686 . . 3  |-  ( g  =  G  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. C ,  D >.Cgr <.
g ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. C ,  D >.Cgr <. G ,  h >. ) ) )
5850, 54, 573anbi123d 1255 . 2  |-  ( g  =  G  ->  (
( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E , 
g >. )  /\  ( <. A ,  C >.Cgr <. E ,  g >.  /\ 
<. B ,  C >.Cgr <. F ,  g >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr
<. g ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr
<. G ,  h >. ) ) ) )
59 opeq2 3987 . . . . 5  |-  ( h  =  H  ->  <. E ,  h >.  =  <. E ,  H >. )
6059breq2d 4226 . . . 4  |-  ( h  =  H  ->  ( <. A ,  D >.Cgr <. E ,  h >.  <->  <. A ,  D >.Cgr <. E ,  H >. ) )
61 opeq2 3987 . . . . 5  |-  ( h  =  H  ->  <. G ,  h >.  =  <. G ,  H >. )
6261breq2d 4226 . . . 4  |-  ( h  =  H  ->  ( <. C ,  D >.Cgr <. G ,  h >.  <->  <. C ,  D >.Cgr <. G ,  H >. ) )
6360, 62anbi12d 693 . . 3  |-  ( h  =  H  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. C ,  D >.Cgr <. G ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. C ,  D >.Cgr <. G ,  H >. ) ) )
64633anbi3d 1261 . 2  |-  ( h  =  H  ->  (
( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. C ,  D >.Cgr
<. G ,  h >. ) )  <->  ( ( B 
Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr
<. G ,  H >. ) ) ) )
65 fveq2 5730 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
66 df-ifs 25975 . 2  |-  InnerFiveSeg  =  { <. p ,  q >.  |  E. n  e.  NN  E. a  e.  ( EE `  n
) E. b  e.  ( EE `  n
) E. c  e.  ( EE `  n
) E. d  e.  ( EE `  n
) E. e  e.  ( EE `  n
) E. f  e.  ( EE `  n
) E. g  e.  ( EE `  n
) E. h  e.  ( EE `  n
) ( p  = 
<. <. a ,  b
>. ,  <. c ,  d >. >.  /\  q  =  <. <. e ,  f
>. ,  <. g ,  h >. >.  /\  ( (
b  Btwn  <. a ,  c >.  /\  f  Btwn  <. e ,  g
>. )  /\  ( <. a ,  c >.Cgr <. e ,  g >.  /\  <. b ,  c
>.Cgr <. f ,  g
>. )  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\ 
<. c ,  d >.Cgr <. g ,  h >. ) ) ) }
679, 15, 26, 32, 41, 47, 58, 64, 65, 66br8 25381 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 >. >. 
InnerFiveSeg  <. <. E ,  F >. ,  <. G ,  H >. >. 
<->  ( ( B  Btwn  <. A ,  C >.  /\  F  Btwn  <. E ,  G >. )  /\  ( <. A ,  C >.Cgr <. E ,  G >.  /\ 
<. B ,  C >.Cgr <. F ,  G >. )  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. C ,  D >.Cgr
<. G ,  H >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4214   ` cfv 5456   NNcn 10002   EEcee 25829    Btwn cbtwn 25830  Cgrccgr 25831    InnerFiveSeg cifs 25971
This theorem is referenced by:  ifscgr  25980  cgrsub  25981  btwnxfr  25992  brifs2  26014  btwnconn1lem6  26028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-iota 5420  df-fv 5464  df-ifs 25975
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