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Theorem brfs 24113
Description: Binary relationship form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
brfs  |-  ( ( ( 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 >. ) ) ) )

Proof of Theorem brfs
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 breq1 4026 . . 3  |-  ( a  =  A  ->  (
a  Colinear  <. b ,  c
>. 
<->  A  Colinear  <. b ,  c
>. ) )
2 opeq1 3796 . . . 4  |-  ( a  =  A  ->  <. a ,  <. b ,  c
>. >.  =  <. A ,  <. b ,  c >. >. )
32breq1d 4033 . . 3  |-  ( a  =  A  ->  ( <. a ,  <. b ,  c >. >.Cgr3 <. e ,  <. f ,  g
>. >. 
<-> 
<. A ,  <. b ,  c >. >.Cgr3 <. e ,  <. f ,  g
>. >. ) )
4 opeq1 3796 . . . . 5  |-  ( a  =  A  ->  <. a ,  d >.  =  <. A ,  d >. )
54breq1d 4033 . . . 4  |-  ( a  =  A  ->  ( <. a ,  d >.Cgr <. e ,  h >.  <->  <. A ,  d >.Cgr <. e ,  h >. ) )
65anbi1d 685 . . 3  |-  ( a  =  A  ->  (
( <. a ,  d
>.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. ) ) )
71, 3, 63anbi123d 1252 . 2  |-  ( a  =  A  ->  (
( a  Colinear  <. b ,  c >.  /\  <. a ,  <. b ,  c
>. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) )  <->  ( A  Colinear  <. b ,  c >.  /\  <. A ,  <. b ,  c
>. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) ) ) )
8 opeq1 3796 . . . 4  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
98breq2d 4035 . . 3  |-  ( b  =  B  ->  ( A  Colinear  <. b ,  c
>. 
<->  A  Colinear  <. B ,  c
>. ) )
108opeq2d 3803 . . . 4  |-  ( b  =  B  ->  <. A ,  <. b ,  c >. >.  =  <. A ,  <. B ,  c >. >. )
1110breq1d 4033 . . 3  |-  ( b  =  B  ->  ( <. A ,  <. b ,  c >. >.Cgr3 <. e ,  <. f ,  g
>. >. 
<-> 
<. A ,  <. B , 
c >. >.Cgr3 <. e ,  <. f ,  g >. >. )
)
12 opeq1 3796 . . . . 5  |-  ( b  =  B  ->  <. b ,  d >.  =  <. B ,  d >. )
1312breq1d 4033 . . . 4  |-  ( b  =  B  ->  ( <. b ,  d >.Cgr <. f ,  h >.  <->  <. B ,  d >.Cgr <. f ,  h >. ) )
1413anbi2d 684 . . 3  |-  ( b  =  B  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. b ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A , 
d >.Cgr <. e ,  h >.  /\  <. B ,  d
>.Cgr <. f ,  h >. ) ) )
159, 11, 143anbi123d 1252 . 2  |-  ( b  =  B  ->  (
( A  Colinear  <. b ,  c >.  /\  <. A ,  <. b ,  c
>. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. b ,  d >.Cgr <. f ,  h >. ) )  <->  ( A  Colinear  <. B ,  c >.  /\  <. A ,  <. B ,  c
>. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) ) ) )
16 opeq2 3797 . . . 4  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
1716breq2d 4035 . . 3  |-  ( c  =  C  ->  ( A  Colinear  <. B ,  c
>. 
<->  A  Colinear  <. B ,  C >. ) )
1816opeq2d 3803 . . . 4  |-  ( c  =  C  ->  <. A ,  <. B ,  c >. >.  =  <. A ,  <. B ,  C >. >. )
1918breq1d 4033 . . 3  |-  ( c  =  C  ->  ( <. A ,  <. B , 
c >. >.Cgr3 <. e ,  <. f ,  g >. >.  <->  <. A ,  <. B ,  C >. >.Cgr3 <.
e ,  <. f ,  g >. >. )
)
2017, 193anbi12d 1253 . 2  |-  ( c  =  C  ->  (
( A  Colinear  <. B , 
c >.  /\  <. A ,  <. B ,  c >. >.Cgr3
<. e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) ) ) )
21 opeq2 3797 . . . . 5  |-  ( d  =  D  ->  <. A , 
d >.  =  <. A ,  D >. )
2221breq1d 4033 . . . 4  |-  ( d  =  D  ->  ( <. A ,  d >.Cgr <. e ,  h >.  <->  <. A ,  D >.Cgr <. e ,  h >. ) )
23 opeq2 3797 . . . . 5  |-  ( d  =  D  ->  <. B , 
d >.  =  <. B ,  D >. )
2423breq1d 4033 . . . 4  |-  ( d  =  D  ->  ( <. B ,  d >.Cgr <. f ,  h >.  <->  <. B ,  D >.Cgr <. f ,  h >. ) )
2522, 24anbi12d 691 . . 3  |-  ( d  =  D  ->  (
( <. A ,  d
>.Cgr <. e ,  h >.  /\  <. B ,  d
>.Cgr <. f ,  h >. )  <->  ( <. A ,  D >.Cgr <. e ,  h >.  /\  <. B ,  D >.Cgr
<. f ,  h >. ) ) )
26253anbi3d 1258 . 2  |-  ( d  =  D  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <.
e ,  <. f ,  g >. >.  /\  ( <. A ,  d >.Cgr <. e ,  h >.  /\ 
<. B ,  d >.Cgr <. f ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. e ,  <. f ,  g >. >.  /\  ( <. A ,  D >.Cgr <.
e ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. ) ) ) )
27 opeq1 3796 . . . 4  |-  ( e  =  E  ->  <. e ,  <. f ,  g
>. >.  =  <. E ,  <. f ,  g >. >. )
2827breq2d 4035 . . 3  |-  ( e  =  E  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. e ,  <. f ,  g >. >.  <->  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. f ,  g >. >. ) )
29 opeq1 3796 . . . . 5  |-  ( e  =  E  ->  <. e ,  h >.  =  <. E ,  h >. )
3029breq2d 4035 . . . 4  |-  ( e  =  E  ->  ( <. A ,  D >.Cgr <.
e ,  h >.  <->  <. A ,  D >.Cgr <. E ,  h >. ) )
3130anbi1d 685 . . 3  |-  ( e  =  E  ->  (
( <. A ,  D >.Cgr
<. e ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. ) ) )
3228, 313anbi23d 1255 . 2  |-  ( e  =  E  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <.
e ,  <. f ,  g >. >.  /\  ( <. A ,  D >.Cgr <.
e ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. f ,  g >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. ) ) ) )
33 opeq1 3796 . . . . 5  |-  ( f  =  F  ->  <. f ,  g >.  =  <. F ,  g >. )
3433opeq2d 3803 . . . 4  |-  ( f  =  F  ->  <. E ,  <. f ,  g >. >.  =  <. E ,  <. F ,  g >. >. )
3534breq2d 4035 . . 3  |-  ( f  =  F  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. f ,  g >. >.  <->  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F , 
g >. >. ) )
36 opeq1 3796 . . . . 5  |-  ( f  =  F  ->  <. f ,  h >.  =  <. F ,  h >. )
3736breq2d 4035 . . . 4  |-  ( f  =  F  ->  ( <. B ,  D >.Cgr <.
f ,  h >.  <->  <. B ,  D >.Cgr <. F ,  h >. ) )
3837anbi2d 684 . . 3  |-  ( f  =  F  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <.
f ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. ) ) )
3935, 383anbi23d 1255 . 2  |-  ( f  =  F  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. f ,  g >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr <. f ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  g >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. ) ) ) )
40 opeq2 3797 . . . . 5  |-  ( g  =  G  ->  <. F , 
g >.  =  <. F ,  G >. )
4140opeq2d 3803 . . . 4  |-  ( g  =  G  ->  <. E ,  <. F ,  g >. >.  =  <. E ,  <. F ,  G >. >. )
4241breq2d 4035 . . 3  |-  ( g  =  G  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  g >. >.  <->  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >. ) )
43423anbi2d 1257 . 2  |-  ( g  =  G  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F , 
g >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr <. F ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. ) ) ) )
44 opeq2 3797 . . . . 5  |-  ( h  =  H  ->  <. E ,  h >.  =  <. E ,  H >. )
4544breq2d 4035 . . . 4  |-  ( h  =  H  ->  ( <. A ,  D >.Cgr <. E ,  h >.  <->  <. A ,  D >.Cgr <. E ,  H >. ) )
46 opeq2 3797 . . . . 5  |-  ( h  =  H  ->  <. F ,  h >.  =  <. F ,  H >. )
4746breq2d 4035 . . . 4  |-  ( h  =  H  ->  ( <. B ,  D >.Cgr <. F ,  h >.  <->  <. B ,  D >.Cgr <. F ,  H >. ) )
4845, 47anbi12d 691 . . 3  |-  ( h  =  H  ->  (
( <. A ,  D >.Cgr
<. E ,  h >.  /\ 
<. B ,  D >.Cgr <. F ,  h >. )  <-> 
( <. A ,  D >.Cgr
<. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) ) )
49483anbi3d 1258 . 2  |-  ( h  =  H  ->  (
( A  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  h >.  /\  <. B ,  D >.Cgr <. F ,  h >. ) )  <->  ( A  Colinear  <. B ,  C >.  /\ 
<. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\ 
<. B ,  D >.Cgr <. F ,  H >. ) ) ) )
50 fveq2 5525 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
51 df-fs 24076 . 2  |-  FiveSeg  =  { <. 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 >. >.  /\  ( a  Colinear  <.
b ,  c >.  /\  <. a ,  <. b ,  c >. >.Cgr3 <. e ,  <. f ,  g
>. >.  /\  ( <. a ,  d >.Cgr <. e ,  h >.  /\  <. b ,  d >.Cgr <. f ,  h >. ) ) ) }
527, 15, 20, 26, 32, 39, 43, 49, 50, 51br8 23524 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  Colinear  <. B ,  C >.  /\  <. A ,  <. B ,  C >. >.Cgr3 <. E ,  <. F ,  G >. >.  /\  ( <. A ,  D >.Cgr <. E ,  H >.  /\  <. B ,  D >.Cgr <. F ,  H >. ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255   NNcn 9746   EEcee 23927  Cgrccgr 23929  Cgr3ccgr3 24070    Colinear ccolin 24071    FiveSeg cfs 24072
This theorem is referenced by:  fscgr  24114  linecgr  24115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-iota 5219  df-fv 5263  df-fs 24076
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