Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brcgr3 Unicode version

Theorem brcgr3 25228
Description: Binary relationship form of the three-place congruence predicate. (Contributed by Scott Fenton, 4-Oct-2013.)
Assertion
Ref Expression
brcgr3  |-  ( ( 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 ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )

Proof of Theorem brcgr3
Dummy variables  a 
b  c  d  e  f  n  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3875 . . . 4  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21breq1d 4112 . . 3  |-  ( a  =  A  ->  ( <. a ,  b >.Cgr <. d ,  e >.  <->  <. A ,  b >.Cgr <.
d ,  e >.
) )
3 opeq1 3875 . . . 4  |-  ( a  =  A  ->  <. a ,  c >.  =  <. A ,  c >. )
43breq1d 4112 . . 3  |-  ( a  =  A  ->  ( <. a ,  c >.Cgr <. d ,  f >.  <->  <. A ,  c >.Cgr <.
d ,  f >.
) )
52, 43anbi12d 1253 . 2  |-  ( a  =  A  ->  (
( <. a ,  b
>.Cgr <. d ,  e
>.  /\  <. a ,  c
>.Cgr <. d ,  f
>.  /\  <. b ,  c
>.Cgr <. e ,  f
>. )  <->  ( <. A , 
b >.Cgr <. d ,  e
>.  /\  <. A ,  c
>.Cgr <. d ,  f
>.  /\  <. b ,  c
>.Cgr <. e ,  f
>. ) ) )
6 opeq2 3876 . . . 4  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76breq1d 4112 . . 3  |-  ( b  =  B  ->  ( <. A ,  b >.Cgr <. d ,  e >.  <->  <. A ,  B >.Cgr <.
d ,  e >.
) )
8 opeq1 3875 . . . 4  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
98breq1d 4112 . . 3  |-  ( b  =  B  ->  ( <. b ,  c >.Cgr <. e ,  f >.  <->  <. B ,  c >.Cgr <.
e ,  f >.
) )
107, 93anbi13d 1254 . 2  |-  ( b  =  B  ->  (
( <. A ,  b
>.Cgr <. d ,  e
>.  /\  <. A ,  c
>.Cgr <. d ,  f
>.  /\  <. b ,  c
>.Cgr <. e ,  f
>. )  <->  ( <. A ,  B >.Cgr <. d ,  e
>.  /\  <. A ,  c
>.Cgr <. d ,  f
>.  /\  <. B ,  c
>.Cgr <. e ,  f
>. ) ) )
11 opeq2 3876 . . . 4  |-  ( c  =  C  ->  <. A , 
c >.  =  <. A ,  C >. )
1211breq1d 4112 . . 3  |-  ( c  =  C  ->  ( <. A ,  c >.Cgr <. d ,  f >.  <->  <. A ,  C >.Cgr <.
d ,  f >.
) )
13 opeq2 3876 . . . 4  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
1413breq1d 4112 . . 3  |-  ( c  =  C  ->  ( <. B ,  c >.Cgr <. e ,  f >.  <->  <. B ,  C >.Cgr <.
e ,  f >.
) )
1512, 143anbi23d 1255 . 2  |-  ( c  =  C  ->  (
( <. A ,  B >.Cgr
<. d ,  e >.  /\  <. A ,  c
>.Cgr <. d ,  f
>.  /\  <. B ,  c
>.Cgr <. e ,  f
>. )  <->  ( <. A ,  B >.Cgr <. d ,  e
>.  /\  <. A ,  C >.Cgr
<. d ,  f >.  /\  <. B ,  C >.Cgr
<. e ,  f >.
) ) )
16 opeq1 3875 . . . 4  |-  ( d  =  D  ->  <. d ,  e >.  =  <. D ,  e >. )
1716breq2d 4114 . . 3  |-  ( d  =  D  ->  ( <. A ,  B >.Cgr <.
d ,  e >.  <->  <. A ,  B >.Cgr <. D ,  e >. ) )
18 opeq1 3875 . . . 4  |-  ( d  =  D  ->  <. d ,  f >.  =  <. D ,  f >. )
1918breq2d 4114 . . 3  |-  ( d  =  D  ->  ( <. A ,  C >.Cgr <.
d ,  f >.  <->  <. A ,  C >.Cgr <. D ,  f >. ) )
2017, 193anbi12d 1253 . 2  |-  ( d  =  D  ->  (
( <. A ,  B >.Cgr
<. d ,  e >.  /\  <. A ,  C >.Cgr
<. d ,  f >.  /\  <. B ,  C >.Cgr
<. e ,  f >.
)  <->  ( <. A ,  B >.Cgr <. D ,  e
>.  /\  <. A ,  C >.Cgr
<. D ,  f >.  /\  <. B ,  C >.Cgr
<. e ,  f >.
) ) )
21 opeq2 3876 . . . 4  |-  ( e  =  E  ->  <. D , 
e >.  =  <. D ,  E >. )
2221breq2d 4114 . . 3  |-  ( e  =  E  ->  ( <. A ,  B >.Cgr <. D ,  e >.  <->  <. A ,  B >.Cgr <. D ,  E >. ) )
23 opeq1 3875 . . . 4  |-  ( e  =  E  ->  <. e ,  f >.  =  <. E ,  f >. )
2423breq2d 4114 . . 3  |-  ( e  =  E  ->  ( <. B ,  C >.Cgr <.
e ,  f >.  <->  <. B ,  C >.Cgr <. E ,  f >. ) )
2522, 243anbi13d 1254 . 2  |-  ( e  =  E  ->  (
( <. A ,  B >.Cgr
<. D ,  e >.  /\  <. A ,  C >.Cgr
<. D ,  f >.  /\  <. B ,  C >.Cgr
<. e ,  f >.
)  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr
<. D ,  f >.  /\  <. B ,  C >.Cgr
<. E ,  f >.
) ) )
26 opeq2 3876 . . . 4  |-  ( f  =  F  ->  <. D , 
f >.  =  <. D ,  F >. )
2726breq2d 4114 . . 3  |-  ( f  =  F  ->  ( <. A ,  C >.Cgr <. D ,  f >.  <->  <. A ,  C >.Cgr <. D ,  F >. ) )
28 opeq2 3876 . . . 4  |-  ( f  =  F  ->  <. E , 
f >.  =  <. E ,  F >. )
2928breq2d 4114 . . 3  |-  ( f  =  F  ->  ( <. B ,  C >.Cgr <. E ,  f >.  <->  <. B ,  C >.Cgr <. E ,  F >. ) )
3027, 293anbi23d 1255 . 2  |-  ( f  =  F  ->  (
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  f >.  /\ 
<. B ,  C >.Cgr <. E ,  f >. )  <-> 
( <. A ,  B >.Cgr
<. D ,  E >.  /\ 
<. A ,  C >.Cgr <. D ,  F >.  /\ 
<. B ,  C >.Cgr <. E ,  F >. ) ) )
31 fveq2 5605 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32 df-cgr3 25222 . 2  |- Cgr3  =  { <. 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 ) ( p  =  <. a ,  <. b ,  c
>. >.  /\  q  =  <. d ,  <. e ,  f >. >.  /\  ( <. a ,  b >.Cgr <. d ,  e >.  /\  <. a ,  c
>.Cgr <. d ,  f
>.  /\  <. b ,  c
>.Cgr <. e ,  f
>. ) ) }
335, 10, 15, 20, 25, 30, 31, 32br6 24672 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 ) ) )  ->  ( <. A ,  <. B ,  C >. >.Cgr3 <. D ,  <. E ,  F >. >.  <->  ( <. A ,  B >.Cgr <. D ,  E >.  /\  <. A ,  C >.Cgr <. D ,  F >.  /\  <. B ,  C >.Cgr
<. E ,  F >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1642    e. wcel 1710   <.cop 3719   class class class wbr 4102   ` cfv 5334   NNcn 9833   EEcee 25075  Cgrccgr 25077  Cgr3ccgr3 25218
This theorem is referenced by:  cgr3permute3  25229  cgr3permute1  25230  cgr3tr4  25234  cgr3com  25235  cgr3rflx  25236  cgrxfr  25237  btwnxfr  25238  lineext  25258  brofs2  25259  brifs2  25260  endofsegid  25267  btwnconn1lem4  25272  btwnconn1lem8  25276  btwnconn1lem11  25279  brsegle2  25291  seglecgr12im  25292  segletr  25296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-iota 5298  df-fv 5342  df-cgr3 25222
  Copyright terms: Public domain W3C validator