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

Theorem brcgr3 26015
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 4013 . . . 4  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21breq1d 4253 . . 3  |-  ( a  =  A  ->  ( <. a ,  b >.Cgr <. d ,  e >.  <->  <. A ,  b >.Cgr <.
d ,  e >.
) )
3 opeq1 4013 . . . 4  |-  ( a  =  A  ->  <. a ,  c >.  =  <. A ,  c >. )
43breq1d 4253 . . 3  |-  ( a  =  A  ->  ( <. a ,  c >.Cgr <. d ,  f >.  <->  <. A ,  c >.Cgr <.
d ,  f >.
) )
52, 43anbi12d 1256 . 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 4014 . . . 4  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76breq1d 4253 . . 3  |-  ( b  =  B  ->  ( <. A ,  b >.Cgr <. d ,  e >.  <->  <. A ,  B >.Cgr <.
d ,  e >.
) )
8 opeq1 4013 . . . 4  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
98breq1d 4253 . . 3  |-  ( b  =  B  ->  ( <. b ,  c >.Cgr <. e ,  f >.  <->  <. B ,  c >.Cgr <.
e ,  f >.
) )
107, 93anbi13d 1257 . 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 4014 . . . 4  |-  ( c  =  C  ->  <. A , 
c >.  =  <. A ,  C >. )
1211breq1d 4253 . . 3  |-  ( c  =  C  ->  ( <. A ,  c >.Cgr <. d ,  f >.  <->  <. A ,  C >.Cgr <.
d ,  f >.
) )
13 opeq2 4014 . . . 4  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
1413breq1d 4253 . . 3  |-  ( c  =  C  ->  ( <. B ,  c >.Cgr <. e ,  f >.  <->  <. B ,  C >.Cgr <.
e ,  f >.
) )
1512, 143anbi23d 1258 . 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 4013 . . . 4  |-  ( d  =  D  ->  <. d ,  e >.  =  <. D ,  e >. )
1716breq2d 4255 . . 3  |-  ( d  =  D  ->  ( <. A ,  B >.Cgr <.
d ,  e >.  <->  <. A ,  B >.Cgr <. D ,  e >. ) )
18 opeq1 4013 . . . 4  |-  ( d  =  D  ->  <. d ,  f >.  =  <. D ,  f >. )
1918breq2d 4255 . . 3  |-  ( d  =  D  ->  ( <. A ,  C >.Cgr <.
d ,  f >.  <->  <. A ,  C >.Cgr <. D ,  f >. ) )
2017, 193anbi12d 1256 . 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 4014 . . . 4  |-  ( e  =  E  ->  <. D , 
e >.  =  <. D ,  E >. )
2221breq2d 4255 . . 3  |-  ( e  =  E  ->  ( <. A ,  B >.Cgr <. D ,  e >.  <->  <. A ,  B >.Cgr <. D ,  E >. ) )
23 opeq1 4013 . . . 4  |-  ( e  =  E  ->  <. e ,  f >.  =  <. E ,  f >. )
2423breq2d 4255 . . 3  |-  ( e  =  E  ->  ( <. B ,  C >.Cgr <.
e ,  f >.  <->  <. B ,  C >.Cgr <. E ,  f >. ) )
2522, 243anbi13d 1257 . 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 4014 . . . 4  |-  ( f  =  F  ->  <. D , 
f >.  =  <. D ,  F >. )
2726breq2d 4255 . . 3  |-  ( f  =  F  ->  ( <. A ,  C >.Cgr <. D ,  f >.  <->  <. A ,  C >.Cgr <. D ,  F >. ) )
28 opeq2 4014 . . . 4  |-  ( f  =  F  ->  <. E , 
f >.  =  <. E ,  F >. )
2928breq2d 4255 . . 3  |-  ( f  =  F  ->  ( <. B ,  C >.Cgr <. E ,  f >.  <->  <. B ,  C >.Cgr <. E ,  F >. ) )
3027, 293anbi23d 1258 . 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 5763 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32 df-cgr3 26009 . 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 25415 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 178    /\ w3a 937    = wceq 1654    e. wcel 1728   <.cop 3846   class class class wbr 4243   ` cfv 5489   NNcn 10038   EEcee 25862  Cgrccgr 25864  Cgr3ccgr3 26005
This theorem is referenced by:  cgr3permute3  26016  cgr3permute1  26017  cgr3tr4  26021  cgr3com  26022  cgr3rflx  26023  cgrxfr  26024  btwnxfr  26025  lineext  26045  brofs2  26046  brifs2  26047  endofsegid  26054  btwnconn1lem4  26059  btwnconn1lem8  26063  btwnconn1lem11  26066  brsegle2  26078  seglecgr12im  26079  segletr  26083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
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 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-iota 5453  df-fv 5497  df-cgr3 26009
  Copyright terms: Public domain W3C validator