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Theorem brcgr3 29671
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 4202 . . . 4  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21breq1d 4447 . . 3  |-  ( a  =  A  ->  ( <. a ,  b >.Cgr <. d ,  e >.  <->  <. A ,  b >.Cgr <.
d ,  e >.
) )
3 opeq1 4202 . . . 4  |-  ( a  =  A  ->  <. a ,  c >.  =  <. A ,  c >. )
43breq1d 4447 . . 3  |-  ( a  =  A  ->  ( <. a ,  c >.Cgr <. d ,  f >.  <->  <. A ,  c >.Cgr <.
d ,  f >.
) )
52, 43anbi12d 1301 . 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 4203 . . . 4  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76breq1d 4447 . . 3  |-  ( b  =  B  ->  ( <. A ,  b >.Cgr <. d ,  e >.  <->  <. A ,  B >.Cgr <.
d ,  e >.
) )
8 opeq1 4202 . . . 4  |-  ( b  =  B  ->  <. b ,  c >.  =  <. B ,  c >. )
98breq1d 4447 . . 3  |-  ( b  =  B  ->  ( <. b ,  c >.Cgr <. e ,  f >.  <->  <. B ,  c >.Cgr <.
e ,  f >.
) )
107, 93anbi13d 1302 . 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 4203 . . . 4  |-  ( c  =  C  ->  <. A , 
c >.  =  <. A ,  C >. )
1211breq1d 4447 . . 3  |-  ( c  =  C  ->  ( <. A ,  c >.Cgr <. d ,  f >.  <->  <. A ,  C >.Cgr <.
d ,  f >.
) )
13 opeq2 4203 . . . 4  |-  ( c  =  C  ->  <. B , 
c >.  =  <. B ,  C >. )
1413breq1d 4447 . . 3  |-  ( c  =  C  ->  ( <. B ,  c >.Cgr <. e ,  f >.  <->  <. B ,  C >.Cgr <.
e ,  f >.
) )
1512, 143anbi23d 1303 . 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 4202 . . . 4  |-  ( d  =  D  ->  <. d ,  e >.  =  <. D ,  e >. )
1716breq2d 4449 . . 3  |-  ( d  =  D  ->  ( <. A ,  B >.Cgr <.
d ,  e >.  <->  <. A ,  B >.Cgr <. D ,  e >. ) )
18 opeq1 4202 . . . 4  |-  ( d  =  D  ->  <. d ,  f >.  =  <. D ,  f >. )
1918breq2d 4449 . . 3  |-  ( d  =  D  ->  ( <. A ,  C >.Cgr <.
d ,  f >.  <->  <. A ,  C >.Cgr <. D ,  f >. ) )
2017, 193anbi12d 1301 . 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 4203 . . . 4  |-  ( e  =  E  ->  <. D , 
e >.  =  <. D ,  E >. )
2221breq2d 4449 . . 3  |-  ( e  =  E  ->  ( <. A ,  B >.Cgr <. D ,  e >.  <->  <. A ,  B >.Cgr <. D ,  E >. ) )
23 opeq1 4202 . . . 4  |-  ( e  =  E  ->  <. e ,  f >.  =  <. E ,  f >. )
2423breq2d 4449 . . 3  |-  ( e  =  E  ->  ( <. B ,  C >.Cgr <.
e ,  f >.  <->  <. B ,  C >.Cgr <. E ,  f >. ) )
2522, 243anbi13d 1302 . 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 4203 . . . 4  |-  ( f  =  F  ->  <. D , 
f >.  =  <. D ,  F >. )
2726breq2d 4449 . . 3  |-  ( f  =  F  ->  ( <. A ,  C >.Cgr <. D ,  f >.  <->  <. A ,  C >.Cgr <. D ,  F >. ) )
28 opeq2 4203 . . . 4  |-  ( f  =  F  ->  <. E , 
f >.  =  <. E ,  F >. )
2928breq2d 4449 . . 3  |-  ( f  =  F  ->  ( <. B ,  C >.Cgr <. E ,  f >.  <->  <. B ,  C >.Cgr <. E ,  F >. ) )
3027, 293anbi23d 1303 . 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 5856 . 2  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
32 df-cgr3 29666 . 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 29161 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1383    e. wcel 1804   <.cop 4020   class class class wbr 4437   ` cfv 5578   NNcn 10542   EEcee 24063  Cgrccgr 24065  Cgr3ccgr3 29661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-iota 5541  df-fv 5586  df-cgr3 29666
This theorem is referenced by:  cgr3permute3  29672  cgr3permute1  29673  cgr3tr4  29677  cgr3com  29678  cgr3rflx  29679  cgrxfr  29680  btwnxfr  29681  lineext  29701  brofs2  29702  brifs2  29703  endofsegid  29710  btwnconn1lem4  29715  btwnconn1lem8  29719  btwnconn1lem11  29722  brsegle2  29734  seglecgr12im  29735  segletr  29739
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