MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  constr3trllem5 Structured version   Unicode version

Theorem constr3trllem5 24856
Description: Lemma for constr3trl 24861. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
constr3cycl.p  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
Assertion
Ref Expression
constr3trllem5  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Distinct variable groups:    k, E    k, F    P, k
Allowed substitution hints:    A( k)    B( k)    C( k)    V( k)

Proof of Theorem constr3trllem5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 usgraedgrnv 24579 . . . . . . 7  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
21ex 432 . . . . . 6  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( A  e.  V  /\  B  e.  V ) ) )
3 usgraedgrnv 24579 . . . . . . 7  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
43ex 432 . . . . . 6  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( B  e.  V  /\  C  e.  V ) ) )
5 usgraedgrnv 24579 . . . . . . 7  |-  ( ( V USGrph  E  /\  { C ,  A }  e.  ran  E )  ->  ( C  e.  V  /\  A  e.  V ) )
65ex 432 . . . . . 6  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  ->  ( C  e.  V  /\  A  e.  V ) ) )
72, 4, 63anim123d 1304 . . . . 5  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V
) ) ) )
87imp 427 . . . 4  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V
) ) )
9 simpll 751 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
10 simprl 754 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  B  e.  V )
11 simprr 755 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
12 constr3cycl.f . . . . . . . 8  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
13 constr3cycl.p . . . . . . . 8  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
1412, 13constr3lem4 24849 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
159, 10, 11, 14syl3anc 1226 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) ) )
16 usgraf 24548 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
17 f1f1orn 5809 . . . . . . . . . . . . 13  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E -1-1-onto-> ran  E )
18 f1ocnvfv2 6158 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { A ,  B }  e.  ran  E )  ->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
)
1918ex 432 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B } ) )
20 f1ocnvfv2 6158 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
)
2120ex 432 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { B ,  C }  e.  ran  E  ->  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
22 f1ocnvfv2 6158 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { C ,  A }  e.  ran  E )  ->  ( E `  ( `' E `  { C ,  A }
) )  =  { C ,  A }
)
2322ex 432 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { C ,  A }  e.  ran  E  ->  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2419, 21, 233anim123d 1304 . . . . . . . . . . . . 13  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  ->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
2516, 17, 243syl 20 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E )  -> 
( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
2625imp 427 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2726adantl 464 . . . . . . . . . 10  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2812, 13constr3lem5 24850 . . . . . . . . . . 11  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )
29 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( `' E `  { A ,  B } ) ) )
30293ad2ant1 1015 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  0 )
)  =  ( E `
 ( `' E `  { A ,  B } ) ) )
3130eqeq1d 2456 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  0 ) )  =  { A ,  B }  <->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
) )
32 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( ( F `  1 )  =  ( `' E `  { B ,  C } )  ->  ( E `  ( F `  1 ) )  =  ( E `  ( `' E `  { B ,  C } ) ) )
33323ad2ant2 1016 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  1 )
)  =  ( E `
 ( `' E `  { B ,  C } ) ) )
3433eqeq1d 2456 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  1 ) )  =  { B ,  C }  <->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
) )
35 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( ( F `  2 )  =  ( `' E `  { C ,  A } )  ->  ( E `  ( F `  2 ) )  =  ( E `  ( `' E `  { C ,  A } ) ) )
36353ad2ant3 1017 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  2 )
)  =  ( E `
 ( `' E `  { C ,  A } ) ) )
3736eqeq1d 2456 . . . . . . . . . . . 12  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( E `  ( F `  2 ) )  =  { C ,  A }  <->  ( E `  ( `' E `  { C ,  A }
) )  =  { C ,  A }
) )
3831, 34, 373anbi123d 1297 . . . . . . . . . . 11  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } )  <->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) ) )
3928, 38ax-mp 5 . . . . . . . . . 10  |-  ( ( ( E `  ( F `  0 )
)  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } )  <->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C }  /\  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
4027, 39sylibr 212 . . . . . . . . 9  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) )
41 simpll 751 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  0
)  =  A )
42 simplr 753 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  1
)  =  B )
4341, 42preq12d 4103 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { A ,  B } )
4443eqeq2d 2468 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  <->  ( E `  ( F `  0
) )  =  { A ,  B }
) )
45 simprl 754 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  2
)  =  C )
4642, 45preq12d 4103 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
1 ) ,  ( P `  2 ) }  =  { B ,  C } )
4746eqeq2d 2468 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P ` 
2 ) }  <->  ( E `  ( F `  1
) )  =  { B ,  C }
) )
48 simprr 755 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  3
)  =  A )
4945, 48preq12d 4103 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { C ,  A } )
5049eqeq2d 2468 . . . . . . . . . . 11  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( E `  ( F `  2 ) )  =  { ( P `  2 ) ,  ( P ` 
3 ) }  <->  ( E `  ( F `  2
) )  =  { C ,  A }
) )
5144, 47, 503anbi123d 1297 . . . . . . . . . 10  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } )  <->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) ) )
5251ad2antrr 723 . . . . . . . . 9  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } )  <->  ( ( E `  ( F `  0 ) )  =  { A ,  B }  /\  ( E `  ( F `  1 ) )  =  { B ,  C }  /\  ( E `  ( F `  2 ) )  =  { C ,  A } ) ) )
5340, 52mpbird 232 . . . . . . . 8  |-  ( ( ( ( ( ( P `  0 )  =  A  /\  ( P `  1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  /\  ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) )  ->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) )
5453ex 432 . . . . . . 7  |-  ( ( ( ( ( P `
 0 )  =  A  /\  ( P `
 1 )  =  B )  /\  (
( P `  2
)  =  C  /\  ( P `  3 )  =  A ) )  /\  ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
) )  ->  (
( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) )
5554ex 432 . . . . . 6  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( ( ( A  e.  V  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  V )
)  ->  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) ) )
5615, 55mpcom 36 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  -> 
( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) ) )
57563adant3 1014 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V )  /\  ( C  e.  V  /\  A  e.  V )
)  ->  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) ) )
588, 57mpcom 36 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) }  /\  ( E `
 ( F ` 
2 ) )  =  { ( P ` 
2 ) ,  ( P `  3 ) } ) )
59 0z 10871 . . . 4  |-  0  e.  ZZ
60 1z 10890 . . . 4  |-  1  e.  ZZ
61 2z 10892 . . . 4  |-  2  e.  ZZ
62 fveq2 5848 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
6362fveq2d 5852 . . . . . 6  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
64 fveq2 5848 . . . . . . 7  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
65 oveq1 6277 . . . . . . . . 9  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
66 0p1e1 10643 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6765, 66syl6eq 2511 . . . . . . . 8  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
6867fveq2d 5852 . . . . . . 7  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
6964, 68preq12d 4103 . . . . . 6  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7063, 69eqeq12d 2476 . . . . 5  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
71 fveq2 5848 . . . . . . 7  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
7271fveq2d 5852 . . . . . 6  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
73 fveq2 5848 . . . . . . 7  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
74 oveq1 6277 . . . . . . . . 9  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
75 1p1e2 10645 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
7674, 75syl6eq 2511 . . . . . . . 8  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
7776fveq2d 5852 . . . . . . 7  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
7873, 77preq12d 4103 . . . . . 6  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
7972, 78eqeq12d 2476 . . . . 5  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
80 fveq2 5848 . . . . . . 7  |-  ( k  =  2  ->  ( F `  k )  =  ( F ` 
2 ) )
8180fveq2d 5852 . . . . . 6  |-  ( k  =  2  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  2 )
) )
82 fveq2 5848 . . . . . . 7  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
83 oveq1 6277 . . . . . . . . 9  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
84 2p1e3 10655 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
8583, 84syl6eq 2511 . . . . . . . 8  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
8685fveq2d 5852 . . . . . . 7  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
8782, 86preq12d 4103 . . . . . 6  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
8881, 87eqeq12d 2476 . . . . 5  |-  ( k  =  2  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  2
) )  =  {
( P `  2
) ,  ( P `
 3 ) } ) )
8970, 79, 88raltpg 4067 . . . 4  |-  ( ( 0  e.  ZZ  /\  1  e.  ZZ  /\  2  e.  ZZ )  ->  ( A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) ) )
9059, 60, 61, 89mp3an 1322 . . 3  |-  ( A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  /\  ( E `  ( F `  2 ) )  =  { ( P `
 2 ) ,  ( P `  3
) } ) )
9158, 90sylibr 212 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  {
0 ,  1 ,  2 }  ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
9212, 13constr3lem2 24848 . . . . . 6  |-  ( # `  F )  =  3
9392oveq2i 6281 . . . . 5  |-  ( 0..^ ( # `  F
) )  =  ( 0..^ 3 )
94 fzo0to3tp 11881 . . . . 5  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
9593, 94eqtri 2483 . . . 4  |-  ( 0..^ ( # `  F
) )  =  {
0 ,  1 ,  2 }
9695a1i 11 . . 3  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( 0..^ (
# `  F )
)  =  { 0 ,  1 ,  2 } )
9796raleqdv 3057 . 2  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 k ) )  =  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  <->  A. k  e.  { 0 ,  1 ,  2 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } ) )
9891, 97mpbird 232 1  |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) )  ->  A. k  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808    \ cdif 3458    u. cun 3459   (/)c0 3783   ~Pcpw 3999   {csn 4016   {cpr 4018   {ctp 4020   <.cop 4022   class class class wbr 4439   `'ccnv 4987   dom cdm 4988   ran crn 4989   -1-1->wf1 5567   -1-1-onto->wf1o 5569   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484   2c2 10581   3c3 10582   ZZcz 10860  ..^cfzo 11799   #chash 12387   USGrph cusg 24532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-usgra 24535
This theorem is referenced by:  constr3trl  24861
  Copyright terms: Public domain W3C validator