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Theorem constr3trllem5 23545
Description: Lemma for constr3trl 23550. (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 23301 . . . . . . 7  |-  ( ( V USGrph  E  /\  { A ,  B }  e.  ran  E )  ->  ( A  e.  V  /\  B  e.  V ) )
21ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { A ,  B }  e.  ran  E  ->  ( A  e.  V  /\  B  e.  V ) ) )
3 usgraedgrnv 23301 . . . . . . 7  |-  ( ( V USGrph  E  /\  { B ,  C }  e.  ran  E )  ->  ( B  e.  V  /\  C  e.  V ) )
43ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { B ,  C }  e.  ran  E  ->  ( B  e.  V  /\  C  e.  V ) ) )
5 usgraedgrnv 23301 . . . . . . 7  |-  ( ( V USGrph  E  /\  { C ,  A }  e.  ran  E )  ->  ( C  e.  V  /\  A  e.  V ) )
65ex 434 . . . . . 6  |-  ( V USGrph  E  ->  ( { C ,  A }  e.  ran  E  ->  ( C  e.  V  /\  A  e.  V ) ) )
72, 4, 63anim123d 1296 . . . . 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 429 . . . 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 753 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  A  e.  V )
10 simprl 755 . . . . . . 7  |-  ( ( ( A  e.  V  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  V ) )  ->  B  e.  V )
11 simprr 756 . . . . . . 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 23538 . . . . . . 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 1218 . . . . . 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 23279 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  E : dom  E
-1-1-> { x  e.  ( ~P V  \  { (/)
} )  |  (
# `  x )  =  2 } )
17 f1f1orn 5657 . . . . . . . . . . . . 13  |-  ( E : dom  E -1-1-> {
x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }  ->  E : dom  E -1-1-onto-> ran  E )
18 f1ocnvfv2 5989 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { A ,  B }  e.  ran  E )  ->  ( E `  ( `' E `  { A ,  B }
) )  =  { A ,  B }
)
1918ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B } ) )
20 f1ocnvfv2 5989 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { B ,  C }  e.  ran  E )  ->  ( E `  ( `' E `  { B ,  C }
) )  =  { B ,  C }
)
2120ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { B ,  C }  e.  ran  E  ->  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
22 f1ocnvfv2 5989 . . . . . . . . . . . . . . 15  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { C ,  A }  e.  ran  E )  ->  ( E `  ( `' E `  { C ,  A }
) )  =  { C ,  A }
)
2322ex 434 . . . . . . . . . . . . . 14  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { C ,  A }  e.  ran  E  ->  ( E `  ( `' E `  { C ,  A } ) )  =  { C ,  A } ) )
2419, 21, 233anim123d 1296 . . . . . . . . . . . . 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 429 . . . . . . . . . . 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 466 . . . . . . . . . 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 23539 . . . . . . . . . . 11  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )
29 fveq2 5696 . . . . . . . . . . . . . 14  |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  ->  ( E `  ( F `  0 ) )  =  ( E `  ( `' E `  { A ,  B } ) ) )
30293ad2ant1 1009 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  0 )
)  =  ( E `
 ( `' E `  { A ,  B } ) ) )
3130eqeq1d 2451 . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . 14  |-  ( ( F `  1 )  =  ( `' E `  { B ,  C } )  ->  ( E `  ( F `  1 ) )  =  ( E `  ( `' E `  { B ,  C } ) ) )
33323ad2ant2 1010 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  1 )
)  =  ( E `
 ( `' E `  { B ,  C } ) ) )
3433eqeq1d 2451 . . . . . . . . . . . 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 5696 . . . . . . . . . . . . . 14  |-  ( ( F `  2 )  =  ( `' E `  { C ,  A } )  ->  ( E `  ( F `  2 ) )  =  ( E `  ( `' E `  { C ,  A } ) ) )
36353ad2ant3 1011 . . . . . . . . . . . . 13  |-  ( ( ( F `  0
)  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E `  { B ,  C } )  /\  ( F `  2 )  =  ( `' E `  { C ,  A } ) )  -> 
( E `  ( F `  2 )
)  =  ( E `
 ( `' E `  { C ,  A } ) ) )
3736eqeq1d 2451 . . . . . . . . . . . 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 1289 . . . . . . . . . . 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 753 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  0
)  =  A )
42 simplr 754 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  1
)  =  B )
4341, 42preq12d 3967 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { A ,  B } )
4443eqeq2d 2454 . . . . . . . . . . 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 755 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  2
)  =  C )
4642, 45preq12d 3967 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
1 ) ,  ( P `  2 ) }  =  { B ,  C } )
4746eqeq2d 2454 . . . . . . . . . . 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 756 . . . . . . . . . . . . 13  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  -> 
( P `  3
)  =  A )
4945, 48preq12d 3967 . . . . . . . . . . . 12  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
1 )  =  B )  /\  ( ( P `  2 )  =  C  /\  ( P `  3 )  =  A ) )  ->  { ( P ` 
2 ) ,  ( P `  3 ) }  =  { C ,  A } )
5049eqeq2d 2454 . . . . . . . . . . 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 1289 . . . . . . . . . 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 725 . . . . . . . . 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 434 . . . . . . 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 434 . . . . . 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 1008 . . . 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 10662 . . . 4  |-  0  e.  ZZ
60 1z 10681 . . . 4  |-  1  e.  ZZ
61 2z 10683 . . . 4  |-  2  e.  ZZ
62 fveq2 5696 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
6362fveq2d 5700 . . . . . 6  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
64 fveq2 5696 . . . . . . 7  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
65 oveq1 6103 . . . . . . . . 9  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
66 0p1e1 10438 . . . . . . . . 9  |-  ( 0  +  1 )  =  1
6765, 66syl6eq 2491 . . . . . . . 8  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
6867fveq2d 5700 . . . . . . 7  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
6964, 68preq12d 3967 . . . . . 6  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
7063, 69eqeq12d 2457 . . . . 5  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
71 fveq2 5696 . . . . . . 7  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
7271fveq2d 5700 . . . . . 6  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
73 fveq2 5696 . . . . . . 7  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
74 oveq1 6103 . . . . . . . . 9  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
75 1p1e2 10440 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
7674, 75syl6eq 2491 . . . . . . . 8  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
7776fveq2d 5700 . . . . . . 7  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
7873, 77preq12d 3967 . . . . . 6  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
7972, 78eqeq12d 2457 . . . . 5  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
80 fveq2 5696 . . . . . . 7  |-  ( k  =  2  ->  ( F `  k )  =  ( F ` 
2 ) )
8180fveq2d 5700 . . . . . 6  |-  ( k  =  2  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  2 )
) )
82 fveq2 5696 . . . . . . 7  |-  ( k  =  2  ->  ( P `  k )  =  ( P ` 
2 ) )
83 oveq1 6103 . . . . . . . . 9  |-  ( k  =  2  ->  (
k  +  1 )  =  ( 2  +  1 ) )
84 2p1e3 10450 . . . . . . . . 9  |-  ( 2  +  1 )  =  3
8583, 84syl6eq 2491 . . . . . . . 8  |-  ( k  =  2  ->  (
k  +  1 )  =  3 )
8685fveq2d 5700 . . . . . . 7  |-  ( k  =  2  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
3 ) )
8782, 86preq12d 3967 . . . . . 6  |-  ( k  =  2  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
2 ) ,  ( P `  3 ) } )
8881, 87eqeq12d 2457 . . . . 5  |-  ( k  =  2  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  2
) )  =  {
( P `  2
) ,  ( P `
 3 ) } ) )
8970, 79, 88raltpg 3932 . . . 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 1314 . . 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 23537 . . . . . 6  |-  ( # `  F )  =  3
9392oveq2i 6107 . . . . 5  |-  ( 0..^ ( # `  F
) )  =  ( 0..^ 3 )
94 fzo0to3tp 11620 . . . . 5  |-  ( 0..^ 3 )  =  {
0 ,  1 ,  2 }
9593, 94eqtri 2463 . . . 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 2928 . 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2720   {crab 2724    \ cdif 3330    u. cun 3331   (/)c0 3642   ~Pcpw 3865   {csn 3882   {cpr 3884   {ctp 3886   <.cop 3888   class class class wbr 4297   `'ccnv 4844   dom cdm 4845   ran crn 4846   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096   0cc0 9287   1c1 9288    + caddc 9290   2c2 10376   3c3 10377   ZZcz 10651  ..^cfzo 11553   #chash 12108   USGrph cusg 23269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-hash 12109  df-usgra 23271
This theorem is referenced by:  constr3trl  23550
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