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Theorem numclwlk1lem2fo 30693
Description: T is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
Hypotheses
Ref Expression
numclwwlk.c  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
numclwwlk.f  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
numclwwlk.g  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
numclwwlk.t  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
Assertion
Ref Expression
numclwlk1lem2fo  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -onto-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Distinct variable groups:    n, E    n, N    n, V    w, C    w, N    C, n, v, w    v, N    n, X, v, w    v, V   
w, E    w, V    w, F    w, G
Allowed substitution hints:    T( w, v, n)    E( v)    F( v, n)    G( v, n)

Proof of Theorem numclwlk1lem2fo
Dummy variables  i  x  p  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 numclwwlk.c . . 3  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . 3  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . 3  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
4 numclwwlk.t . . 3  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
51, 2, 3, 4numclwlk1lem2f 30690 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
6 elxp 4862 . . . . 5  |-  ( p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  ( X F ( N  -  2 ) )  /\  b  e.  (
<. V ,  E >. Neighbors  X
) ) ) )
71, 2, 3numclwlk1lem2foa 30689 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( (
a concat  <" X "> ) concat  <" b "> )  e.  ( X G N ) ) )
87com12 31 . . . . . . . . . 10  |-  ( ( a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
a concat  <" X "> ) concat  <" b "> )  e.  ( X G N ) ) )
98adantl 466 . . . . . . . . 9  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N ) ) )
109imp 429 . . . . . . . 8  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N ) )
11 simprll 761 . . . . . . . . . 10  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  p  =  <. a ,  b >. )
12 nbgraisvtx 23347 . . . . . . . . . . . . . . . . 17  |-  ( V USGrph  E  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  b  e.  V ) )
13123ad2ant1 1009 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  b  e.  V ) )
14 simp1 988 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  V USGrph  E )
15 uzuzle23 30198 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  ( ZZ>= `  2 )
)
16 uznn0sub 10897 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  ( ZZ>= `  2
)  ->  ( N  -  2 )  e. 
NN0 )
1715, 16syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( N  -  2 )  e. 
NN0 )
18173ad2ant3 1011 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( N  -  2 )  e. 
NN0 )
19 simp2 989 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  X  e.  V )
201, 2numclwwlkovfel2 30681 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( V USGrph  E  /\  ( N  -  2 )  e.  NN0  /\  X  e.  V )  ->  (
a  e.  ( X F ( N  - 
2 ) )  <->  ( (
a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X ) ) )
2114, 18, 19, 20syl3anc 1218 . . . . . . . . . . . . . . . . . . 19  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  <->  ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X ) ) )
22 df-3an 967 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a ) ,  ( a `  0 ) }  e.  ran  E
)  /\  ( # `  a
)  =  ( N  -  2 )  /\  ( a `  0
)  =  X )  <-> 
( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X ) )
2321, 22syl6bb 261 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  <->  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X ) ) )
24 simplll 757 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  a  e. Word  V )
25 s1cl 12298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( X  e.  V  ->  <" X ">  e. Word  V )
2625adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  <" X ">  e. Word  V )
2726adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  <" X ">  e. Word  V )
2827adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <" X ">  e. Word  V )
29 s1cl 12298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( b  e.  V  ->  <" b ">  e. Word  V )
3029adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <" b ">  e. Word  V )
31 ccatass 12291 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( ( a concat  <" X "> ) concat  <" b "> )  =  ( a concat  (
<" X "> concat  <" b "> ) ) )
3231oveq1d 6111 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. )  =  ( ( a concat  ( <" X "> concat  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >. ) )
3324, 28, 30, 32syl3anc 1218 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. )  =  (
( a concat  ( <" X "> concat  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >.
) )
34 ccatcl 12279 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
<" X ">  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( <" X "> concat  <" b "> )  e. Word  V
)
3527, 29, 34syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( <" X "> concat  <" b "> )  e. Word  V )
36 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( # `  a )  =  ( N  - 
2 ) )
3736eqcomd 2448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( N  -  2 )  =  ( # `  a ) )
3837adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  ( N  -  2 )  =  ( # `  a
) )
3938adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( N  -  2 )  =  ( # `  a
) )
40 swrdccatid 12393 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a  e. Word  V  /\  ( <" X "> concat 
<" b "> )  e. Word  V  /\  ( N  -  2 )  =  ( # `  a
) )  ->  (
( a concat  ( <" X "> concat  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >.
)  =  a )
4124, 35, 39, 40syl3anc 1218 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( a concat  ( <" X "> concat  <" b "> ) ) substr  <. 0 ,  ( N  -  2 ) >.
)  =  a )
4233, 41eqtr2d 2476 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  a  =  ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) )
43 ovex 6121 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( a concat  <" X "> ) concat  <" b "> )  e.  _V
44 lsw 12271 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  _V  ->  ( lastS  `  ( (
a concat  <" X "> ) concat  <" b "> ) )  =  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  (
( # `  ( ( a concat  <" X "> ) concat  <" b "> ) )  - 
1 ) ) )
4543, 44ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( lastS  `  (
( a concat  <" X "> ) concat  <" b "> ) )  =  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  (
( # `  ( ( a concat  <" X "> ) concat  <" b "> ) )  - 
1 ) )
46 simpl 457 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
a  e. Word  V )
47 ccatcl 12279 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V )  ->  (
a concat  <" X "> )  e. Word  V )
4846, 26, 47syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
a concat  <" X "> )  e. Word  V )
49 lswccats1 12317 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( a concat  <" X "> )  e. Word  V  /\  b  e.  V
)  ->  ( lastS  `  (
( a concat  <" X "> ) concat  <" b "> ) )  =  b )
5048, 49sylan 471 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( lastS  `  ( ( a concat  <" X "> ) concat  <" b "> ) )  =  b )
51 ccatlen 12280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( a concat  <" X "> )  e. Word  V  /\  <" b ">  e. Word  V )  ->  ( # `  (
( a concat  <" X "> ) concat  <" b "> ) )  =  ( ( # `  (
a concat  <" X "> ) )  +  (
# `  <" b "> ) ) )
5248, 29, 51syl2an 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( ( a concat  <" X "> ) concat  <" b "> ) )  =  ( ( # `  (
a concat  <" X "> ) )  +  (
# `  <" b "> ) ) )
5346, 26anim12i 566 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
a  e. Word  V  /\  <" X ">  e. Word  V ) )
5453adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
a  e. Word  V  /\  <" X ">  e. Word  V ) )
55 ccatlen 12280 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( a  e. Word  V  /\  <" X ">  e. Word  V )  ->  ( # `
 ( a concat  <" X "> )
)  =  ( (
# `  a )  +  ( # `  <" X "> )
) )
5654, 55syl 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( a concat  <" X "> )
)  =  ( (
# `  a )  +  ( # `  <" X "> )
) )
57 s1len 12301 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( # `  <" b "> )  =  1
5857a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 <" b "> )  =  1 )
5956, 58oveq12d 6114 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( # `  ( a concat  <" X "> ) )  +  (
# `  <" b "> ) )  =  ( ( ( # `  a )  +  (
# `  <" X "> ) )  +  1 ) )
60 s1len 12301 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( # `  <" X "> )  =  1
6160a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( # `  <" X "> )  =  1 )
6236, 61oveqan12d 6115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( # `  a )  +  ( # `  <" X "> )
)  =  ( ( N  -  2 )  +  1 ) )
6362oveq1d 6111 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  ( ( ( N  -  2 )  +  1 )  +  1 ) )
64 eluzelcn 10877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  ( ZZ>= `  3
)  ->  N  e.  CC )
65 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( N  e.  CC  ->  N  e.  CC )
66 2cnd 10399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( N  e.  CC  ->  2  e.  CC )
6765, 66subcld 9724 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  ( N  -  2 )  e.  CC )
68 ax-1cn 9345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  1  e.  CC
6968a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  1  e.  CC )
7067, 69, 69addassd 9413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( N  e.  CC  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  ( ( N  -  2 )  +  ( 1  +  1 ) ) )
71 1p1e2 10440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37  |-  ( 1  +  1 )  =  2
7271a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  |-  ( N  e.  CC  ->  (
1  +  1 )  =  2 )
7372oveq2d 6112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( N  e.  CC  ->  (
( N  -  2 )  +  ( 1  +  1 ) )  =  ( ( N  -  2 )  +  2 ) )
7470, 73eqtrd 2475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  CC  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  ( ( N  -  2 )  +  2 ) )
7564, 74syl 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( (
( N  -  2 )  +  1 )  +  1 )  =  ( ( N  - 
2 )  +  2 ) )
76 2cnd 10399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( N  e.  ( ZZ>= `  3
)  ->  2  e.  CC )
7764, 76npcand 9728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( ( N  -  2 )  +  2 )  =  N )
7875, 77eqtrd 2475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( N  e.  ( ZZ>= `  3
)  ->  ( (
( N  -  2 )  +  1 )  +  1 )  =  N )
7978adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( ( N  -  2 )  +  1 )  +  1 )  =  N )
8079adantl 466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( N  - 
2 )  +  1 )  +  1 )  =  N )
8163, 80eqtrd 2475 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  N )
8281adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( # `  a
)  +  ( # `  <" X "> ) )  +  1 )  =  N )
8352, 59, 823eqtrd 2479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  ( # `
 ( ( a concat  <" X "> ) concat  <" b "> ) )  =  N )
8483oveq1d 6111 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( # `  ( ( a concat  <" X "> ) concat  <" b "> ) )  - 
1 )  =  ( N  -  1 ) )
8584fveq2d 5700 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  (
( ( a concat  <" X "> ) concat  <" b "> ) `  ( ( # `
 ( ( a concat  <" X "> ) concat  <" b "> ) )  - 
1 ) )  =  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) )
8645, 50, 853eqtr3a 2499 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  b  =  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) )
8742, 86opeq12d 4072 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( a  e. Word  V  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  /\  b  e.  V )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. )
8887exp31 604 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( a  e. Word  V  /\  ( # `  a )  =  ( N  - 
2 ) )  -> 
( ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
89883ad2antl1 1150 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a ) ,  ( a `  0 ) }  e.  ran  E
)  /\  ( # `  a
)  =  ( N  -  2 ) )  ->  ( ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9089adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( b  e.  V  -> 
<. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. ) ) )
9190com12 31 . . . . . . . . . . . . . . . . . . 19  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  -> 
( ( ( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  a
)  -  1 ) ) { ( a `
 i ) ,  ( a `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
92913adant1 1006 . . . . . . . . . . . . . . . . . 18  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
( ( a  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  a )  -  1 ) ) { ( a `  i ) ,  ( a `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  a
) ,  ( a `
 0 ) }  e.  ran  E )  /\  ( # `  a
)  =  ( N  -  2 ) )  /\  ( a ` 
0 )  =  X )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9323, 92sylbid 215 . . . . . . . . . . . . . . . . 17  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( a  e.  ( X F ( N  -  2 ) )  ->  ( b  e.  V  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9493com23 78 . . . . . . . . . . . . . . . 16  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  V  ->  ( a  e.  ( X F ( N  -  2 ) )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9513, 94syld 44 . . . . . . . . . . . . . . 15  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( b  e.  ( <. V ,  E >. Neighbors  X )  ->  (
a  e.  ( X F ( N  - 
2 ) )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. ) ) )
9695com13 80 . . . . . . . . . . . . . 14  |-  ( a  e.  ( X F ( N  -  2 ) )  ->  (
b  e.  ( <. V ,  E >. Neighbors  X
)  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) ) )
9796imp 429 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) )  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) )
9897adantl 466 . . . . . . . . . . . 12  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. ) )
9998imp 429 . . . . . . . . . . 11  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. )
10099adantl 466 . . . . . . . . . 10  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  <. a ,  b >.  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. )
10111, 100eqtrd 2475 . . . . . . . . 9  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  p  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. )
102 simpl 457 . . . . . . . . . 10  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N ) )
103 fveq2 5696 . . . . . . . . . . . 12  |-  ( x  =  ( ( a concat  <" X "> ) concat  <" b "> )  ->  ( T `  x )  =  ( T `  ( ( a concat  <" X "> ) concat  <" b "> ) ) )
104103eqeq2d 2454 . . . . . . . . . . 11  |-  ( x  =  ( ( a concat  <" X "> ) concat  <" b "> )  ->  (
p  =  ( T `
 x )  <->  p  =  ( T `  ( ( a concat  <" X "> ) concat  <" b "> ) ) ) )
105 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) ) )
1061, 2, 3, 4numclwlk1lem2fv 30691 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  ->  ( T `  ( ( a concat  <" X "> ) concat  <" b "> ) )  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) )
107105, 102, 106sylc 60 . . . . . . . . . . . 12  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( T `  (
( a concat  <" X "> ) concat  <" b "> ) )  = 
<. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. )
108107eqeq2d 2454 . . . . . . . . . . 11  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( p  =  ( T `  ( ( a concat  <" X "> ) concat  <" b "> ) )  <->  p  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) )
>. ) )
109104, 108sylan9bbr 700 . . . . . . . . . 10  |-  ( ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  ( ( p  =  <. a ,  b
>.  /\  ( a  e.  ( X F ( N  -  2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) ) ) )  /\  x  =  ( ( a concat  <" X "> ) concat  <" b "> ) )  -> 
( p  =  ( T `  x )  <-> 
p  =  <. (
( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >. ) )
110102, 109rspcedv 3082 . . . . . . . . 9  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  -> 
( p  =  <. ( ( ( a concat  <" X "> ) concat  <" b "> ) substr  <. 0 ,  ( N  -  2 )
>. ) ,  ( ( ( a concat  <" X "> ) concat  <" b "> ) `  ( N  -  1 ) ) >.  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
111101, 110mpd 15 . . . . . . . 8  |-  ( ( ( ( a concat  <" X "> ) concat  <" b "> )  e.  ( X G N )  /\  (
( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) )
11210, 111mpancom 669 . . . . . . 7  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  /\  ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) )
113112ex 434 . . . . . 6  |-  ( ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
114113exlimivv 1689 . . . . 5  |-  ( E. a E. b ( p  =  <. a ,  b >.  /\  (
a  e.  ( X F ( N  - 
2 ) )  /\  b  e.  ( <. V ,  E >. Neighbors  X ) ) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
1156, 114sylbi 195 . . . 4  |-  ( p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  -> 
( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
`  3 ) )  ->  E. x  e.  ( X G N ) p  =  ( T `
 x ) ) )
116115impcom 430 . . 3  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  p  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )  ->  E. x  e.  ( X G N ) p  =  ( T `
 x ) )
117116ralrimiva 2804 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  A. p  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) E. x  e.  ( X G N ) p  =  ( T `  x ) )
118 dffo3 5863 . 2  |-  ( T : ( X G N ) -onto-> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  ( T : ( X G N ) --> ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  /\  A. p  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) E. x  e.  ( X G N ) p  =  ( T `  x ) ) )
1195, 117, 118sylanbrc 664 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) -onto-> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724   _Vcvv 2977   {cpr 3884   <.cop 3888   class class class wbr 4297    e. cmpt 4355    X. cxp 4843   ran crn 4846   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   CCcc 9285   0cc0 9287   1c1 9288    + caddc 9290    - cmin 9600   2c2 10376   3c3 10377   NN0cn0 10584   ZZ>=cuz 10866  ..^cfzo 11553   #chash 12108  Word cword 12226   lastS clsw 12227   concat cconcat 12228   <"cs1 12229   substr csubstr 12230   USGrph cusg 23269   Neighbors cnbgra 23334   ClWWalksN cclwwlkn 30419
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-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-map 7221  df-pm 7222  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  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-word 12234  df-lsw 12235  df-concat 12236  df-s1 12237  df-substr 12238  df-s2 12480  df-usgra 23271  df-nbgra 23337  df-clwwlk 30421  df-clwwlkn 30422
This theorem is referenced by:  numclwlk1lem2f1o  30694
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