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Theorem numclwlk1lem2f 24866
Description: T is a function. (Contributed by Alexander van der Vekens, 19-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
numclwlk1lem2f  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( 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 numclwlk1lem2f
StepHypRef Expression
1 numclwwlk.c . . . . . . 7  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . . . . . 7  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . . . . . 7  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
41, 2, 3extwwlkfab 24864 . . . . . 6  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( X G N )  =  {
w  e.  ( C `
 N )  |  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) } )
54eleq2d 2537 . . . . 5  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( X G N )  <->  w  e.  { w  e.  ( C `  N
)  |  ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) } ) )
6 rabid 3038 . . . . 5  |-  ( w  e.  { w  e.  ( C `  N
)  |  ( ( w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) }  <->  ( w  e.  ( C `  N
)  /\  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) ) )
75, 6syl6bb 261 . . . 4  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( X G N )  <->  ( w  e.  ( C `  N
)  /\  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
)  /\  ( w `  ( N  -  2 ) )  =  X ) ) ) )
8 simprr1 1044 . . . . . 6  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( w  e.  ( C `  N )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )  ->  ( w substr  <.
0 ,  ( N  -  2 ) >.
)  e.  ( X F ( N  - 
2 ) ) )
9 simprr2 1045 . . . . . 6  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( w  e.  ( C `  N )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )  ->  ( w `  ( N  -  1 ) )  e.  (
<. V ,  E >. Neighbors  X
) )
10 opelxp 5029 . . . . . 6  |-  ( <.
( w substr  <. 0 ,  ( N  -  2 ) >. ) ,  ( w `  ( N  -  1 ) )
>.  e.  ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) )  <->  ( (
w substr  <. 0 ,  ( N  -  2 )
>. )  e.  ( X F ( N  - 
2 ) )  /\  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X
) ) )
118, 9, 10sylanbrc 664 . . . . 5  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  ( w  e.  ( C `  N )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) ) )  ->  <. ( w substr  <. 0 ,  ( N  -  2 ) >.
) ,  ( w `
 ( N  - 
1 ) ) >.  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) )
1211ex 434 . . . 4  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( (
w  e.  ( C `
 N )  /\  ( ( w substr  <. 0 ,  ( N  - 
2 ) >. )  e.  ( X F ( N  -  2 ) )  /\  ( w `
 ( N  - 
1 ) )  e.  ( <. V ,  E >. Neighbors  X )  /\  (
w `  ( N  -  2 ) )  =  X ) )  ->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >.  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) ) )
137, 12sylbid 215 . . 3  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  ( w  e.  ( X G N )  ->  <. ( w substr  <. 0 ,  ( N  -  2 ) >.
) ,  ( w `
 ( N  - 
1 ) ) >.  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) ) )
1413imp 429 . 2  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
3 ) )  /\  w  e.  ( X G N ) )  ->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >.  e.  ( ( X F ( N  -  2 ) )  X.  ( <. V ,  E >. Neighbors  X
) ) )
15 numclwwlk.t . 2  |-  T  =  ( w  e.  ( X G N ) 
|->  <. ( w substr  <. 0 ,  ( N  - 
2 ) >. ) ,  ( w `  ( N  -  1
) ) >. )
1614, 15fmptd 6046 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
)  ->  T :
( X G N ) --> ( ( X F ( N  - 
2 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   0cc0 9493   1c1 9494    - cmin 9806   2c2 10586   3c3 10587   NN0cn0 10796   ZZ>=cuz 11083   substr csubstr 12505   USGrph cusg 24103   Neighbors cnbgra 24190   ClWWalksN cclwwlkn 24522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-map 7423  df-pm 7424  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-fz 11674  df-fzo 11794  df-hash 12375  df-word 12509  df-lsw 12510  df-substr 12513  df-usgra 24106  df-nbgra 24193  df-clwwlk 24524  df-clwwlkn 24525
This theorem is referenced by:  numclwlk1lem2f1  24868  numclwlk1lem2fo  24869
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