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

Theorem numclwwlkovgelim 25829
Description: Properties of an element of the value of operation  G. (Contributed by Alexander van der Vekens, 24-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 ) ) } )
Assertion
Ref Expression
numclwwlkovgelim  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( X G N )  ->  ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  (
( P `  0
)  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) ) )
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, P
Allowed substitution hints:    P( v, n)    E( v)    F( v, n)    G( w, v, n)

Proof of Theorem numclwwlkovgelim
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 numclwwlk.c . . . 4  |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `
 n ) )
2 numclwwlk.f . . . 4  |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w ` 
0 )  =  v } )
3 numclwwlk.g . . . 4  |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>=
`  2 )  |->  { w  e.  ( C `
 n )  |  ( ( w ` 
0 )  =  v  /\  ( w `  ( n  -  2
) )  =  ( w `  0 ) ) } )
41, 2, 3numclwwlkovgel 25828 . . 3  |-  ( ( X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e.  ( X G N )  <-> 
( P  e.  ( ( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) )
543adant1 1027 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( X G N )  <->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) )
6 usgrav 25077 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
7 eluzge2nn0 11205 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  2
)  ->  N  e.  NN0 )
86, 7anim12i 570 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
9 df-3an 988 . . . . . . . . . . . 12  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  <->  ( ( V  e.  _V  /\  E  e.  _V )  /\  N  e.  NN0 ) )
108, 9sylibr 216 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( V  e.  _V  /\  E  e. 
_V  /\  N  e.  NN0 ) )
11 isclwwlkn 25509 . . . . . . . . . . 11  |-  ( ( V  e.  _V  /\  E  e.  _V  /\  N  e.  NN0 )  ->  ( P  e.  ( ( V ClWWalksN  E ) `  N
)  <->  ( P  e.  ( V ClWWalks  E )  /\  ( # `  P
)  =  N ) ) )
1210, 11syl 17 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( P  e.  ( V ClWWalks  E )  /\  ( # `  P
)  =  N ) ) )
13 isclwwlk 25508 . . . . . . . . . . . . 13  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( P  e.  ( V ClWWalks  E )  <->  ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( (
# `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
146, 13syl 17 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( P  e.  ( V ClWWalks  E )  <->  ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
1514adantr 467 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( V ClWWalks  E )  <->  ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E ) ) )
1615anbi1d 712 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( P  e.  ( V ClWWalks  E )  /\  ( # `  P )  =  N )  <->  ( ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  /\  ( # `  P
)  =  N ) ) )
1712, 16bitrd 257 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  <->  ( ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  /\  ( # `  P
)  =  N ) ) )
18 simp1 1009 . . . . . . . . . . 11  |-  ( ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  ->  P  e. Word  V
)
1918a1i 11 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P
)  -  1 ) ) { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P
) ,  ( P `
 0 ) }  e.  ran  E )  ->  P  e. Word  V
) )
2019anim1d 568 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( (
( P  e. Word  V  /\  A. i  e.  ( 0..^ ( ( # `  P )  -  1 ) ) { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  e.  ran  E  /\  { ( lastS  `  P ) ,  ( P `  0 ) }  e.  ran  E
)  /\  ( # `  P
)  =  N )  ->  ( P  e. Word  V  /\  ( # `  P
)  =  N ) ) )
2117, 20sylbid 219 . . . . . . . 8  |-  ( ( V USGrph  E  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( P  e. Word  V  /\  ( # `  P
)  =  N ) ) )
22213adant2 1028 . . . . . . 7  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( ( V ClWWalksN  E ) `
 N )  -> 
( P  e. Word  V  /\  ( # `  P
)  =  N ) ) )
2322com12 32 . . . . . 6  |-  ( P  e.  ( ( V ClWWalksN  E ) `  N
)  ->  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e. Word  V  /\  ( # `  P )  =  N ) ) )
24233ad2ant1 1030 . . . . 5  |-  ( ( P  e.  ( ( V ClWWalksN  E ) `  N
)  /\  ( P `  0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) )  ->  (
( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  -> 
( P  e. Word  V  /\  ( # `  P
)  =  N ) ) )
2524impcom 432 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( P  e.  (
( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )  ->  ( P  e. Word  V  /\  ( # `  P
)  =  N ) )
26 3simpc 1008 . . . . 5  |-  ( ( P  e.  ( ( V ClWWalksN  E ) `  N
)  /\  ( P `  0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) )  ->  (
( P `  0
)  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )
2726adantl 468 . . . 4  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( P  e.  (
( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )  ->  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) )
2825, 27jca 535 . . 3  |-  ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= ` 
2 ) )  /\  ( P  e.  (
( V ClWWalksN  E ) `  N )  /\  ( P `  0 )  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) )  ->  ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  (
( P `  0
)  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) )
2928ex 436 . 2  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( ( P  e.  ( ( V ClWWalksN  E ) `  N
)  /\  ( P `  0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) )  ->  (
( P  e. Word  V  /\  ( # `  P
)  =  N )  /\  ( ( P `
 0 )  =  X  /\  ( P `
 ( N  - 
2 ) )  =  ( P `  0
) ) ) ) )
305, 29sylbid 219 1  |-  ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  2 )
)  ->  ( P  e.  ( X G N )  ->  ( ( P  e. Word  V  /\  ( # `
 P )  =  N )  /\  (
( P `  0
)  =  X  /\  ( P `  ( N  -  2 ) )  =  ( P ` 
0 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   {crab 2743   _Vcvv 3047   {cpr 3972   class class class wbr 4405    |-> cmpt 4464   ran crn 4838   ` cfv 5585  (class class class)co 6295    |-> cmpt2 6297   0cc0 9544   1c1 9545    + caddc 9547    - cmin 9865   2c2 10666   NN0cn0 10876   ZZ>=cuz 11166  ..^cfzo 11922   #chash 12522  Word cword 12663   lastS clsw 12664   USGrph cusg 25069   ClWWalks cclwwlk 25488   ClWWalksN cclwwlkn 25489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12523  df-word 12671  df-usgra 25072  df-clwwlk 25491  df-clwwlkn 25492
This theorem is referenced by:  numclwlk1lem2f1  25834
  Copyright terms: Public domain W3C validator