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Theorem rusgraprop4 30724
Description: The properties of a k-regular undirected simple graph expressed with trailing edges of walks (as words). (Contributed by Alexander van der Vekens, 2-Aug-2018.)
Assertion
Ref Expression
rusgraprop4  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e. Word  V (
p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) )
Distinct variable groups:    n, E, p    K, p    n, V, p
Allowed substitution hint:    K( n)

Proof of Theorem rusgraprop4
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 rusgraprop3 30723 . 2  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  ( # `
 { n  e.  V  |  { v ,  n }  e.  ran  E } )  =  K ) )
2 simp1 988 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K )  ->  V USGrph  E )
3 simp2 989 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K )  ->  K  e.  NN0 )
4 lswcl 12391 . . . . . . . . . . . 12  |-  ( ( p  e. Word  V  /\  p  =/=  (/) )  ->  ( lastS  `  p )  e.  V
)
54adantll 713 . . . . . . . . . . 11  |-  ( ( ( V USGrph  E  /\  p  e. Word  V )  /\  p  =/=  (/) )  -> 
( lastS  `  p )  e.  V )
6 preq1 4065 . . . . . . . . . . . . . . . . 17  |-  ( v  =  ( lastS  `  p
)  ->  { v ,  n }  =  {
( lastS  `  p ) ,  n } )
76eleq1d 2523 . . . . . . . . . . . . . . . 16  |-  ( v  =  ( lastS  `  p
)  ->  ( {
v ,  n }  e.  ran  E  <->  { ( lastS  `  p ) ,  n }  e.  ran  E ) )
87rabbidv 3070 . . . . . . . . . . . . . . 15  |-  ( v  =  ( lastS  `  p
)  ->  { n  e.  V  |  {
v ,  n }  e.  ran  E }  =  { n  e.  V  |  { ( lastS  `  p
) ,  n }  e.  ran  E } )
98fveq2d 5806 . . . . . . . . . . . . . 14  |-  ( v  =  ( lastS  `  p
)  ->  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  ( # `  { n  e.  V  |  { ( lastS  `  p
) ,  n }  e.  ran  E } ) )
109eqeq1d 2456 . . . . . . . . . . . . 13  |-  ( v  =  ( lastS  `  p
)  ->  ( ( # `
 { n  e.  V  |  { v ,  n }  e.  ran  E } )  =  K  <->  ( # `  {
n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E } )  =  K ) )
1110rspcva 3177 . . . . . . . . . . . 12  |-  ( ( ( lastS  `  p )  e.  V  /\  A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K )  ->  ( # `  {
n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E } )  =  K )
1211ex 434 . . . . . . . . . . 11  |-  ( ( lastS  `  p )  e.  V  ->  ( A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( # `  { n  e.  V  |  {
( lastS  `  p ) ,  n }  e.  ran  E } )  =  K ) )
135, 12syl 16 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  p  e. Word  V )  /\  p  =/=  (/) )  -> 
( A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( # `  { n  e.  V  |  {
( lastS  `  p ) ,  n }  e.  ran  E } )  =  K ) )
1413ex 434 . . . . . . . . 9  |-  ( ( V USGrph  E  /\  p  e. Word  V )  ->  (
p  =/=  (/)  ->  ( A. v  e.  V  ( # `  { n  e.  V  |  {
v ,  n }  e.  ran  E } )  =  K  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) )
1514com23 78 . . . . . . . 8  |-  ( ( V USGrph  E  /\  p  e. Word  V )  ->  ( A. v  e.  V  ( # `  { n  e.  V  |  {
v ,  n }  e.  ran  E } )  =  K  ->  (
p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) )
1615ex 434 . . . . . . 7  |-  ( V USGrph  E  ->  ( p  e. Word  V  ->  ( A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) ) )
1716com23 78 . . . . . 6  |-  ( V USGrph  E  ->  ( A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( p  e. Word  V  ->  ( p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) ) )
1817a1d 25 . . . . 5  |-  ( V USGrph  E  ->  ( K  e. 
NN0  ->  ( A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K  -> 
( p  e. Word  V  ->  ( p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) ) ) )
19183imp1 1201 . . . 4  |-  ( ( ( V USGrph  E  /\  K  e.  NN0  /\  A. v  e.  V  ( # `
 { n  e.  V  |  { v ,  n }  e.  ran  E } )  =  K )  /\  p  e. Word  V )  ->  (
p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) )
2019ralrimiva 2830 . . 3  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K )  ->  A. p  e. Word  V
( p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) )
212, 3, 203jca 1168 . 2  |-  ( ( V USGrph  E  /\  K  e. 
NN0  /\  A. v  e.  V  ( # `  {
n  e.  V  |  { v ,  n }  e.  ran  E }
)  =  K )  ->  ( V USGrph  E  /\  K  e.  NN0  /\ 
A. p  e. Word  V
( p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) )
221, 21syl 16 1  |-  ( <. V ,  E >. RegUSGrph  K  ->  ( V USGrph  E  /\  K  e.  NN0  /\  A. p  e. Word  V (
p  =/=  (/)  ->  ( # `
 { n  e.  V  |  { ( lastS  `  p ) ,  n }  e.  ran  E }
)  =  K ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   (/)c0 3748   {cpr 3990   <.cop 3994   class class class wbr 4403   ran crn 4952   ` cfv 5529   NN0cn0 10693   #chash 12223  Word cword 12342   lastS clsw 12343   USGrph cusg 23436   RegUSGrph crusgra 30708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8223  df-cda 8451  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-n0 10694  df-z 10761  df-uz 10976  df-xadd 11204  df-fz 11558  df-fzo 11669  df-hash 12224  df-word 12350  df-lsw 12351  df-usgra 23438  df-nbgra 23504  df-vdgr 23736  df-rgra 30709  df-rusgra 30710
This theorem is referenced by: (None)
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