MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgrcyclnl1 Structured version   Unicode version

Theorem usgrcyclnl1 23494
Description: In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
Assertion
Ref Expression
usgrcyclnl1  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  1 )

Proof of Theorem usgrcyclnl1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cycliswlk 23486 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 23401 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 23479 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P 
<->  ( F ( V Paths 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
433adant1 1006 . . . . 5  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P 
<->  ( F ( V Paths 
E ) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
5 pthistrl 23439 . . . . . . . 8  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
6 trliswlk 23406 . . . . . . . 8  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
75, 6syl 16 . . . . . . 7  |-  ( F ( V Paths  E ) P  ->  F ( V Walks  E ) P )
8 usgrnloop 23430 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
9 nne 2607 . . . . . . . . . . . . . . . 16  |-  ( -.  ( # `  F
)  =/=  1  <->  ( # `
 F )  =  1 )
10 0z 10649 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  e.  ZZ
11 1z 10668 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  ZZ
12 0lt1 9854 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  <  1
13 fzolb 11550 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0  e.  ( 0..^ 1 )  <->  ( 0  e.  ZZ  /\  1  e.  ZZ  /\  0  <  1 ) )
1410, 11, 12, 13mpbir3an 1170 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  0  e.  ( 0..^ 1 )
15 oveq2 6094 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  1  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 1 ) )
1614, 15syl5eleqr 2525 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  F )  =  1  ->  0  e.  ( 0..^ ( # `  F ) ) )
1716anim2i 569 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  ( # `  F )  =  1 )  ->  ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  0  e.  ( 0..^ ( # `  F ) ) ) )
1817ex 434 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  ->  ( ( # `
 F )  =  1  ->  ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  0  e.  ( 0..^ ( # `  F ) ) ) ) )
1918adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  ( # `  F )  e.  NN0 )  ->  ( ( # `  F )  =  1  ->  ( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  0  e.  ( 0..^ ( # `  F ) ) ) ) )
2019impcom 430 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( # `  F
)  =  1  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 ) )  -> 
( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  0  e.  ( 0..^ ( # `  F ) ) ) )
21 fveq2 5686 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
22 oveq1 6093 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
2322fveq2d 5690 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
2421, 23neeq12d 2618 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  0  ->  (
( P `  k
)  =/=  ( P `
 ( k  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  (
0  +  1 ) ) ) )
2524rspccva 3067 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  0  e.  ( 0..^ ( # `  F ) ) )  ->  ( P ` 
0 )  =/=  ( P `  ( 0  +  1 ) ) )
2620, 25syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  =  1  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 ) )  -> 
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) ) )
27 0p1e1 10425 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0  +  1 )  =  1
28 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  =  ( # `  F
)  ->  1  =  ( # `  F ) )
2928eqcoms 2441 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  1  ->  1  =  ( # `  F
) )
3027, 29syl5eq 2482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  F )  =  1  ->  (
0  +  1 )  =  ( # `  F
) )
3130fveq2d 5690 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  1  ->  ( P `  ( 0  +  1 ) )  =  ( P `  ( # `  F ) ) )
3231neeq2d 2617 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  1  ->  (
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )
33 df-ne 2603 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( P `  0 )  =/=  ( P `  ( # `  F ) )  <->  -.  ( P `  0 )  =  ( P `  ( # `
 F ) ) )
3432, 33syl6bb 261 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  =  1  ->  (
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) )  <->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
3526, 34syl5ibcom 220 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  =  1  /\  ( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 ) )  -> 
( ( # `  F
)  =  1  ->  -.  ( P `  0
)  =  ( P `
 ( # `  F
) ) ) )
3635ex 434 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  1  ->  (
( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 )  ->  (
( # `  F )  =  1  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) ) )
3736pm2.43a 49 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  1  ->  (
( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 )  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
389, 37sylbi 195 . . . . . . . . . . . . . . 15  |-  ( -.  ( # `  F
)  =/=  1  -> 
( ( A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) )  /\  ( # `
 F )  e. 
NN0 )  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
3938com12 31 . . . . . . . . . . . . . 14  |-  ( ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  ( # `  F )  e.  NN0 )  ->  ( -.  ( # `
 F )  =/=  1  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
4039con4d 105 . . . . . . . . . . . . 13  |-  ( ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  /\  ( # `  F )  e.  NN0 )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( # `  F
)  =/=  1 ) )
4140ex 434 . . . . . . . . . . . 12  |-  ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  ->  ( ( # `
 F )  e. 
NN0  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( # `  F
)  =/=  1 ) ) )
4241com23 78 . . . . . . . . . . 11  |-  ( A. k  e.  ( 0..^ ( # `  F
) ) ( P `
 k )  =/=  ( P `  (
k  +  1 ) )  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( ( # `
 F )  e. 
NN0  ->  ( # `  F
)  =/=  1 ) ) )
438, 42syl 16 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( ( # `
 F )  e. 
NN0  ->  ( # `  F
)  =/=  1 ) ) )
4443ex 434 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( F ( V Walks  E ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( ( # `  F )  e.  NN0  ->  ( # `  F
)  =/=  1 ) ) ) )
4544com14 88 . . . . . . . 8  |-  ( (
# `  F )  e.  NN0  ->  ( F
( V Walks  E ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) ) )
46453ad2ant1 1009 . . . . . . 7  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Walks  E ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) ) )
477, 46syl5 32 . . . . . 6  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Paths  E ) P  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) ) )
4847impd 431 . . . . 5  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) )
494, 48sylbid 215 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) )
502, 49syl 16 . . 3  |-  ( F ( V Walks  E ) P  ->  ( F
( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) ) )
511, 50mpcom 36 . 2  |-  ( F ( V Cycles  E ) P  ->  ( V USGrph  E  ->  ( # `  F
)  =/=  1 ) )
5251impcom 430 1  |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F
)  =/=  1 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   _Vcvv 2967   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   0cc0 9274   1c1 9275    + caddc 9277    < clt 9410   NN0cn0 10571   ZZcz 10638  ..^cfzo 11540   #chash 12095   USGrph cusg 23232   Walks cwalk 23373   Trails ctrail 23374   Paths cpath 23375   Cycles ccycl 23382
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-fzo 11541  df-hash 12096  df-word 12221  df-usgra 23234  df-wlk 23383  df-trail 23384  df-pth 23385  df-cycl 23388
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator