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Theorem usgrcyclnl1 23677
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 23669 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 23584 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 23662 . . . . . 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 23622 . . . . . . . 8  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
6 trliswlk 23589 . . . . . . . 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 23613 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
9 nne 2653 . . . . . . . . . . . . . . . 16  |-  ( -.  ( # `  F
)  =/=  1  <->  ( # `
 F )  =  1 )
10 0z 10767 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  e.  ZZ
11 1z 10786 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  ZZ
12 0lt1 9972 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  <  1
13 fzolb 11674 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6207 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  1  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 1 ) )
1614, 15syl5eleqr 2549 . . . . . . . . . . . . . . . . . . . . . . . 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 5798 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
22 oveq1 6206 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
2322fveq2d 5802 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
2421, 23neeq12d 2730 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  0  ->  (
( P `  k
)  =/=  ( P `
 ( k  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  (
0  +  1 ) ) ) )
2524rspccva 3176 . . . . . . . . . . . . . . . . . . . 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 10543 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0  +  1 )  =  1
28 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  =  ( # `  F
)  ->  1  =  ( # `  F ) )
2928eqcoms 2466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  1  ->  1  =  ( # `  F
) )
3027, 29syl5eq 2507 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  F )  =  1  ->  (
0  +  1 )  =  ( # `  F
) )
3130fveq2d 5802 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  1  ->  ( P `  ( 0  +  1 ) )  =  ( P `  ( # `  F ) ) )
3231neeq2d 2729 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  1  ->  (
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )
33 df-ne 2649 . . . . . . . . . . . . . . . . . . . 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 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   _Vcvv 3076   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393    + caddc 9395    < clt 9528   NN0cn0 10689   ZZcz 10756  ..^cfzo 11664   #chash 12219   USGrph cusg 23415   Walks cwalk 23556   Trails ctrail 23557   Paths cpath 23558   Cycles ccycl 23565
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 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
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 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417  df-wlk 23566  df-trail 23567  df-pth 23568  df-cycl 23571
This theorem is referenced by: (None)
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