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Theorem usgrcyclnl1 24842
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 24834 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 24725 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 24827 . . . . . 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 1012 . . . . 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 24776 . . . . . . . 8  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
6 trliswlk 24743 . . . . . . . 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 24767 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
9 nne 2655 . . . . . . . . . . . . . . . 16  |-  ( -.  ( # `  F
)  =/=  1  <->  ( # `
 F )  =  1 )
10 0z 10871 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  e.  ZZ
11 1z 10890 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  ZZ
12 0lt1 10071 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  <  1
13 fzolb 11810 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0  e.  ( 0..^ 1 )  <->  ( 0  e.  ZZ  /\  1  e.  ZZ  /\  0  <  1 ) )
1410, 11, 12, 13mpbir3an 1176 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  0  e.  ( 0..^ 1 )
15 oveq2 6278 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  1  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 1 ) )
1614, 15syl5eleqr 2549 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  F )  =  1  ->  0  e.  ( 0..^ ( # `  F ) ) )
1716anim2i 567 . . . . . . . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . . . . . 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 428 . . . . . . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
22 oveq1 6277 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
2322fveq2d 5852 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
2421, 23neeq12d 2733 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  0  ->  (
( P `  k
)  =/=  ( P `
 ( k  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  (
0  +  1 ) ) ) )
2524rspccva 3206 . . . . . . . . . . . . . . . . . . . 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 10643 . . . . . . . . . . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  1  ->  ( P `  ( 0  +  1 ) )  =  ( P `  ( # `  F ) ) )
3231neeq2d 2732 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  1  ->  (
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )
33 df-ne 2651 . . . . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . 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 432 . . . . . . . . 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 1015 . . . . . . 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 429 . . . . 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 428 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617   NN0cn0 10791   ZZcz 10860  ..^cfzo 11799   #chash 12387   USGrph cusg 24532   Walks cwalk 24700   Trails ctrail 24701   Paths cpath 24702   Cycles ccycl 24709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388  df-word 12526  df-usgra 24535  df-wlk 24710  df-trail 24711  df-pth 24712  df-cycl 24715
This theorem is referenced by: (None)
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