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Theorem usgrcyclnl1 23349
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 23341 . . 3  |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
2 wlkbprop 23256 . . . 4  |-  ( F ( V Walks  E ) P  ->  ( ( # `
 F )  e. 
NN0  /\  ( V  e.  _V  /\  E  e. 
_V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 23334 . . . . . 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 999 . . . . 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 23294 . . . . . . . 8  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
6 trliswlk 23261 . . . . . . . 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 23285 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1 ) ) )
9 nne 2602 . . . . . . . . . . . . . . . 16  |-  ( -.  ( # `  F
)  =/=  1  <->  ( # `
 F )  =  1 )
10 0z 10645 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  e.  ZZ
11 1z 10664 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  e.  ZZ
12 0lt1 9850 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  0  <  1
13 fzolb 11542 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 0  e.  ( 0..^ 1 )  <->  ( 0  e.  ZZ  /\  1  e.  ZZ  /\  0  <  1 ) )
1410, 11, 12, 13mpbir3an 1163 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  0  e.  ( 0..^ 1 )
15 oveq2 6088 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  1  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 1 ) )
1614, 15syl5eleqr 2520 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
# `  F )  =  1  ->  0  e.  ( 0..^ ( # `  F ) ) )
1716anim2i 564 . . . . . . . . . . . . . . . . . . . . . . 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 462 . . . . . . . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
22 oveq1 6087 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
2322fveq2d 5683 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P `  ( 0  +  1 ) ) )
2421, 23neeq12d 2613 . . . . . . . . . . . . . . . . . . . . 21  |-  ( k  =  0  ->  (
( P `  k
)  =/=  ( P `
 ( k  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  (
0  +  1 ) ) ) )
2524rspccva 3061 . . . . . . . . . . . . . . . . . . . 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 10421 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 0  +  1 )  =  1
28 id 22 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  =  ( # `  F
)  ->  1  =  ( # `  F ) )
2928eqcoms 2436 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( (
# `  F )  =  1  ->  1  =  ( # `  F
) )
3027, 29syl5eq 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
# `  F )  =  1  ->  (
0  +  1 )  =  ( # `  F
) )
3130fveq2d 5683 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  =  1  ->  ( P `  ( 0  +  1 ) )  =  ( P `  ( # `  F ) ) )
3231neeq2d 2612 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  =  1  ->  (
( P `  0
)  =/=  ( P `
 ( 0  +  1 ) )  <->  ( P `  0 )  =/=  ( P `  ( # `
 F ) ) ) )
33 df-ne 2598 . . . . . . . . . . . . . . . . . . . 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 1002 . . . . . . 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 ) ) ) )
4847imp3a 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   _Vcvv 2962   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   0cc0 9270   1c1 9271    + caddc 9273    < clt 9406   NN0cn0 10567   ZZcz 10634  ..^cfzo 11532   #chash 12087   USGrph cusg 23087   Walks cwalk 23228   Trails ctrail 23229   Paths cpath 23230   Cycles ccycl 23237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-hash 12088  df-word 12213  df-usgra 23089  df-wlk 23238  df-trail 23239  df-pth 23240  df-cycl 23243
This theorem is referenced by: (None)
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