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Theorem cyclnspth 24307
Description: A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Assertion
Ref Expression
cyclnspth  |-  ( F  =/=  (/)  ->  ( F
( V Cycles  E ) P  ->  -.  F ( V SPaths  E ) P ) )

Proof of Theorem cyclnspth
Dummy variables  e 
f  p  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cycl 24189 . . . 4  |- Cycles  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Paths 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
21brovmpt2ex 6948 . . 3  |-  ( F ( V Cycles  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 24301 . . . 4  |-  ( ( ( 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 ) ) ) ) )
4 pthistrl 24250 . . . . . . . . . . . 12  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
5 trliswlk 24217 . . . . . . . . . . . 12  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
6 2mwlk 24197 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
7 lennncl 12525 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  F  =/=  (/) )  -> 
( # `  F )  e.  NN )
8 df-f1 5591 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  <->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P ) )
9 nnne0 10564 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN  ->  ( # `  F
)  =/=  0 )
109necomd 2738 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  e.  NN  ->  0  =/=  ( # `  F ) )
1110neneqd 2669 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  e.  NN  ->  -.  0  =  ( # `  F
) )
1211adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  -.  0  =  ( # `  F ) )
13 simpr 461 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  P : ( 0 ... ( # `  F ) ) -1-1-> V
)
14 nnnn0 10798 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN  ->  ( # `  F
)  e.  NN0 )
15 0nn0 10806 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  0  e.  NN0
1615a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  0  e.  NN0 )
17 id 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  NN0 )
18 nn0ge0 10817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  0  <_  (
# `  F )
)
19 elfz2nn0 11764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( 0  e.  ( 0 ... ( # `  F
) )  <->  ( 0  e.  NN0  /\  ( # `
 F )  e. 
NN0  /\  0  <_  (
# `  F )
) )
2016, 17, 18, 19syl3anbrc 1180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
21 nn0re 10800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  RR )
2221leidd 10115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  <_  ( # `  F
) )
23 elfz2nn0 11764 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  ( 0 ... ( # `
 F ) )  <-> 
( ( # `  F
)  e.  NN0  /\  ( # `  F )  e.  NN0  /\  ( # `
 F )  <_ 
( # `  F ) ) )
2417, 17, 22, 23syl3anbrc 1180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
2520, 24jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN0  ->  ( 0  e.  ( 0 ... ( # `  F
) )  /\  ( # `
 F )  e.  ( 0 ... ( # `
 F ) ) ) )
2614, 25syl 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  e.  NN  ->  ( 0  e.  ( 0 ... ( # `  F
) )  /\  ( # `
 F )  e.  ( 0 ... ( # `
 F ) ) ) )
2726adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  ( 0  e.  ( 0 ... ( # `
 F ) )  /\  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) ) )
28 f1fveq 6156 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( 0  e.  ( 0 ... ( # `  F ) )  /\  ( # `  F )  e.  ( 0 ... ( # `  F
) ) ) )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  <->  0  =  ( # `  F ) ) )
2913, 27, 28syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  <->  0  =  ( # `  F ) ) )
3012, 29mtbird 301 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  -.  ( P `  0 )  =  ( P `  ( # `
 F ) ) )
3130expcom 435 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( # `  F
)  e.  NN  ->  -.  ( P `  0
)  =  ( P `
 ( # `  F
) ) ) )
328, 31sylbir 213 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P )  ->  (
( # `  F )  e.  NN  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
3332expcom 435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Fun  `' P  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  e.  NN  ->  -.  ( P `  0
)  =  ( P `
 ( # `  F
) ) ) ) )
3433com13 80 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( Fun  `' P  ->  -.  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
3534imp 429 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( Fun  `' P  ->  -.  ( P `  0 )  =  ( P `  ( # `
 F ) ) ) )
3635con2d 115 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  -.  Fun  `' P
) )
3736ex 434 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  e.  NN  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  ->  -.  Fun  `' P ) ) )
3837com23 78 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  NN  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  -.  Fun  `' P ) ) )
397, 38syl 16 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  F  =/=  (/) )  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  -.  Fun  `' P
) ) )
4039ex 434 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  ( F  =/=  (/)  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  -.  Fun  `' P ) ) ) )
4140com24 87 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( F  =/=  (/)  ->  -.  Fun  `' P ) ) ) )
4241imp 429 . . . . . . . . . . . 12  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P
) ) )
434, 5, 6, 424syl 21 . . . . . . . . . . 11  |-  ( F ( V Paths  E ) P  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P ) ) )
4443imp 429 . . . . . . . . . 10  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P ) )
4544adantl 466 . . . . . . . . 9  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  -> 
( F  =/=  (/)  ->  -.  Fun  `' P ) )
4645imp 429 . . . . . . . 8  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  /\  F  =/=  (/) )  ->  -.  Fun  `' P )
4746intnand 914 . . . . . . 7  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  /\  F  =/=  (/) )  ->  -.  ( F ( V Trails  E
) P  /\  Fun  `' P ) )
48 isspth 24247 . . . . . . . . 9  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
4948adantr 465 . . . . . . . 8  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  -> 
( F ( V SPaths  E ) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
5049adantr 465 . . . . . . 7  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  /\  F  =/=  (/) )  ->  ( F ( V SPaths  E
) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
5147, 50mtbird 301 . . . . . 6  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  /\  F  =/=  (/) )  ->  -.  F ( V SPaths  E
) P )
5251ex 434 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) )  /\  ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )  -> 
( F  =/=  (/)  ->  -.  F ( V SPaths  E
) P ) )
5352ex 434 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( F  =/=  (/)  ->  -.  F
( V SPaths  E ) P ) ) )
543, 53sylbid 215 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( F  =/=  (/)  ->  -.  F ( V SPaths  E ) P ) ) )
552, 54mpcom 36 . 2  |-  ( F ( V Cycles  E ) P  ->  ( F  =/=  (/)  ->  -.  F
( V SPaths  E ) P ) )
5655com12 31 1  |-  ( F  =/=  (/)  ->  ( F
( V Cycles  E ) P  ->  -.  F ( V SPaths  E ) P ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   Fun wfun 5580   -->wf 5582   -1-1->wf1 5583   ` cfv 5586  (class class class)co 6282   0cc0 9488    <_ cle 9625   NNcn 10532   NN0cn0 10791   ...cfz 11668   #chash 12369  Word cword 12496   Walks cwalk 24174   Trails ctrail 24175   Paths cpath 24176   SPaths cspath 24177   Cycles ccycl 24183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504  df-wlk 24184  df-trail 24185  df-pth 24186  df-spth 24187  df-cycl 24189
This theorem is referenced by: (None)
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