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Theorem cyclnspth 25060
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 24942 . . . 4  |- Cycles  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Paths 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
21brovmpt2ex 6956 . . 3  |-  ( F ( V Cycles  E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
3 iscycl 25054 . . . 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 25003 . . . . . . . . . . . 12  |-  ( F ( V Paths  E ) P  ->  F ( V Trails  E ) P )
5 trliswlk 24970 . . . . . . . . . . . 12  |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
6 2mwlk 24950 . . . . . . . . . . . 12  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
7 lennncl 12617 . . . . . . . . . . . . . . . 16  |-  ( ( F  e. Word  dom  E  /\  F  =/=  (/) )  -> 
( # `  F )  e.  NN )
8 df-f1 5576 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  <->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P ) )
9 nnne0 10611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN  ->  ( # `  F
)  =/=  0 )
109necomd 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  e.  NN  ->  0  =/=  ( # `  F ) )
1110neneqd 2607 . . . . . . . . . . . . . . . . . . . . . . . . . 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 10845 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN  ->  ( # `  F
)  e.  NN0 )
15 0elfz 11830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
16 id 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  NN0 )
17 nn0re 10847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  RR )
1817leidd 10161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  <_  ( # `  F
) )
19 elfz2nn0 11826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( (
# `  F )  e.  ( 0 ... ( # `
 F ) )  <-> 
( ( # `  F
)  e.  NN0  /\  ( # `  F )  e.  NN0  /\  ( # `
 F )  <_ 
( # `  F ) ) )
2016, 16, 18, 19syl3anbrc 1183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
2115, 20jca 532 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  e.  NN0  ->  ( 0  e.  ( 0 ... ( # `  F
) )  /\  ( # `
 F )  e.  ( 0 ... ( # `
 F ) ) ) )
2214, 21syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  e.  NN  ->  ( 0  e.  ( 0 ... ( # `  F
) )  /\  ( # `
 F )  e.  ( 0 ... ( # `
 F ) ) ) )
2322adantr 465 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  ( 0  e.  ( 0 ... ( # `
 F ) )  /\  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) ) )
24 f1fveq 6153 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( 0  e.  ( 0 ... ( # `  F ) )  /\  ( # `  F )  e.  ( 0 ... ( # `  F
) ) ) )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  <->  0  =  ( # `  F ) ) )
2513, 23, 24syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  <->  0  =  ( # `  F ) ) )
2612, 25mtbird 301 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) -1-1-> V )  ->  -.  ( P `  0 )  =  ( P `  ( # `
 F ) ) )
2726expcom 435 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( # `  F
)  e.  NN  ->  -.  ( P `  0
)  =  ( P `
 ( # `  F
) ) ) )
288, 27sylbir 215 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P )  ->  (
( # `  F )  e.  NN  ->  -.  ( P `  0 )  =  ( P `  ( # `  F ) ) ) )
2928expcom 435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Fun  `' P  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( # `  F
)  e.  NN  ->  -.  ( P `  0
)  =  ( P `
 ( # `  F
) ) ) ) )
3029com13 82 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( Fun  `' P  ->  -.  ( P ` 
0 )  =  ( P `  ( # `  F ) ) ) ) )
3130imp 429 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( Fun  `' P  ->  -.  ( P `  0 )  =  ( P `  ( # `
 F ) ) ) )
3231con2d 117 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( # `  F
)  e.  NN  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  -.  Fun  `' P
) )
3332ex 434 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  e.  NN  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  ->  -.  Fun  `' P ) ) )
3433com23 80 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  e.  NN  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  -.  Fun  `' P ) ) )
357, 34syl 17 . . . . . . . . . . . . . . 15  |-  ( ( F  e. Word  dom  E  /\  F  =/=  (/) )  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  -.  Fun  `' P
) ) )
3635ex 434 . . . . . . . . . . . . . 14  |-  ( F  e. Word  dom  E  ->  ( F  =/=  (/)  ->  (
( P `  0
)  =  ( P `
 ( # `  F
) )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  -.  Fun  `' P ) ) ) )
3736com24 89 . . . . . . . . . . . . 13  |-  ( F  e. Word  dom  E  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  ( P `  ( # `  F ) )  -> 
( F  =/=  (/)  ->  -.  Fun  `' P ) ) ) )
3837imp 429 . . . . . . . . . . . 12  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  ( P `  ( # `
 F ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P
) ) )
394, 5, 6, 384syl 19 . . . . . . . . . . 11  |-  ( F ( V Paths  E ) P  ->  ( ( P `  0 )  =  ( P `  ( # `  F ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P ) ) )
4039imp 429 . . . . . . . . . 10  |-  ( ( F ( V Paths  E
) P  /\  ( P `  0 )  =  ( P `  ( # `  F ) ) )  ->  ( F  =/=  (/)  ->  -.  Fun  `' P ) )
4140adantl 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 ) )
4241imp 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 )
4342intnand 919 . . . . . . 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 ) )
44 isspth 25000 . . . . . . . . 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 ) ) )
4544adantr 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
) ) )
4645adantr 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
) ) )
4743, 46mtbird 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 )
4847ex 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 ) )
4948ex 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 ) ) )
503, 49sylbid 217 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V Cycles  E ) P  ->  ( F  =/=  (/)  ->  -.  F ( V SPaths  E ) P ) ) )
512, 50mpcom 36 . 2  |-  ( F ( V Cycles  E ) P  ->  ( F  =/=  (/)  ->  -.  F
( V SPaths  E ) P ) )
5251com12 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 186    /\ wa 369    = wceq 1407    e. wcel 1844    =/= wne 2600   _Vcvv 3061   (/)c0 3740   class class class wbr 4397   `'ccnv 4824   dom cdm 4825   Fun wfun 5565   -->wf 5567   -1-1->wf1 5568   ` cfv 5571  (class class class)co 6280   0cc0 9524    <_ cle 9661   NNcn 10578   NN0cn0 10838   ...cfz 11728   #chash 12454  Word cword 12585   Walks cwalk 24927   Trails ctrail 24928   Paths cpath 24929   SPaths cspath 24930   Cycles ccycl 24936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-map 7461  df-pm 7462  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8354  df-cda 8582  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-n0 10839  df-z 10908  df-uz 11130  df-fz 11729  df-fzo 11857  df-hash 12455  df-word 12593  df-wlk 24937  df-trail 24938  df-pth 24939  df-spth 24940  df-cycl 24942
This theorem is referenced by: (None)
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