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.

Step	Hyp	Ref	Expression
1		df-cycl 24942	. . . 4 |- Cycles = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Paths e ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
2	1	brovmpt2ex 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 )
10	9	necomd 2676	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) e. NN -> 0 =/= ( # ` F ) )
11	10	neneqd 2607	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` F ) e. NN -> -. 0 = ( # ` F ) )
12	11	adantr 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 )
18	17	leidd 10161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. NN0 -> ( # ` F ) <_ ( # ` F ) )
19		elfz2nn0 11826	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. ( 0 ... ( # ` F ) ) <-> ( ( # ` F ) e. NN0 /\ ( # ` F ) e. NN0 /\ ( # ` F ) <_ ( # ` F ) ) )
20	16, 16, 18, 19	syl3anbrc 1183	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. ( 0 ... ( # ` F ) ) )
21	15, 20	jca 532	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) e. NN0 -> ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) )
22	14, 21	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) e. NN -> ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) )
23	22	adantr 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 ) ) )
25	13, 23, 24	syl2anc 661	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) -1-1-> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> 0 = ( # ` F ) ) )
26	12, 25	mtbird 301	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) -1-1-> V ) -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) )
27	26	expcom 435	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> V -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
28	8, 27	sylbir 215	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ Fun `' P ) -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
29	28	expcom 435	. . . . . . . . . . . . . . . . . . . . 21 |- ( Fun `' P -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
30	29	com13 82	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) e. NN -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( Fun `' P -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
31	30	imp 429	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( Fun `' P -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
32	31	con2d 117	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> -. Fun `' P ) )
33	32	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) e. NN -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> -. Fun `' P ) ) )
34	33	com23 80	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. NN -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) )
35	7, 34	syl 17	. . . . . . . . . . . . . . 15 |- ( ( F e. Word dom E /\ F =/= (/) ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) )
36	35	ex 434	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> ( F =/= (/) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) ) )
37	36	com24 89	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) ) )
38	37	imp 429	. . . . . . . . . . . 12 |- ( ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) )
39	4, 5, 6, 38	4syl 19	. . . . . . . . . . 11 |- ( F ( V Paths E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) )
40	39	imp 429	. . . . . . . . . 10 |- ( ( F ( V Paths E ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( F =/= (/) -> -. Fun `' P ) )
41	40	adantl 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 ) )
42	41	imp 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 )
43	42	intnand 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 ) ) )
45	44	adantr 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 ) ) )
46	45	adantr 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 ) ) )
47	43, 46	mtbird 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 )
48	47	ex 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 ) )
49	48	ex 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 ) ) )
50	3, 49	sylbid 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 ) ) )
51	2, 50	mpcom 36	. 2 |- ( F ( V Cycles E ) P -> ( F =/= (/) -> -. F ( V SPaths E ) P ) )
52	51	com12 31	1 |- ( F =/= (/) -> ( F ( V Cycles E ) P -> -. F ( V SPaths E ) P ) )

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)