Theorem cyclnspth 23654

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 23557	. . . 4 |- Cycles = ( v e. _V , e e. _V |-> { <. f , p >. | ( f ( v Paths e ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
2	1	brovmpt2ex 6843	. . 3 |- ( F ( V Cycles E ) P -> ( ( V e. _V /\ E e. _V ) /\ ( F e. _V /\ P e. _V ) ) )
3		iscycl 23648	. . . 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 23608	. . . . . . . . . . . 12 |- ( F ( V Paths E ) P -> F ( V Trails E ) P )
5		trliswlk 23575	. . . . . . . . . . . 12 |- ( F ( V Trails E ) P -> F ( V Walks E ) P )
6		2mwlk 23564	. . . . . . . . . . . 12 |- ( F ( V Walks E ) P -> ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) )
7		lennncl 12354	. . . . . . . . . . . . . . . 16 |- ( ( F e. Word dom E /\ F =/= (/) ) -> ( # ` F ) e. NN )
8		df-f1 5523	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> V <-> ( P : ( 0 ... ( # ` F ) ) --> V /\ Fun `' P ) )
9		nnne0 10457	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) e. NN -> ( # ` F ) =/= 0 )
10	9	necomd 2719	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) e. NN -> 0 =/= ( # ` F ) )
11	10	neneqd 2651	. . . . . . . . . . . . . . . . . . . . . . . . . 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 10689	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) e. NN -> ( # ` F ) e. NN0 )
15		0nn0 10697	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- 0 e. NN0
16	15	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. NN0 -> 0 e. NN0 )
17		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. NN0 )
18		nn0ge0 10708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. NN0 -> 0 <_ ( # ` F ) )
19		elfz2nn0 11583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( 0 e. ( 0 ... ( # ` F ) ) <-> ( 0 e. NN0 /\ ( # ` F ) e. NN0 /\ 0 <_ ( # ` F ) ) )
20	16, 17, 18, 19	syl3anbrc 1172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` F ) e. NN0 -> 0 e. ( 0 ... ( # ` F ) ) )
21		nn0re 10691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. RR )
22	21	leidd 10009	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. NN0 -> ( # ` F ) <_ ( # ` F ) )
23		elfz2nn0 11583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` F ) e. ( 0 ... ( # ` F ) ) <-> ( ( # ` F ) e. NN0 /\ ( # ` F ) e. NN0 /\ ( # ` F ) <_ ( # ` F ) ) )
24	17, 17, 22, 23	syl3anbrc 1172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. ( 0 ... ( # ` F ) ) )
25	20, 24	jca 532	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` F ) e. NN0 -> ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) )
26	14, 25	syl 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` F ) e. NN -> ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) )
27	26	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) -1-1-> V ) -> ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) )
28		f1fveq 6076	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( P : ( 0 ... ( # ` F ) ) -1-1-> V /\ ( 0 e. ( 0 ... ( # ` F ) ) /\ ( # ` F ) e. ( 0 ... ( # ` F ) ) ) ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> 0 = ( # ` F ) ) )
29	13, 27, 28	syl2anc 661	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) -1-1-> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) <-> 0 = ( # ` F ) ) )
30	12, 29	mtbird 301	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) -1-1-> V ) -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) )
31	30	expcom 435	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> V -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
32	8, 31	sylbir 213	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P : ( 0 ... ( # ` F ) ) --> V /\ Fun `' P ) -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
33	32	expcom 435	. . . . . . . . . . . . . . . . . . . . 21 |- ( Fun `' P -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( # ` F ) e. NN -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
34	33	com13 80	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) e. NN -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( Fun `' P -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) )
35	34	imp 429	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( Fun `' P -> -. ( P ` 0 ) = ( P ` ( # ` F ) ) ) )
36	35	con2d 115	. . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` F ) e. NN /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> -. Fun `' P ) )
37	36	ex 434	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) e. NN -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> -. Fun `' P ) ) )
38	37	com23 78	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) e. NN -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) )
39	7, 38	syl 16	. . . . . . . . . . . . . . 15 |- ( ( F e. Word dom E /\ F =/= (/) ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) )
40	39	ex 434	. . . . . . . . . . . . . 14 |- ( F e. Word dom E -> ( F =/= (/) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V -> -. Fun `' P ) ) ) )
41	40	com24 87	. . . . . . . . . . . . 13 |- ( F e. Word dom E -> ( P : ( 0 ... ( # ` F ) ) --> V -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) ) )
42	41	imp 429	. . . . . . . . . . . 12 |- ( ( F e. Word dom E /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) )
43	4, 5, 6, 42	4syl 21	. . . . . . . . . . 11 |- ( F ( V Paths E ) P -> ( ( P ` 0 ) = ( P ` ( # ` F ) ) -> ( F =/= (/) -> -. Fun `' P ) ) )
44	43	imp 429	. . . . . . . . . 10 |- ( ( F ( V Paths E ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( F =/= (/) -> -. Fun `' P ) )
45	44	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 ) )
46	45	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 )
47	46	intnand 907	. . . . . . 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 23605	. . . . . . . . 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 ) ) )
49	48	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 ) ) )
50	49	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 ) ) )
51	47, 50	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 )
52	51	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 ) )
53	52	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 ) ) )
54	3, 53	sylbid 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 ) ) )
55	2, 54	mpcom 36	. 2 |- ( F ( V Cycles E ) P -> ( F =/= (/) -> -. F ( V SPaths E ) P ) )
56	55	com12 31	1 |- ( F =/= (/) -> ( F ( V Cycles E ) P -> -. F ( V SPaths E ) P ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2644   _Vcvv 3070   (/)c0 3737   class class class wbr 4392   `'ccnv 4939   dom cdm 4940   Fun wfun 5512   -->wf 5514   -1-1->wf1 5515   ` cfv 5518  (class class class)co 6192   0cc0 9385   <_ cle 9522   NNcn 10425   NN0cn0 10682   ...cfz 11540   #chash 12206  Word cword 12325   Walks cwalk 23542   Trails ctrail 23543   Paths cpath 23544   SPaths cspath 23545   Cycles ccycl 23551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-card 8212  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-fzo 11652  df-hash 12207  df-word 12333  df-wlk 23552  df-trail 23553  df-pth 23554  df-spth 23555  df-cycl 23557

This theorem is referenced by: (None)