Theorem clwwlkprop 25174

Description: Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)

Assertion

Ref	Expression
clwwlkprop	|- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )

Proof of Theorem clwwlkprop

Dummy variables e i v w p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlk 25155	. . 3 |- ClWWalks = ( v e. _V , e e. _V |-> { w e. Word v | ( A. i e. ( 0..^ ( ( # ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. ran e /\ { ( lastS ` w ) , ( w ` 0 ) } e. ran e ) } )
2	1	elmpt2cl 6497	. 2 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V ) )
3		pm3.22 447	. . . . . 6 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( ( V e. _V /\ E e. _V ) /\ P e. Word V ) )
4		df-3an 976	. . . . . 6 |- ( ( V e. _V /\ E e. _V /\ P e. Word V ) <-> ( ( V e. _V /\ E e. _V ) /\ P e. Word V ) )
5	3, 4	sylibr 212	. . . . 5 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )
6	5	a1d 25	. . . 4 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
7	6	ex 432	. . 3 |- ( P e. Word V -> ( ( V e. _V /\ E e. _V ) -> ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) ) )
8		clwwlk 25170	. . . . . 6 |- ( ( V e. _V /\ E e. _V ) -> ( V ClWWalks E ) = { p e. Word V | ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` p ) , ( p ` 0 ) } e. ran E ) } )
9	8	eleq2d 2472	. . . . 5 |- ( ( V e. _V /\ E e. _V ) -> ( P e. ( V ClWWalks E ) <-> P e. { p e. Word V | ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` p ) , ( p ` 0 ) } e. ran E ) } ) )
10		fveq2 5848	. . . . . . . . . . 11 |- ( p = P -> ( # ` p ) = ( # ` P ) )
11	10	oveq1d 6292	. . . . . . . . . 10 |- ( p = P -> ( ( # ` p ) - 1 ) = ( ( # ` P ) - 1 ) )
12	11	oveq2d 6293	. . . . . . . . 9 |- ( p = P -> ( 0..^ ( ( # ` p ) - 1 ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
13		fveq1 5847	. . . . . . . . . . 11 |- ( p = P -> ( p ` i ) = ( P ` i ) )
14		fveq1 5847	. . . . . . . . . . 11 |- ( p = P -> ( p ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
15	13, 14	preq12d 4058	. . . . . . . . . 10 |- ( p = P -> { ( p ` i ) , ( p ` ( i + 1 ) ) } = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
16	15	eleq1d 2471	. . . . . . . . 9 |- ( p = P -> ( { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
17	12, 16	raleqbidv 3017	. . . . . . . 8 |- ( p = P -> ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E <-> A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
18		fveq2 5848	. . . . . . . . . 10 |- ( p = P -> ( lastS ` p ) = ( lastS ` P ) )
19		fveq1 5847	. . . . . . . . . 10 |- ( p = P -> ( p ` 0 ) = ( P ` 0 ) )
20	18, 19	preq12d 4058	. . . . . . . . 9 |- ( p = P -> { ( lastS ` p ) , ( p ` 0 ) } = { ( lastS ` P ) , ( P ` 0 ) } )
21	20	eleq1d 2471	. . . . . . . 8 |- ( p = P -> ( { ( lastS ` p ) , ( p ` 0 ) } e. ran E <-> { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) )
22	17, 21	anbi12d 709	. . . . . . 7 |- ( p = P -> ( ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` p ) , ( p ` 0 ) } e. ran E ) <-> ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) ) )
23	22	elrab 3206	. . . . . 6 |- ( P e. { p e. Word V | ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` p ) , ( p ` 0 ) } e. ran E ) } <-> ( P e. Word V /\ ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) ) )
24		pm2.24 109	. . . . . . 7 |- ( P e. Word V -> ( -. P e. Word V -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
25	24	adantr 463	. . . . . 6 |- ( ( P e. Word V /\ ( A. i e. ( 0..^ ( ( # ` P ) - 1 ) ) { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) ) -> ( -. P e. Word V -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
26	23, 25	sylbi 195	. . . . 5 |- ( P e. { p e. Word V | ( A. i e. ( 0..^ ( ( # ` p ) - 1 ) ) { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E /\ { ( lastS ` p ) , ( p ` 0 ) } e. ran E ) } -> ( -. P e. Word V -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
27	9, 26	syl6bi 228	. . . 4 |- ( ( V e. _V /\ E e. _V ) -> ( P e. ( V ClWWalks E ) -> ( -. P e. Word V -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) ) )
28	27	com3r 79	. . 3 |- ( -. P e. Word V -> ( ( V e. _V /\ E e. _V ) -> ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) ) )
29	7, 28	pm2.61i 164	. 2 |- ( ( V e. _V /\ E e. _V ) -> ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
30	2, 29	mpcom 34	1 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2753   {crab 2757   _Vcvv 3058   {cpr 3973   ran crn 4823   ` cfv 5568  (class class class)co 6277   0cc0 9521   1c1 9522   + caddc 9524   - cmin 9840  ..^cfzo 11852   #chash 12450  Word cword 12581   lastS clsw 12582   ClWWalks cclwwlk 25152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-clwwlk 25155

This theorem is referenced by:  clwwlkgt0  25175  clwwlknprop  25176  clwwlkn0  25178  clwwisshclww  25211  clwwisshclwwn  25212  erclwwlkeqlen  25216  erclwwlkref  25217  erclwwlksym  25218  erclwwlktr  25219