Theorem clwwlkprop 25577

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 25558	. . 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 6530	. 2 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V ) )
3		pm3.22 456	. . . . . 6 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( ( V e. _V /\ E e. _V ) /\ P e. Word V ) )
4		df-3an 1009	. . . . . 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 217	. . . . 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 441	. . 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 25573	. . . . . 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 2534	. . . . 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 5879	. . . . . . . . . . 11 |- ( p = P -> ( # ` p ) = ( # ` P ) )
11	10	oveq1d 6323	. . . . . . . . . 10 |- ( p = P -> ( ( # ` p ) - 1 ) = ( ( # ` P ) - 1 ) )
12	11	oveq2d 6324	. . . . . . . . 9 |- ( p = P -> ( 0..^ ( ( # ` p ) - 1 ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
13		fveq1 5878	. . . . . . . . . . 11 |- ( p = P -> ( p ` i ) = ( P ` i ) )
14		fveq1 5878	. . . . . . . . . . 11 |- ( p = P -> ( p ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
15	13, 14	preq12d 4050	. . . . . . . . . 10 |- ( p = P -> { ( p ` i ) , ( p ` ( i + 1 ) ) } = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
16	15	eleq1d 2533	. . . . . . . . 9 |- ( p = P -> ( { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
17	12, 16	raleqbidv 2987	. . . . . . . 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 5879	. . . . . . . . . 10 |- ( p = P -> ( lastS ` p ) = ( lastS ` P ) )
19		fveq1 5878	. . . . . . . . . 10 |- ( p = P -> ( p ` 0 ) = ( P ` 0 ) )
20	18, 19	preq12d 4050	. . . . . . . . 9 |- ( p = P -> { ( lastS ` p ) , ( p ` 0 ) } = { ( lastS ` P ) , ( P ` 0 ) } )
21	20	eleq1d 2533	. . . . . . . 8 |- ( p = P -> ( { ( lastS ` p ) , ( p ` 0 ) } e. ran E <-> { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) )
22	17, 21	anbi12d 725	. . . . . . 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 3184	. . . . . 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		pm2.24 112	. . . . . . 7 |- ( P e. Word V -> ( -. P e. Word V -> ( V e. _V /\ E e. _V /\ P e. Word V ) ) )
25	24	adantr 472	. . . . . 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 200	. . . . 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 236	. . . 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 81	. . 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 169	. 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 36	1 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   {cpr 3961   ran crn 4840   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560   - cmin 9880  ..^cfzo 11942   #chash 12553  Word cword 12703   lastS clsw 12704   ClWWalks cclwwlk 25555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-clwwlk 25558

This theorem is referenced by:  clwwlkgt0  25578  clwwlknprop  25579  clwwlkn0  25581  clwwisshclww  25614  clwwisshclwwn  25615  erclwwlkeqlen  25619  erclwwlkref  25620  erclwwlksym  25621  erclwwlktr  25622