Theorem clwwlkprop 25496

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 25477	. . 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 6525	. 2 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V ) )
3		pm3.22 450	. . . . . 6 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( ( V e. _V /\ E e. _V ) /\ P e. Word V ) )
4		df-3an 984	. . . . . 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 215	. . . . 5 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )
6	5	a1d 26	. . . 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 435	. . 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 25492	. . . . . 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 2492	. . . . 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 5881	. . . . . . . . . . 11 |- ( p = P -> ( # ` p ) = ( # ` P ) )
11	10	oveq1d 6320	. . . . . . . . . 10 |- ( p = P -> ( ( # ` p ) - 1 ) = ( ( # ` P ) - 1 ) )
12	11	oveq2d 6321	. . . . . . . . 9 |- ( p = P -> ( 0..^ ( ( # ` p ) - 1 ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
13		fveq1 5880	. . . . . . . . . . 11 |- ( p = P -> ( p ` i ) = ( P ` i ) )
14		fveq1 5880	. . . . . . . . . . 11 |- ( p = P -> ( p ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
15	13, 14	preq12d 4087	. . . . . . . . . 10 |- ( p = P -> { ( p ` i ) , ( p ` ( i + 1 ) ) } = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
16	15	eleq1d 2491	. . . . . . . . 9 |- ( p = P -> ( { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
17	12, 16	raleqbidv 3036	. . . . . . . 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 5881	. . . . . . . . . 10 |- ( p = P -> ( lastS ` p ) = ( lastS ` P ) )
19		fveq1 5880	. . . . . . . . . 10 |- ( p = P -> ( p ` 0 ) = ( P ` 0 ) )
20	18, 19	preq12d 4087	. . . . . . . . 9 |- ( p = P -> { ( lastS ` p ) , ( p ` 0 ) } = { ( lastS ` P ) , ( P ` 0 ) } )
21	20	eleq1d 2491	. . . . . . . 8 |- ( p = P -> ( { ( lastS ` p ) , ( p ` 0 ) } e. ran E <-> { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) )
22	17, 21	anbi12d 715	. . . . . . 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 3228	. . . . . 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 466	. . . . . 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 198	. . . . 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 231	. . . 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 82	. . 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 167	. 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 37	1 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   A.wral 2771   {crab 2775   _Vcvv 3080   {cpr 4000   ran crn 4854   ` cfv 5601  (class class class)co 6305   0cc0 9546   1c1 9547   + caddc 9549   - cmin 9867  ..^cfzo 11922   #chash 12521  Word cword 12660   lastS clsw 12661   ClWWalks cclwwlk 25474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-pm 7486  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-card 8381  df-cda 8605  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923  df-hash 12522  df-word 12668  df-clwwlk 25477

This theorem is referenced by:  clwwlkgt0  25497  clwwlknprop  25498  clwwlkn0  25500  clwwisshclww  25533  clwwisshclwwn  25534  erclwwlkeqlen  25538  erclwwlkref  25539  erclwwlksym  25540  erclwwlktr  25541