Theorem clwwlkprop 30604

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 30587	. . 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 6417	. 2 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V ) )
3		pm3.22 449	. . . . . 6 |- ( ( P e. Word V /\ ( V e. _V /\ E e. _V ) ) -> ( ( V e. _V /\ E e. _V ) /\ P e. Word V ) )
4		df-3an 967	. . . . . 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 434	. . 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 30600	. . . . . 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 2524	. . . . 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 5802	. . . . . . . . . . 11 |- ( p = P -> ( # ` p ) = ( # ` P ) )
11	10	oveq1d 6218	. . . . . . . . . 10 |- ( p = P -> ( ( # ` p ) - 1 ) = ( ( # ` P ) - 1 ) )
12	11	oveq2d 6219	. . . . . . . . 9 |- ( p = P -> ( 0..^ ( ( # ` p ) - 1 ) ) = ( 0..^ ( ( # ` P ) - 1 ) ) )
13		fveq1 5801	. . . . . . . . . . 11 |- ( p = P -> ( p ` i ) = ( P ` i ) )
14		fveq1 5801	. . . . . . . . . . 11 |- ( p = P -> ( p ` ( i + 1 ) ) = ( P ` ( i + 1 ) ) )
15	13, 14	preq12d 4073	. . . . . . . . . 10 |- ( p = P -> { ( p ` i ) , ( p ` ( i + 1 ) ) } = { ( P ` i ) , ( P ` ( i + 1 ) ) } )
16	15	eleq1d 2523	. . . . . . . . 9 |- ( p = P -> ( { ( p ` i ) , ( p ` ( i + 1 ) ) } e. ran E <-> { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ran E ) )
17	12, 16	raleqbidv 3037	. . . . . . . 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 5802	. . . . . . . . . 10 |- ( p = P -> ( lastS ` p ) = ( lastS ` P ) )
19		fveq1 5801	. . . . . . . . . 10 |- ( p = P -> ( p ` 0 ) = ( P ` 0 ) )
20	18, 19	preq12d 4073	. . . . . . . . 9 |- ( p = P -> { ( lastS ` p ) , ( p ` 0 ) } = { ( lastS ` P ) , ( P ` 0 ) } )
21	20	eleq1d 2523	. . . . . . . 8 |- ( p = P -> ( { ( lastS ` p ) , ( p ` 0 ) } e. ran E <-> { ( lastS ` P ) , ( P ` 0 ) } e. ran E ) )
22	17, 21	anbi12d 710	. . . . . . 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 3224	. . . . . 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 465	. . . . . 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 36	1 |- ( P e. ( V ClWWalks E ) -> ( V e. _V /\ E e. _V /\ P e. Word V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2799   {crab 2803   _Vcvv 3078   {cpr 3990   ran crn 4952   ` cfv 5529  (class class class)co 6203   0cc0 9397   1c1 9398   + caddc 9400   - cmin 9710  ..^cfzo 11669   #chash 12224  Word cword 12343   lastS clsw 12344   ClWWalks cclwwlk 30584

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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-word 12351  df-clwwlk 30587

This theorem is referenced by:  clwwlkgt0  30605  clwwlknprop  30606  clwwlkn0  30608  clwwisshclww  30642  clwwisshclwwn  30643  erclwwlkeqlen  30653  erclwwlkref  30654  erclwwlksym  30655  erclwwlktr  30656