Theorem wlklniswwlkn1 26227

Description: The sequence of vertices in a walk of length n is a walk as word of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

Assertion

Ref	Expression
wlklniswwlkn1	⊢ (𝑉 USGrph 𝐸 → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Proof of Theorem wlklniswwlkn1

Step	Hyp	Ref	Expression
1		wlkbprop 26051	. . . 4 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
2	1	adantr 480	. . 3 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
3		wlkiswwlk1 26218	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (𝐹(𝑉 Walks 𝐸)𝑃 → 𝑃 ∈ (𝑉 WWalks 𝐸)))
4	3	com12 32	. . . . . . . 8 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → (𝑉 USGrph 𝐸 → 𝑃 ∈ (𝑉 WWalks 𝐸)))
5	4	adantr 480	. . . . . . 7 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → (𝑉 USGrph 𝐸 → 𝑃 ∈ (𝑉 WWalks 𝐸)))
6	5	adantl 481	. . . . . 6 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → (𝑉 USGrph 𝐸 → 𝑃 ∈ (𝑉 WWalks 𝐸)))
7	6	imp 444	. . . . 5 ⊢ (((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) ∧ 𝑉 USGrph 𝐸) → 𝑃 ∈ (𝑉 WWalks 𝐸))
8		2mwlk 26049	. . . . . . . . . 10 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉))
9		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → 𝑃 Fn (0...(#‘𝐹)))
10		hashfn 13025	. . . . . . . . . . . 12 ⊢ (𝑃 Fn (0...(#‘𝐹)) → (#‘𝑃) = (#‘(0...(#‘𝐹))))
11		hashfz0 13079	. . . . . . . . . . . . . 14 ⊢ ((#‘𝐹) ∈ ℕ0 → (#‘(0...(#‘𝐹))) = ((#‘𝐹) + 1))
12	11	3ad2ant1 1075	. . . . . . . . . . . . 13 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘(0...(#‘𝐹))) = ((#‘𝐹) + 1))
13		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ ((#‘(0...(#‘𝐹))) = (#‘𝑃) → ((#‘(0...(#‘𝐹))) = ((#‘𝐹) + 1) ↔ (#‘𝑃) = ((#‘𝐹) + 1)))
14	13	eqcoms 2618	. . . . . . . . . . . . 13 ⊢ ((#‘𝑃) = (#‘(0...(#‘𝐹))) → ((#‘(0...(#‘𝐹))) = ((#‘𝐹) + 1) ↔ (#‘𝑃) = ((#‘𝐹) + 1)))
15	12, 14	syl5ib 233	. . . . . . . . . . . 12 ⊢ ((#‘𝑃) = (#‘(0...(#‘𝐹))) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = ((#‘𝐹) + 1)))
16	9, 10, 15	3syl 18	. . . . . . . . . . 11 ⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = ((#‘𝐹) + 1)))
17	16	adantl 481	. . . . . . . . . 10 ⊢ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = ((#‘𝐹) + 1)))
18	8, 17	syl 17	. . . . . . . . 9 ⊢ (𝐹(𝑉 Walks 𝐸)𝑃 → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = ((#‘𝐹) + 1)))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = ((#‘𝐹) + 1)))
20		oveq1 6556	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) + 1) = (𝑁 + 1))
21	20	eqeq2d 2620	. . . . . . . . 9 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝑃) = ((#‘𝐹) + 1) ↔ (#‘𝑃) = (𝑁 + 1)))
22	21	adantl 481	. . . . . . . 8 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → ((#‘𝑃) = ((#‘𝐹) + 1) ↔ (#‘𝑃) = (𝑁 + 1)))
23	19, 22	sylibd 228	. . . . . . 7 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (#‘𝑃) = (𝑁 + 1)))
24	23	impcom 445	. . . . . 6 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → (#‘𝑃) = (𝑁 + 1))
25	24	adantr 480	. . . . 5 ⊢ (((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) ∧ 𝑉 USGrph 𝐸) → (#‘𝑃) = (𝑁 + 1))
26		simpl2l 1107	. . . . . . . 8 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → 𝑉 ∈ V)
27		simpl2r 1108	. . . . . . . 8 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → 𝐸 ∈ V)
28		eleq1 2676	. . . . . . . . . . . . 13 ⊢ ((#‘𝐹) = 𝑁 → ((#‘𝐹) ∈ ℕ0 ↔ 𝑁 ∈ ℕ0))
29	28	biimpcd 238	. . . . . . . . . . . 12 ⊢ ((#‘𝐹) ∈ ℕ0 → ((#‘𝐹) = 𝑁 → 𝑁 ∈ ℕ0))
30	29	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((#‘𝐹) = 𝑁 → 𝑁 ∈ ℕ0))
31	30	com12 32	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 𝑁 → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝑁 ∈ ℕ0))
32	31	adantl 481	. . . . . . . . 9 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → (((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → 𝑁 ∈ ℕ0))
33	32	impcom 445	. . . . . . . 8 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → 𝑁 ∈ ℕ0)
34	26, 27, 33	3jca 1235	. . . . . . 7 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
35	34	adantr 480	. . . . . 6 ⊢ (((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) ∧ 𝑉 USGrph 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
36		iswwlkn 26212	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑃 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑃) = (𝑁 + 1))))
37	35, 36	syl 17	. . . . 5 ⊢ (((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) ∧ 𝑉 USGrph 𝐸) → (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑃 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑃) = (𝑁 + 1))))
38	7, 25, 37	mpbir2and 959	. . . 4 ⊢ (((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) ∧ 𝑉 USGrph 𝐸) → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
39	38	ex 449	. . 3 ⊢ ((((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁)) → (𝑉 USGrph 𝐸 → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
40	2, 39	mpancom 700	. 2 ⊢ ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → (𝑉 USGrph 𝐸 → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))
41	40	com12 32	1 ⊢ (𝑉 USGrph 𝐸 → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   WWalks cwwlk 26205   WWalksN cwwlkn 26206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wlklniswwlkn  26229