Theorem wwlksnwwlksnon 26810

Description: A walk of fixed length is a walk of fixed length between two vertices. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypothesis

Ref	Expression
wwlksnwwlksnon.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
wwlksnwwlksnon	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))

Distinct variable groups:   𝐺,𝑎,𝑏   𝑁,𝑎,𝑏   𝑈,𝑎,𝑏   𝑉,𝑎,𝑏   𝑊,𝑎,𝑏

Proof of Theorem wwlksnwwlksnon

Step	Hyp	Ref	Expression
1		wwlknbp2 26752	. . . . . 6 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
2	1	adantl 482	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
3		wwlksnwwlksnon.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	eqcomi 2631	. . . . . . . . . . 11 ⊢ (Vtx‘𝐺) = 𝑉
5	4	wrdeqi 13328	. . . . . . . . . 10 ⊢ Word (Vtx‘𝐺) = Word 𝑉
6	5	eleq2i 2693	. . . . . . . . 9 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) ↔ 𝑊 ∈ Word 𝑉)
7	6	biimpi 206	. . . . . . . 8 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → 𝑊 ∈ Word 𝑉)
8	7	adantr 481	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → 𝑊 ∈ Word 𝑉)
9		nn0p1nn 11332	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
10		lbfzo0 12507	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(𝑁 + 1)) ↔ (𝑁 + 1) ∈ ℕ)
11	9, 10	sylibr 224	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0..^(𝑁 + 1)))
12	11	ad3antrrr 766	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 0 ∈ (0..^(𝑁 + 1)))
13		oveq2 6658	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0..^(#‘𝑊)) = (0..^(𝑁 + 1)))
14	13	eleq2d 2687	. . . . . . . . 9 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0 ∈ (0..^(#‘𝑊)) ↔ 0 ∈ (0..^(𝑁 + 1))))
15	14	ad2antll 765	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (0 ∈ (0..^(#‘𝑊)) ↔ 0 ∈ (0..^(𝑁 + 1))))
16	12, 15	mpbird 247	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 0 ∈ (0..^(#‘𝑊)))
17		wrdsymbcl 13318	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 0 ∈ (0..^(#‘𝑊))) → (𝑊‘0) ∈ 𝑉)
18	8, 16, 17	syl2an2 875	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑊‘0) ∈ 𝑉)
19		fzonn0p1 12544	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ (0..^(𝑁 + 1)))
20	19	ad3antrrr 766	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑁 ∈ (0..^(𝑁 + 1)))
21	13	eleq2d 2687	. . . . . . . . 9 ⊢ ((#‘𝑊) = (𝑁 + 1) → (𝑁 ∈ (0..^(#‘𝑊)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
22	21	ad2antll 765	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑁 ∈ (0..^(#‘𝑊)) ↔ 𝑁 ∈ (0..^(𝑁 + 1))))
23	20, 22	mpbird 247	. . . . . . 7 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑁 ∈ (0..^(#‘𝑊)))
24		wrdsymbcl 13318	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑁) ∈ 𝑉)
25	8, 23, 24	syl2an2 875	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑊‘𝑁) ∈ 𝑉)
26		simplr 792	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
27		eqidd 2623	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑊‘0) = (𝑊‘0))
28		eqidd 2623	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → (𝑊‘𝑁) = (𝑊‘𝑁))
29		eqeq2 2633	. . . . . . . 8 ⊢ (𝑎 = (𝑊‘0) → ((𝑊‘0) = 𝑎 ↔ (𝑊‘0) = (𝑊‘0)))
30	29	3anbi2d 1404	. . . . . . 7 ⊢ (𝑎 = (𝑊‘0) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = (𝑊‘0) ∧ (𝑊‘𝑁) = 𝑏)))
31		eqeq2 2633	. . . . . . . 8 ⊢ (𝑏 = (𝑊‘𝑁) → ((𝑊‘𝑁) = 𝑏 ↔ (𝑊‘𝑁) = (𝑊‘𝑁)))
32	31	3anbi3d 1405	. . . . . . 7 ⊢ (𝑏 = (𝑊‘𝑁) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = (𝑊‘0) ∧ (𝑊‘𝑁) = 𝑏) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = (𝑊‘0) ∧ (𝑊‘𝑁) = (𝑊‘𝑁))))
33	30, 32	rspc2ev 3324	. . . . . 6 ⊢ (((𝑊‘0) ∈ 𝑉 ∧ (𝑊‘𝑁) ∈ 𝑉 ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = (𝑊‘0) ∧ (𝑊‘𝑁) = (𝑊‘𝑁))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏))
34	18, 25, 26, 27, 28, 33	syl113anc 1338	. . . . 5 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏))
35	2, 34	mpdan 702	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏))
36	35	ex 450	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏)))
37		simp1 1061	. . . . 5 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) → 𝑊 ∈ (𝑁 WWalksN 𝐺))
38	37	a1i 11	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) → 𝑊 ∈ (𝑁 WWalksN 𝐺)))
39	38	rexlimdvva 3038	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) → 𝑊 ∈ (𝑁 WWalksN 𝐺)))
40	36, 39	impbid 202	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏)))
41	3	wwlknon 26742	. . . . 5 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏)))
42	41	bicomd 213	. . . 4 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) ↔ 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))
43	42	adantl 482	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) ↔ 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))
44	43	2rexbidva 3056	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝑎 ∧ (𝑊‘𝑁) = 𝑏) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))
45	40, 44	bitrd 268	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292  ..^cfzo 12465  #chash 13117  Word cword 13291  Vtxcvtx 25874   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  wspthsnwspthsnon  26811  elwwlks2  26861