Theorem wwlksnextprop 26807

Description: Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextprop.x	⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextprop.y	⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}

Assertion

Ref	Expression
wwlksnextprop	⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)})

Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑃   𝑦,𝐸   𝑥,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌   𝑥,𝑤

Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥,𝑤)   𝐺(𝑥,𝑦)   𝑋(𝑥,𝑤)   𝑌(𝑥,𝑤)

Proof of Theorem wwlksnextprop

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2		wwlksnextprop.x	. . . . . . . . 9 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
3	2	wwlksnextproplem1 26804	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
4	3	ancoms 469	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
5	4	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0))
6		eqeq2 2633	. . . . . . 7 ⊢ ((𝑥‘0) = 𝑃 → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
7	6	adantl 482	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑥‘0) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
8	5, 7	mpbid 222	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃)
9		wwlksnextprop.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
10	2, 9	wwlksnextproplem2 26805	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸)
11	10	ancoms 469	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸)
12	11	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸)
13		simpr 477	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
14	13	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥 ∈ 𝑋)
15		simpr 477	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16		simpll 790	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑁 ∈ ℕ0)
17		wwlksnextprop.y	. . . . . . . 8 ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
18	2, 9, 17	wwlksnextproplem3 26806	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑥‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
19	14, 15, 16, 18	syl3anc 1326	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)
20		eqeq2 2633	. . . . . . . 8 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩)))
21		fveq1 6190	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
22	21	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
23		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ( lastS ‘𝑦) = ( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)))
24	23	preq1d 4274	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → {( lastS ‘𝑦), ( lastS ‘𝑥)} = {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)})
25	24	eleq1d 2686	. . . . . . . 8 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ({( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸 ↔ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸))
26	20, 22, 25	3anbi123d 1399	. . . . . . 7 ⊢ (𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸)))
27	26	adantl 482	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 substr ⟨0, (𝑁 + 1)⟩)) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸)))
28	19, 27	rspcedv 3313	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ∧ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 ∧ {( lastS ‘(𝑥 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑥)} ∈ 𝐸) → ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)))
29	1, 8, 12, 28	mp3and 1427	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸))
30	29	ex 450	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)))
31	21	eqcoms 2630	. . . . . . . . 9 ⊢ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → (𝑦‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
32	31	eqeq1d 2624	. . . . . . . 8 ⊢ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃))
33	3	eqcomd 2628	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
34	33	ancoms 469	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
35	34	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0))
36		eqeq2 2633	. . . . . . . . . 10 ⊢ (𝑃 = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
37	36	eqcoms 2630	. . . . . . . . 9 ⊢ (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0)))
38	35, 37	syl5ibr 236	. . . . . . . 8 ⊢ (((𝑥 substr ⟨0, (𝑁 + 1)⟩)‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
39	32, 38	syl6bi 243	. . . . . . 7 ⊢ ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃)))
40	39	imp 445	. . . . . 6 ⊢ (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
41	40	3adant3 1081	. . . . 5 ⊢ (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
42	41	com12 32	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
43	42	rexlimdva 3031	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
44	30, 43	impbid 202	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)))
45	44	rabbidva 3188	1 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916  {cpr 4179  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕ₀cn0 11292   lastS clsw 13292   substr csubstr 13295  Edgcedg 25939   WWalksN cwwlksn 26718

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-oadd 7564  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by: (None)