Theorem wwlksnextprop 27698

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.) (Revised by AV, 29-Oct-2022.)

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 prefix (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 2799	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
2		wwlksnextprop.x	. . . . . . . . 9 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
3	2	wwlksnextproplem1 27695	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
4	3	ancoms 462	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
5	4	adantr 484	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
6		eqeq2 2810	. . . . . . 7 ⊢ ((𝑥‘0) = 𝑃 → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
7	6	adantl 485	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
8	5, 7	mpbid 235	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃)
9		wwlksnextprop.e	. . . . . . . 8 ⊢ 𝐸 = (Edg‘𝐺)
10	2, 9	wwlksnextproplem2 27696	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
11	10	ancoms 462	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
12	11	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
13		simpr 488	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
14	13	adantr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥 ∈ 𝑋)
15		simpr 488	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16		simpll 766	. . . . . . 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 27697	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑥‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
19	14, 15, 16, 18	syl3anc 1368	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
20		eqeq2 2810	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑥 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1))))
21		fveq1 6644	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
22	21	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
23		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑥 prefix (𝑁 + 1))))
24	23	preq1d 4635	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)})
25	24	eleq1d 2874	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸))
26	20, 22, 25	3anbi123d 1433	. . . . . . 7 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
27	26	adantl 485	. . . . . 6 ⊢ ((((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) ∧ 𝑦 = (𝑥 prefix (𝑁 + 1))) → (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) ↔ ((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)))
28	19, 27	rspcedv 3564	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (((𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ∧ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 ∧ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸) → ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
29	1, 8, 12, 28	mp3and 1461	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸))
30	29	ex 416	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥‘0) = 𝑃 → ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
31	21	eqcoms 2806	. . . . . . . . 9 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
32	31	eqeq1d 2800	. . . . . . . 8 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
33	3	eqcomd 2804	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
34	33	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
35	34	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
36		eqeq2 2810	. . . . . . . . . 10 ⊢ (𝑃 = ((𝑥 prefix (𝑁 + 1))‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
37	36	eqcoms 2806	. . . . . . . . 9 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
38	35, 37	syl5ibr 249	. . . . . . . 8 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
39	32, 38	syl6bi 256	. . . . . . 7 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃)))
40	39	imp 410	. . . . . 6 ⊢ (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
41	40	3adant3 1129	. . . . 5 ⊢ (((𝑥 prefix (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 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
43	42	rexlimdva 3243	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥‘0) = 𝑃))
44	30, 43	impbid 215	. 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥‘0) = 𝑃 ↔ ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)))
45	44	rabbidva 3425	1 ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  {cpr 4527  ‘cfv 6324  (class class class)co 7135  0cc0 10526  1c1 10527   + caddc 10529  ℕ₀cn0 11885  lastSclsw 13905   prefix cpfx 14023  Edgcedg 26840   WWalksN cwwlksn 27612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-lsw 13906  df-substr 13994  df-pfx 14024  df-wwlks 27616  df-wwlksn 27617

This theorem is referenced by: (None)