Theorem wwlksnextprop 29890

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 2732	. . . . 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 29887	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
4	3	ancoms 458	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
5	4	adantr 480	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
6		eqeq2 2743	. . . . . . 7 ⊢ ((𝑥‘0) = 𝑃 → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
7	6	adantl 481	. . . . . 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 232	. . . . 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 29888	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
11	10	ancoms 458	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
12	11	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
13		simpr 484	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
14	13	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥 ∈ 𝑋)
15		simpr 484	. . . . . . 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 29889	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑥‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
19	14, 15, 16, 18	syl3anc 1373	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
20		eqeq2 2743	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑥 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1))))
21		fveq1 6821	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
22	21	eqeq1d 2733	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
23		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑥 prefix (𝑁 + 1))))
24	23	preq1d 4689	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)})
25	24	eleq1d 2816	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸))
26	20, 22, 25	3anbi123d 1438	. . . . . . 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 481	. . . . . 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 3565	. . . . 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 1466	. . . 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 412	. . 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 2739	. . . . . . . . 9 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
32	31	eqeq1d 2733	. . . . . . . 8 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
33	3	eqcomd 2737	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
34	33	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
35	34	adantr 480	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
36		eqeq2 2743	. . . . . . . . . 10 ⊢ (𝑃 = ((𝑥 prefix (𝑁 + 1))‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
37	36	eqcoms 2739	. . . . . . . . 9 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
38	35, 37	imbitrrid 246	. . . . . . . 8 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
39	32, 38	biimtrdi 253	. . . . . . 7 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃)))
40	39	imp 406	. . . . . 6 ⊢ (((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃) → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
41	40	3adant3 1132	. . . . 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 3133	. . 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 212	. 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 3401	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  {crab 3395  {cpr 4575  ‘cfv 6481  (class class class)co 7346  0cc0 11006  1c1 11007   + caddc 11009  ℕ₀cn0 12381  lastSclsw 14469   prefix cpfx 14578  Edgcedg 29025   WWalksN cwwlksn 29804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-hash 14238  df-word 14421  df-lsw 14470  df-substr 14549  df-pfx 14579  df-wwlks 29808  df-wwlksn 29809

This theorem is referenced by: (None)