Theorem wwlksnextprop 28857

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 2737	. . . . 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 28854	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
4	3	ancoms 459	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
5	4	adantr 481	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → ((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0))
6		eqeq2 2748	. . . . . . 7 ⊢ ((𝑥‘0) = 𝑃 → (((𝑥 prefix (𝑁 + 1))‘0) = (𝑥‘0) ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
7	6	adantl 482	. . . . . 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 231	. . . . 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 28855	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
11	10	ancoms 459	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
12	11	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸)
13		simpr 485	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
14	13	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → 𝑥 ∈ 𝑋)
15		simpr 485	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥‘0) = 𝑃)
16		simpll 765	. . . . . . 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 28856	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋 ∧ (𝑥‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
19	14, 15, 16, 18	syl3anc 1371	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ (𝑥‘0) = 𝑃) → (𝑥 prefix (𝑁 + 1)) ∈ 𝑌)
20		eqeq2 2748	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑥 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1))))
21		fveq1 6841	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
22	21	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
23		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑥 prefix (𝑁 + 1))))
24	23	preq1d 4700	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑥)} = {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)})
25	24	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = (𝑥 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸 ↔ {(lastS‘(𝑥 prefix (𝑁 + 1))), (lastS‘𝑥)} ∈ 𝐸))
26	20, 22, 25	3anbi123d 1436	. . . . . . 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 482	. . . . . 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 3574	. . . . 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 1464	. . . 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 413	. . 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 2744	. . . . . . . . 9 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → (𝑦‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
32	31	eqeq1d 2738	. . . . . . . 8 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 ↔ ((𝑥 prefix (𝑁 + 1))‘0) = 𝑃))
33	3	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
34	33	ancoms 459	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
35	34	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0))
36		eqeq2 2748	. . . . . . . . . 10 ⊢ (𝑃 = ((𝑥 prefix (𝑁 + 1))‘0) → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
37	36	eqcoms 2744	. . . . . . . . 9 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → ((𝑥‘0) = 𝑃 ↔ (𝑥‘0) = ((𝑥 prefix (𝑁 + 1))‘0)))
38	35, 37	syl5ibr 245	. . . . . . . 8 ⊢ (((𝑥 prefix (𝑁 + 1))‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃))
39	32, 38	syl6bi 252	. . . . . . 7 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑦 → ((𝑦‘0) = 𝑃 → (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝑥‘0) = 𝑃)))
40	39	imp 407	. . . . . 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 3152	. . 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 211	. 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 3414	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3073  {crab 3407  {cpr 4588  ‘cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  ℕ₀cn0 12413  lastSclsw 14450   prefix cpfx 14558  Edgcedg 27998   WWalksN cwwlksn 28771

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-lsw 14451  df-substr 14529  df-pfx 14559  df-wwlks 28775  df-wwlksn 28776

This theorem is referenced by: (None)