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Theorem wwlksnextproplem2 29153

Description: Lemma 2 for wwlksnextprop 29155. (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)

**Assertion**
Ref	Expression
wwlksnextproplem2	⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)

Proof of Theorem wwlksnextproplem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		wwlksnextprop.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	wwlknp 29086	. . . 4 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
4		fzonn0p1 13705	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(𝑁 + 1)))
5	4	adantl 482	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ (0..^(𝑁 + 1)))
6		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
7		fvoveq1 7428	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
8	6, 7	preq12d 4744	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
9	8	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
10	9	rspcv 3608	. . . . . . . . . 10 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
11	5, 10	syl 17	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
12	11	imp 407	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸)
13		simpll 765	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 𝑊 ∈ Word (Vtx‘𝐺))
14		1zzd 12589	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ∈ ℤ)
15		lencl 14479	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) ∈ ℕ₀)
16	15	nn0zd 12580	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) ∈ ℤ)
17	16	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (♯‘𝑊) ∈ ℤ)
18		peano2nn0 12508	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
19	18	nn0zd 12580	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℤ)
20	19	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ ℤ)
21		nn0ge0 12493	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
22		1red 11211	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℝ)
23		nn0re 12477	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
24	22, 23	addge02d 11799	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → (0 ≤ 𝑁 ↔ 1 ≤ (𝑁 + 1)))
25	21, 24	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 1 ≤ (𝑁 + 1))
26	25	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → 1 ≤ (𝑁 + 1))
27	18	nn0red 12529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
28	27	lep1d 12141	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ ((𝑁 + 1) + 1))
29		breq2 5151	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ≤ (♯‘𝑊) ↔ (𝑁 + 1) ≤ ((𝑁 + 1) + 1)))
30	28, 29	syl5ibrcom 246	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (♯‘𝑊)))
31	30	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ∈ ℕ₀ → (𝑁 ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 + 1) ≤ (♯‘𝑊))))
32	31	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑊) ∈ ℕ₀ → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ (♯‘𝑊))))
33	15, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑁 ∈ ℕ₀ → (𝑁 + 1) ≤ (♯‘𝑊))))
34	33	imp31 418	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ≤ (♯‘𝑊))
35	14, 17, 20, 26, 34	elfzd 13488	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
36		pfxfvlsw 14641	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix (𝑁 + 1))) = (𝑊‘((𝑁 + 1) − 1)))
37	13, 35, 36	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘(𝑊 prefix (𝑁 + 1))) = (𝑊‘((𝑁 + 1) − 1)))
38		nn0cn 12478	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
39		1cnd 11205	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
40	38, 39	pncand 11568	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
41	40	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
42	41	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
43	37, 42	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘(𝑊 prefix (𝑁 + 1))) = (𝑊‘𝑁))
44		lsw 14510	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
45	44	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
46		fvoveq1 7428	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
47	46	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
48	18	nn0cnd 12530	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
49	48, 39	pncand 11568	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
50	49	fveq2d 6892	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
51	47, 50	sylan9eq 2792	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(𝑁 + 1)))
52	45, 51	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → (lastS‘𝑊) = (𝑊‘(𝑁 + 1)))
53	43, 52	preq12d 4744	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
54	53	eleq1d 2818	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) → ({(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
55	54	adantr 481	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → ({(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
56	12, 55	mpbird 256	. . . . . . 7 ⊢ ((((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) ∧ 𝑁 ∈ ℕ₀) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
57	56	exp31 420	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (𝑁 ∈ ℕ₀ → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)))
58	57	com23 86	. . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)))
59	58	3impia 1117	. . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
60	3, 59	syl 17	. . 3 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
61		wwlksnextprop.x	. . 3 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
62	60, 61	eleq2s 2851	. 2 ⊢ (𝑊 ∈ 𝑋 → (𝑁 ∈ ℕ₀ → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
63	62	imp 407	1 ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ₀) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {cpr 4629 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11245 − cmin 11440 ℕ₀cn0 12468 ℤcz 12554 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 Word cword 14460 lastSclsw 14508 prefix cpfx 14616 Vtxcvtx 28245 Edgcedg 28296 WWalksN cwwlksn 29069

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-lsw 14509 df-substr 14587 df-pfx 14617 df-wwlks 29073 df-wwlksn 29074

This theorem is referenced by: wwlksnextprop 29155

W3C validator