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Theorem wwlksnredwwlkn 28260

Description: For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)

**Hypothesis**
Ref	Expression
wwlksnredwwlkn.e	⊢ 𝐸 = (Edg‘𝐺)

**Assertion**
Ref	Expression
wwlksnredwwlkn	⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))

Distinct variable groups: 𝑦,𝐸 𝑦,𝐺 𝑦,𝑁 𝑦,𝑊

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

Step	Hyp	Ref	Expression
1		eqidd 2739	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
2		eqid 2738	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		wwlksnredwwlkn.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	wwlknp 28208	. . . 4 ⊢ (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
5		simprl 768	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
6		peano2nn0 12273	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7		peano2nn0 12273	. . . . . . . . . . . . 13 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ₀)
9		id 22	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℕ₀)
10		nn0p1nn 12272	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
11	6, 10	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
12		nn0re 12242	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
13		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
14		peano2re 11148	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
15		peano2re 11148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 + 1) ∈ ℝ → ((𝑁 + 1) + 1) ∈ ℝ)
16	14, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → ((𝑁 + 1) + 1) ∈ ℝ)
17	13, 14, 16	3jca 1127	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℝ → (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
18	12, 17	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ))
19	12	ltp1d 11905	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (𝑁 + 1))
20		nn0re 12242	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
21	6, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
22	21	ltp1d 11905	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) < ((𝑁 + 1) + 1))
23		lttr 11051	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) → ((𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1)) → 𝑁 < ((𝑁 + 1) + 1)))
24	23	imp 407	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ ∧ ((𝑁 + 1) + 1) ∈ ℝ) ∧ (𝑁 < (𝑁 + 1) ∧ (𝑁 + 1) < ((𝑁 + 1) + 1))) → 𝑁 < ((𝑁 + 1) + 1))
25	18, 19, 22, 24	syl12anc 834	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < ((𝑁 + 1) + 1))
26		elfzo0 13428	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ ∧ 𝑁 < ((𝑁 + 1) + 1)))
27	9, 11, 25, 26	syl3anbrc 1342	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28		fz0add1fz1 13457	. . . . . . . . . . . 12 ⊢ ((((𝑁 + 1) + 1) ∈ ℕ₀ ∧ 𝑁 ∈ (0..^((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
29	8, 27, 28	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
30	29	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
31		oveq2 7283	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (1...(♯‘𝑊)) = (1...((𝑁 + 1) + 1)))
32	31	eleq2d 2824	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
33	32	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
34	33	adantl 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
35	30, 34	mpbird 256	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
36	5, 35	jca 512	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
37	36	3adantr3 1170	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
38		pfxfvlsw 14408	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))) → (lastS‘(𝑊 prefix (𝑁 + 1))) = (𝑊‘((𝑁 + 1) − 1)))
39	37, 38	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → (lastS‘(𝑊 prefix (𝑁 + 1))) = (𝑊‘((𝑁 + 1) − 1)))
40		lsw 14267	. . . . . . . 8 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
41	40	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
42	41	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
43	39, 42	preq12d 4677	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} = {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((♯‘𝑊) − 1))})
44		oveq1 7282	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
45	44	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
46	45	adantl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
47	46	fveq2d 6778	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘(((𝑁 + 1) + 1) − 1)))
48	47	preq2d 4676	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((♯‘𝑊) − 1))} = {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))})
49		nn0cn 12243	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
50		1cnd 10970	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
51	49, 50	pncand 11333	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
52	51	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
53	6	nn0cnd 12295	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
54	53, 50	pncand 11333	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
55	54	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
56	52, 55	preq12d 4677	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
57	56	adantr 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
58	48, 57	eqtrd 2778	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((♯‘𝑊) − 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
59		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
60		fvoveq1 7298	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
61	59, 60	preq12d 4677	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
62	61	eleq1d 2823	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
63	62	rspcv 3557	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
64		fzonn0p1 13464	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^(𝑁 + 1)))
65	63, 64	syl11 33	. . . . . . . 8 ⊢ (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → (𝑁 ∈ ℕ₀ → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
66	65	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸) → (𝑁 ∈ ℕ₀ → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
67	66	impcom 408	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸)
68	58, 67	eqeltrd 2839	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘((♯‘𝑊) − 1))} ∈ 𝐸)
69	43, 68	eqeltrd 2839	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1) ∧ ∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸)) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
70	4, 69	sylan2 593	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
71		wwlksnred 28257	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
72	71	imp 407	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))
73		eqeq2 2750	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1))))
74		fveq2 6774	. . . . . . . 8 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑊 prefix (𝑁 + 1))))
75	74	preq1d 4675	. . . . . . 7 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑊)} = {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)})
76	75	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸 ↔ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
77	73, 76	anbi12d 631	. . . . 5 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) ↔ ((𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)) ∧ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)))
78	77	adantl 482	. . . 4 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) ∧ 𝑦 = (𝑊 prefix (𝑁 + 1))) → (((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸) ↔ ((𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)) ∧ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)))
79	72, 78	rspcedv 3554	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (((𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)) ∧ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
80	1, 70, 79	mp2and 696	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸))
81	80	ex 413	1 ⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {cpr 4563 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 < clt 11009 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ...cfz 13239 ..^cfzo 13382 ♯chash 14044 Word cword 14217 lastSclsw 14265 prefix cpfx 14383 Vtxcvtx 27366 Edgcedg 27417 WWalksN cwwlksn 28191

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-lsw 14266 df-substr 14354 df-pfx 14384 df-wwlks 28195 df-wwlksn 28196

This theorem is referenced by: wwlksnredwwlkn0 28261

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