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Theorem wwlksnredwwlkn 29972

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 2738	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
2		eqid 2737	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		wwlksnredwwlkn.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	wwlknp 29920	. . . 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 771	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
6		peano2nn0 12445	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7		peano2nn0 12445	. . . . . . . . . . . . 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 12444	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
11	6, 10	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
12		nn0re 12414	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
13		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
14		peano2re 11310	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
15		peano2re 11310	. . . . . . . . . . . . . . . . 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 1129	. . . . . . . . . . . . . . 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 12076	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (𝑁 + 1))
20		nn0re 12414	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
21	6, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
22	21	ltp1d 12076	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) < ((𝑁 + 1) + 1))
23		lttr 11213	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < ((𝑁 + 1) + 1))
26		elfzo0 13620	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ ∧ 𝑁 < ((𝑁 + 1) + 1)))
27	9, 11, 25, 26	syl3anbrc 1345	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28		fz0add1fz1 13655	. . . . . . . . . . . 12 ⊢ ((((𝑁 + 1) + 1) ∈ ℕ₀ ∧ 𝑁 ∈ (0..^((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
29	8, 27, 28	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
31		oveq2 7368	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (1...(♯‘𝑊)) = (1...((𝑁 + 1) + 1)))
32	31	eleq2d 2823	. . . . . . . . . . . 12 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1)) → ((𝑁 + 1) ∈ (1...(♯‘𝑊)) ↔ (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1))))
34	33	adantl 481	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...(♯‘𝑊)))
36	5, 35	jca 511	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (𝑁 + 1) ∈ (1...(♯‘𝑊))))
37	36	3adantr3 1173	. . . . . . 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 14622	. . . . . . 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 14491	. . . . . . . 8 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
41	40	3ad2ant1 1134	. . . . . . 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 481	. . . . . 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 4699	. . . . 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 7367	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
45	44	3ad2ant2 1135	. . . . . . . . . 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 481	. . . . . . . . 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 6839	. . . . . . . 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 4698	. . . . . . 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 12415	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
50		1cnd 11131	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
51	49, 50	pncand 11497	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
52	51	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
53	6	nn0cnd 12468	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
54	53, 50	pncand 11497	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
55	54	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
56	52, 55	preq12d 4699	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → {(𝑊‘((𝑁 + 1) − 1)), (𝑊‘(((𝑁 + 1) + 1) − 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
57	56	adantr 480	. . . . . . 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 2772	. . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
60		fvoveq1 7383	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
61	59, 60	preq12d 4699	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
62	61	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
63	62	rspcv 3573	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
64		fzonn0p1 13662	. . . . . . . . 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 1136	. . . . . . 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 407	. . . . . 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 2837	. . . . 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 2837	. . . 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 594	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
71		wwlksnred 29969	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
72	71	imp 406	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺))
73		eqeq2 2749	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1))))
74		fveq2 6835	. . . . . . . 8 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑊 prefix (𝑁 + 1))))
75	74	preq1d 4697	. . . . . . 7 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑊)} = {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)})
76	75	eleq1d 2822	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸 ↔ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
77	73, 76	anbi12d 633	. . . . 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 481	. . . 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 3570	. . 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 700	. 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 412	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3061 {cpr 4583 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 < clt 11170 − cmin 11368 ℕcn 12149 ℕ₀cn0 12405 ...cfz 13427 ..^cfzo 13574 ♯chash 14257 Word cword 14440 lastSclsw 14489 prefix cpfx 14598 Vtxcvtx 29073 Edgcedg 29124 WWalksN cwwlksn 29903

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-lsw 14490 df-substr 14569 df-pfx 14599 df-wwlks 29907 df-wwlksn 29908

This theorem is referenced by: wwlksnredwwlkn0 29973

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