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Theorem wwlksnredwwlkn 29877

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 2736	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
2		eqid 2735	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		wwlksnredwwlkn.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	wwlknp 29825	. . . 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 770	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
6		peano2nn0 12541	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7		peano2nn0 12541	. . . . . . . . . . . . 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 12540	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
11	6, 10	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
12		nn0re 12510	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
13		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
14		peano2re 11408	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
15		peano2re 11408	. . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . 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 12172	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (𝑁 + 1))
20		nn0re 12510	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
21	6, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
22	21	ltp1d 12172	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) < ((𝑁 + 1) + 1))
23		lttr 11311	. . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < ((𝑁 + 1) + 1))
26		elfzo0 13717	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ ∧ 𝑁 < ((𝑁 + 1) + 1)))
27	9, 11, 25, 26	syl3anbrc 1344	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28		fz0add1fz1 13751	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → (𝑁 + 1) ∈ (1...((𝑁 + 1) + 1)))
31		oveq2 7413	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → (1...(♯‘𝑊)) = (1...((𝑁 + 1) + 1)))
32	31	eleq2d 2820	. . . . . . . . . . . 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 1172	. . . . . . 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 14713	. . . . . . 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 14582	. . . . . . . 8 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
41	40	3ad2ant1 1133	. . . . . . 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 4717	. . . . 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 7412	. . . . . . . . . . 11 ⊢ ((♯‘𝑊) = ((𝑁 + 1) + 1) → ((♯‘𝑊) − 1) = (((𝑁 + 1) + 1) − 1))
45	44	3ad2ant2 1134	. . . . . . . . . 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 6880	. . . . . . . 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 4716	. . . . . . 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 12511	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
50		1cnd 11230	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
51	49, 50	pncand 11595	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
52	51	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
53	6	nn0cnd 12564	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
54	53, 50	pncand 11595	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
55	54	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
56	52, 55	preq12d 4717	. . . . . . . 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 2770	. . . . . 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 6876	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
60		fvoveq1 7428	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
61	59, 60	preq12d 4717	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))})
62	61	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ↔ {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
63	62	rspcv 3597	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
64		fzonn0p1 13758	. . . . . . . . 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 1135	. . . . . . 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 2834	. . . . 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 2834	. . . 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 29874	. . . . 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 2747	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1))))
74		fveq2 6876	. . . . . . . 8 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑊 prefix (𝑁 + 1))))
75	74	preq1d 4715	. . . . . . 7 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → {(lastS‘𝑦), (lastS‘𝑊)} = {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)})
76	75	eleq1d 2819	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ({(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸 ↔ {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸))
77	73, 76	anbi12d 632	. . . . 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 3594	. . 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 699	. 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 1086 = wceq 1540 ∈ wcel 2108 ∀wral 3051 ∃wrex 3060 {cpr 4603 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 < clt 11269 − cmin 11466 ℕcn 12240 ℕ₀cn0 12501 ...cfz 13524 ..^cfzo 13671 ♯chash 14348 Word cword 14531 lastSclsw 14580 prefix cpfx 14688 Vtxcvtx 28975 Edgcedg 29026 WWalksN cwwlksn 29808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-hash 14349 df-word 14532 df-lsw 14581 df-substr 14659 df-pfx 14689 df-wwlks 29812 df-wwlksn 29813

This theorem is referenced by: wwlksnredwwlkn0 29878

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