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Theorem wwlksnredwwlkn 28840

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 2737	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1)))
2		eqid 2736	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		wwlksnredwwlkn.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
4	2, 3	wwlknp 28788	. . . 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 769	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = ((𝑁 + 1) + 1))) → 𝑊 ∈ Word (Vtx‘𝐺))
6		peano2nn0 12453	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
7		peano2nn0 12453	. . . . . . . . . . . . 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 12452	. . . . . . . . . . . . . 14 ⊢ ((𝑁 + 1) ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
11	6, 10	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) + 1) ∈ ℕ)
12		nn0re 12422	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
13		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
14		peano2re 11328	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℝ → (𝑁 + 1) ∈ ℝ)
15		peano2re 11328	. . . . . . . . . . . . . . . . 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 12085	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < (𝑁 + 1))
20		nn0re 12422	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 + 1) ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
21	6, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℝ)
22	21	ltp1d 12085	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) < ((𝑁 + 1) + 1))
23		lttr 11231	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 < ((𝑁 + 1) + 1))
26		elfzo0 13613	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (0..^((𝑁 + 1) + 1)) ↔ (𝑁 ∈ ℕ₀ ∧ ((𝑁 + 1) + 1) ∈ ℕ ∧ 𝑁 < ((𝑁 + 1) + 1)))
27	9, 11, 25, 26	syl3anbrc 1343	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ (0..^((𝑁 + 1) + 1)))
28		fz0add1fz1 13642	. . . . . . . . . . . 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 7365	. . . . . . . . . . . . 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 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 1171	. . . . . . 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 14583	. . . . . . 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 14452	. . . . . . . 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 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 4702	. . . . 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 7364	. . . . . . . . . . 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 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 6846	. . . . . . . 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 4701	. . . . . . 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 12423	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
50		1cnd 11150	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
51	49, 50	pncand 11513	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
52	51	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘((𝑁 + 1) − 1)) = (𝑊‘𝑁))
53	6	nn0cnd 12475	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℂ)
54	53, 50	pncand 11513	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → (((𝑁 + 1) + 1) − 1) = (𝑁 + 1))
55	54	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (𝑊‘(((𝑁 + 1) + 1) − 1)) = (𝑊‘(𝑁 + 1)))
56	52, 55	preq12d 4702	. . . . . . . 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 2776	. . . . . 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 6842	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘𝑖) = (𝑊‘𝑁))
60		fvoveq1 7380	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑁 → (𝑊‘(𝑖 + 1)) = (𝑊‘(𝑁 + 1)))
61	59, 60	preq12d 4702	. . . . . . . . . . 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 3577	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(𝑁 + 1)) → (∀𝑖 ∈ (0..^(𝑁 + 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 → {(𝑊‘𝑁), (𝑊‘(𝑁 + 1))} ∈ 𝐸))
64		fzonn0p1 13649	. . . . . . . . 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 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 2838	. . . . 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 2838	. . . 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 28837	. . . . 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 2748	. . . . . 6 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → ((𝑊 prefix (𝑁 + 1)) = 𝑦 ↔ (𝑊 prefix (𝑁 + 1)) = (𝑊 prefix (𝑁 + 1))))
74		fveq2 6842	. . . . . . . 8 ⊢ (𝑦 = (𝑊 prefix (𝑁 + 1)) → (lastS‘𝑦) = (lastS‘(𝑊 prefix (𝑁 + 1))))
75	74	preq1d 4700	. . . . . . 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 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 3574	. . 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 697	. 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 {cpr 4588 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 − cmin 11385 ℕcn 12153 ℕ₀cn0 12413 ...cfz 13424 ..^cfzo 13567 ♯chash 14230 Word cword 14402 lastSclsw 14450 prefix cpfx 14558 Vtxcvtx 27947 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-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: wwlksnredwwlkn0 28841

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