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Theorem wwlksnextinj 27390

Description: Lemma for wwlksnextbij 27398. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)

**Hypotheses**
Ref	Expression
wwlksnextbij0.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e	⊢ 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d	⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0.r	⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.f	⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))

**Assertion**
Ref	Expression
wwlksnextinj	⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷–1-1→𝑅)

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑊 𝑡,𝐷 𝑛,𝐸 𝑤,𝐸 𝑡,𝑁,𝑤 𝑡,𝑅 𝑛,𝑉 𝑤,𝑉 𝑛,𝑊 𝑡,𝑛 Allowed substitution hints: 𝐷(𝑤,𝑛) 𝑅(𝑤,𝑛) 𝐸(𝑡) 𝐹(𝑤,𝑡,𝑛) 𝐺(𝑡,𝑛) 𝑁(𝑛) 𝑉(𝑡) 𝑊(𝑡)

Proof of Theorem wwlksnextinj Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		wwlksnextbij0.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
3		wwlksnextbij0.d	. . 3 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
4		wwlksnextbij0.r	. . 3 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
5		wwlksnextbij0.f	. . 3 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))
6	1, 2, 3, 4, 5	wwlksnextfun 27389	. 2 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷⟶𝑅)
7		fveq2 6499	. . . . . . 7 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8		fvex 6512	. . . . . . 7 ⊢ (lastS‘𝑑) ∈ V
9	7, 5, 8	fvmpt 6595	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
10		fveq2 6499	. . . . . . 7 ⊢ (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11		fvex 6512	. . . . . . 7 ⊢ (lastS‘𝑥) ∈ V
12	10, 5, 11	fvmpt 6595	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (lastS‘𝑥))
13	9, 12	eqeqan12d 2794	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
14	13	adantl 474	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15		fveqeq2 6508	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16		oveq1 6983	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
17	16	eqeq1d 2780	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
18		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
19	18	preq2d 4550	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
20	19	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
21	15, 17, 20	3anbi123d 1415	. . . . . . 7 ⊢ (𝑤 = 𝑑 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
22	21, 3	elrab2 3599	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23		fveqeq2 6508	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24		oveq1 6983	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
25	24	eqeq1d 2780	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑥 prefix (𝑁 + 1)) = 𝑊))
26		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
27	26	preq2d 4550	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
28	27	eleq1d 2850	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
29	23, 25, 28	3anbi123d 1415	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
30	29, 3	elrab2 3599	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31		eqtr3 2801	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
32	31	expcom 406	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33	32	3ad2ant1 1113	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
34	33	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
35	34	com12 32	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
36	35	3ad2ant1 1113	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
37	36	adantl 474	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
38	37	imp 398	. . . . . . . . . 10 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (♯‘𝑑) = (♯‘𝑥))
39	38	adantr 473	. . . . . . . . 9 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → (♯‘𝑑) = (♯‘𝑥))
40	39	adantr 473	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (♯‘𝑑) = (♯‘𝑥))
41		simpr 477	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (lastS‘𝑑) = (lastS‘𝑥))
42		eqtr3 2801	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → (𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
43		1e2m1 11574	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 = (2 − 1)
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 1 = (2 − 1))
45	44	oveq2d 6992	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) = (𝑁 + (2 − 1)))
46		nn0cn 11718	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
47		2cnd 11518	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℂ)
48		1cnd 10434	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
49	46, 47, 48	addsubassd 10818	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
50	45, 49	eqtr4d 2817	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((𝑁 + 2) − 1))
51	50	adantr 473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52		oveq1 6983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
53	52	eqeq2d 2788	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
54	53	adantl 474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
55	51, 54	mpbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56		oveq2 6984	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((♯‘𝑑) − 1)))
57		oveq2 6984	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
58	56, 57	eqeq12d 2793	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
59	55, 58	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) ↔ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
60	59	biimpd 221	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
61	60	ex 405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → ((♯‘𝑑) = (𝑁 + 2) → ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
62	61	com13 88	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
63	42, 62	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
64	63	ex 405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
65	64	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 prefix (𝑁 + 1)) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))))
66	65	impcom 399	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
67	66	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 prefix (𝑁 + 1)) = 𝑊 → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
68	67	3ad2ant2 1114	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
69	68	adantl 474	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
70	69	com12 32	. . . . . . . . . . . 12 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
71	70	3adant3 1112	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
72	71	adantl 474	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
73	72	imp31 410	. . . . . . . . 9 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
74	73	adantr 473	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
75		simpl 475	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
76		simpl 475	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
77	75, 76	anim12i 603	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉))
78	77	adantr 473	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → (𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉))
79		nn0re 11717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
80		2re 11514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
82		nn0ge0 11734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
83		2pos 11550	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 < 2
84	83	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 < 2)
85	79, 81, 82, 84	addgegt0d 11014	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 2))
86	85	adantl 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 2))
87		breq2 4933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
88	87	adantr 473	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
89	86, 88	mpbird 249	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (♯‘𝑑))
90		hashgt0n0 13541	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
91	89, 90	sylan2 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀)) → 𝑑 ≠ ∅)
92	91	exp32 413	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅)))
93	92	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅)))
94	93	3ad2ant1 1113	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅)))
95	94	impcom 399	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅))
96	95	adantr 473	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅))
97	96	imp 398	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → 𝑑 ≠ ∅)
98	85	adantl 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 2))
99		breq2 4933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
100	99	adantr 473	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
101	98, 100	mpbird 249	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (♯‘𝑥))
102		hashgt0n0 13541	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
103	101, 102	sylan2 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀)) → 𝑥 ≠ ∅)
104	103	exp32 413	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ Word 𝑉 → ((♯‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅)))
105	104	com12 32	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅)))
106	105	3ad2ant1 1113	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅)))
107	106	impcom 399	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅))
108	107	adantl 474	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅))
109	108	imp 398	. . . . . . . . . . 11 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → 𝑥 ≠ ∅)
110	78, 97, 109	jca32 508	. . . . . . . . . 10 ⊢ ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
111	110	adantr 473	. . . . . . . . 9 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
112		simpl 475	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
113	112	adantr 473	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
114		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
115	114	adantr 473	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
116		hashneq0 13540	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
117	116	biimprd 240	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
118	117	adantr 473	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
119	118	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
120	119	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121	120	impcom 399	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
122		pfxsuff1eqwrdeq 13881	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
123	113, 115, 121, 122	syl3anc 1351	. . . . . . . . . 10 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
124		ancom 453	. . . . . . . . . . . 12 ⊢ (((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥)) ↔ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
125	124	anbi2i 613	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
126		3anass 1076	. . . . . . . . . . 11 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
127	125, 126	bitr4i 270	. . . . . . . . . 10 ⊢ (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1))))
128	123, 127	syl6bb 279	. . . . . . . . 9 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
129	111, 128	syl 17	. . . . . . . 8 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 prefix ((♯‘𝑑) − 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))))
130	40, 41, 74, 129	mpbir3and 1322	. . . . . . 7 ⊢ (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ₀) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → 𝑑 = 𝑥)
131	130	exp31 412	. . . . . 6 ⊢ (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ₀ → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
132	22, 30, 131	syl2anb 588	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑁 ∈ ℕ₀ → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
133	132	impcom 399	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
134	14, 133	sylbid 232	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
135	134	ralrimivva 3141	. 2 ⊢ (𝑁 ∈ ℕ₀ → ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
136		dff13 6838	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥)))
137	6, 135, 136	sylanbrc 575	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷–1-1→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 {crab 3092 ∅c0 4178 {cpr 4443 class class class wbr 4929 ↦ cmpt 5008 ⟶wf 6184 –1-1→wf1 6185 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 < clt 10474 − cmin 10670 2c2 11495 ℕ₀cn0 11707 ♯chash 13505 Word cword 13672 lastSclsw 13725 prefix cpfx 13852 Vtxcvtx 26484 Edgcedg 26535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-n0 11708 df-xnn0 11780 df-z 11794 df-uz 12059 df-fz 12709 df-fzo 12850 df-hash 13506 df-word 13673 df-lsw 13726 df-s1 13759 df-substr 13804 df-pfx 13853

This theorem is referenced by: wwlksnextbij0 27392

W3C validator