Theorem wwlksnextinj 29932

Description: Lemma for wwlksnextbij 29935. (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 29931	. 2 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
7		fveq2 6920	. . . . . . 7 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8		fvex 6933	. . . . . . 7 ⊢ (lastS‘𝑑) ∈ V
9	7, 5, 8	fvmpt 7029	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
10		fveq2 6920	. . . . . . 7 ⊢ (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11		fvex 6933	. . . . . . 7 ⊢ (lastS‘𝑥) ∈ V
12	10, 5, 11	fvmpt 7029	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (lastS‘𝑥))
13	9, 12	eqeqan12d 2754	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
14	13	adantl 481	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15		fveqeq2 6929	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16		oveq1 7455	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
17	16	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
18		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
19	18	preq2d 4765	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
20	19	eleq1d 2829	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
21	15, 17, 20	3anbi123d 1436	. . . . . . 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 3711	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23		fveqeq2 6929	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24		oveq1 7455	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
25	24	eqeq1d 2742	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑥 prefix (𝑁 + 1)) = 𝑊))
26		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
27	26	preq2d 4765	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
28	27	eleq1d 2829	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
29	23, 25, 28	3anbi123d 1436	. . . . . . 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 3711	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31		eqtr3 2766	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
32	31	expcom 413	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33	32	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
34	33	adantl 481	. . . . . . . . . . . . . 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 1133	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 406	. . . . . . . . . 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 480	. . . . . . . . 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 480	. . . . . . . 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 484	. . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → (𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
43		1e2m1 12420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 = (2 − 1)
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 1 = (2 − 1))
45	44	oveq2d 7464	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
46		nn0cn 12563	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ)
47		2cnd 12371	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
48		1cnd 11285	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
49	46, 47, 48	addsubassd 11667	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
50	45, 49	eqtr4d 2783	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
51	50	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52		oveq1 7455	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
53	52	eqeq2d 2751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
54	53	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
55	51, 54	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((♯‘𝑑) − 1)))
57		oveq2 7456	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
58	56, 57	eqeq12d 2756	. . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 1132	. . . . . . . . . . 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 481	. . . . . . . . . 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 417	. . . . . . . . 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 480	. . . . . . . 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 482	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
76		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
77	75, 76	anim12i 612	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 12562	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ)
80		2re 12367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
82		nn0ge0 12578	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
83		2pos 12396	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 < 2
84	83	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 → 0 < 2)
85	79, 81, 82, 84	addgegt0d 11863	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
86	85	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
87		breq2 5170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
88	87	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
89	86, 88	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑑))
90		hashgt0n0 14414	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
91	89, 90	sylan2 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
92	91	exp32 420	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅)))
95	94	impcom 407	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → 𝑑 ≠ ∅))
96	95	adantr 480	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
99		breq2 5170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
100	99	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
101	98, 100	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑥))
102		hashgt0n0 14414	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
103	101, 102	sylan2 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
104	103	exp32 420	. . . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅)))
107	106	impcom 407	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → 𝑥 ≠ ∅))
108	107	adantl 481	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 515	. . . . . . . . . 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 480	. . . . . . . . 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 482	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
113	112	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
114		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
115	114	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
116		hashneq0 14413	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
117	116	biimprd 248	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
118	117	adantr 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121	120	impcom 407	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
122		pfxsuff1eqwrdeq 14747	. . . . . . . . . . 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 1371	. . . . . . . . . 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 460	. . . . . . . . . . . 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 622	. . . . . . . . . . 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 1095	. . . . . . . . . . 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 278	. . . . . . . . . 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	bitrdi 287	. . . . . . . . 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 1342	. . . . . . 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 419	. . . . . 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 597	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
133	132	impcom 407	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
134	14, 133	sylbid 240	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
135	134	ralrimivva 3208	. 2 ⊢ (𝑁 ∈ ℕ0 → ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
136		dff13 7292	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥)))
137	6, 135, 136	sylanbrc 582	1 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  ∅c0 4352  {cpr 4650   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   − cmin 11520  2c2 12348  ℕ₀cn0 12553  ♯chash 14379  Word cword 14562  lastSclsw 14610   prefix cpfx 14718  Vtxcvtx 29031  Edgcedg 29082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-lsw 14611  df-s1 14644  df-substr 14689  df-pfx 14719

This theorem is referenced by:  wwlksnextbij0  29934