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Theorem wwlksnextinj 29749

Description: Lemma for wwlksnextbij 29752. (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 29748	. 2 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷⟶𝑅)
7		fveq2 6890	. . . . . . 7 ⊢ (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8		fvex 6903	. . . . . . 7 ⊢ (lastS‘𝑑) ∈ V
9	7, 5, 8	fvmpt 6998	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 → (𝐹‘𝑑) = (lastS‘𝑑))
10		fveq2 6890	. . . . . . 7 ⊢ (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11		fvex 6903	. . . . . . 7 ⊢ (lastS‘𝑥) ∈ V
12	10, 5, 11	fvmpt 6998	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → (𝐹‘𝑥) = (lastS‘𝑥))
13	9, 12	eqeqan12d 2739	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
14	13	adantl 480	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15		fveqeq2 6899	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16		oveq1 7420	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → (𝑤 prefix (𝑁 + 1)) = (𝑑 prefix (𝑁 + 1)))
17	16	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑑 prefix (𝑁 + 1)) = 𝑊))
18		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
19	18	preq2d 4741	. . . . . . . . 9 ⊢ (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
20	19	eleq1d 2810	. . . . . . . 8 ⊢ (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
21	15, 17, 20	3anbi123d 1432	. . . . . . 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 3679	. . . . . 6 ⊢ (𝑑 ∈ 𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23		fveqeq2 6899	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24		oveq1 7420	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (𝑤 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
25	24	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑥 prefix (𝑁 + 1)) = 𝑊))
26		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
27	26	preq2d 4741	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
28	27	eleq1d 2810	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
29	23, 25, 28	3anbi123d 1432	. . . . . . 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 3679	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31		eqtr3 2751	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
32	31	expcom 412	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33	32	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
34	33	adantl 480	. . . . . . . . . . . . . 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 1130	. . . . . . . . . . . 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 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)) → (♯‘𝑑) = (♯‘𝑥)))
38	37	imp 405	. . . . . . . . . 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 479	. . . . . . . . 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 479	. . . . . . . 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 483	. . . . . . . 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 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊) → (𝑑 prefix (𝑁 + 1)) = (𝑥 prefix (𝑁 + 1)))
43		1e2m1 12364	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 1 = (2 − 1)
44	43	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 1 = (2 − 1))
45	44	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) = (𝑁 + (2 − 1)))
46		nn0cn 12507	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
47		2cnd 12315	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℂ)
48		1cnd 11234	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑁 ∈ ℕ₀ → 1 ∈ ℂ)
49	46, 47, 48	addsubassd 11616	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
50	45, 49	eqtr4d 2768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) = ((𝑁 + 2) − 1))
51	50	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52		oveq1 7420	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
53	52	eqeq2d 2736	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
54	53	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
55	51, 54	mpbird 256	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ ℕ₀ ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56		oveq2 7421	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 prefix (𝑁 + 1)) = (𝑑 prefix ((♯‘𝑑) − 1)))
57		oveq2 7421	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 prefix (𝑁 + 1)) = (𝑥 prefix ((♯‘𝑑) − 1)))
58	56, 57	eqeq12d 2741	. . . . . . . . . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 1129	. . . . . . . . . . 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 480	. . . . . . . . . 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 416	. . . . . . . . 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 479	. . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
76		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
77	75, 76	anim12i 611	. . . . . . . . . . . 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 479	. . . . . . . . . . 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 12506	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
80		2re 12311	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 2 ∈ ℝ
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 2 ∈ ℝ)
82		nn0ge0 12522	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 ≤ 𝑁)
83		2pos 12340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 < 2
84	83	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ₀ → 0 < 2)
85	79, 81, 82, 84	addgegt0d 11812	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 2))
86	85	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 2))
87		breq2 5148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
88	87	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
89	86, 88	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (♯‘𝑑))
90		hashgt0n0 14351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
91	89, 90	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀)) → 𝑑 ≠ ∅)
92	91	exp32 419	. . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅)))
95	94	impcom 406	. . . . . . . . . . . . 13 ⊢ ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → 𝑑 ≠ ∅))
96	95	adantr 479	. . . . . . . . . . . 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 405	. . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (𝑁 + 2))
99		breq2 5148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
100	99	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
101	98, 100	mpbird 256	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀) → 0 < (♯‘𝑥))
102		hashgt0n0 14351	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
103	101, 102	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ₀)) → 𝑥 ≠ ∅)
104	103	exp32 419	. . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅)))
107	106	impcom 406	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ₀ → 𝑥 ≠ ∅))
108	107	adantl 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 → 𝑥 ≠ ∅))
109	108	imp 405	. . . . . . . . . . 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 514	. . . . . . . . . 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 479	. . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
113	112	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
114		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
115	114	adantr 479	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
116		hashneq0 14350	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
117	116	biimprd 247	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
118	117	adantr 479	. . . . . . . . . . . . . 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 479	. . . . . . . . . . . 12 ⊢ ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121	120	impcom 406	. . . . . . . . . . 11 ⊢ (((𝑑 ∈ Word 𝑉 ∧ 𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
122		pfxsuff1eqwrdeq 14676	. . . . . . . . . . 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 1368	. . . . . . . . . 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 459	. . . . . . . . . . . 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 621	. . . . . . . . . . 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 1092	. . . . . . . . . . 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 277	. . . . . . . . . 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 286	. . . . . . . . 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 1339	. . . . . . 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 418	. . . . . 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 596	. . . . 5 ⊢ ((𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷) → (𝑁 ∈ ℕ₀ → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
133	132	impcom 406	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
134	14, 133	sylbid 239	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ (𝑑 ∈ 𝐷 ∧ 𝑥 ∈ 𝐷)) → ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
135	134	ralrimivva 3191	. 2 ⊢ (𝑁 ∈ ℕ₀ → ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥))
136		dff13 7259	. 2 ⊢ (𝐹:𝐷–1-1→𝑅 ↔ (𝐹:𝐷⟶𝑅 ∧ ∀𝑑 ∈ 𝐷 ∀𝑥 ∈ 𝐷 ((𝐹‘𝑑) = (𝐹‘𝑥) → 𝑑 = 𝑥)))
137	6, 135, 136	sylanbrc 581	1 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷–1-1→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ∀wral 3051 {crab 3419 ∅c0 4319 {cpr 4627 class class class wbr 5144 ↦ cmpt 5227 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7413 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 − cmin 11469 2c2 12292 ℕ₀cn0 12497 ♯chash 14316 Word cword 14491 lastSclsw 14539 prefix cpfx 14647 Vtxcvtx 28848 Edgcedg 28899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-lsw 14540 df-s1 14573 df-substr 14618 df-pfx 14648

This theorem is referenced by: wwlksnextbij0 29751

W3C validator