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Theorem wwlksnextinjOLD 27176
 Description: Obsolete version of wwlksnextinj 27171 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
wwlksnextbij0OLD.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0OLD.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0OLD.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0OLD.r 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0OLD.f 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
Assertion
Ref Expression
wwlksnextinjOLD (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
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 wwlksnextinjOLD
Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0OLD.v . . 3 𝑉 = (Vtx‘𝐺)
2 wwlksnextbij0OLD.e . . 3 𝐸 = (Edg‘𝐺)
3 wwlksnextbij0OLD.d . . 3 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
4 wwlksnextbij0OLD.r . . 3 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
5 wwlksnextbij0OLD.f . . 3 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
61, 2, 3, 4, 5wwlksnextfunOLD 27175 . 2 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
7 fveq2 6411 . . . . . . 7 (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8 fvex 6424 . . . . . . 7 (lastS‘𝑑) ∈ V
97, 5, 8fvmpt 6507 . . . . . 6 (𝑑𝐷 → (𝐹𝑑) = (lastS‘𝑑))
10 fveq2 6411 . . . . . . 7 (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11 fvex 6424 . . . . . . 7 (lastS‘𝑥) ∈ V
1210, 5, 11fvmpt 6507 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (lastS‘𝑥))
139, 12eqeqan12d 2815 . . . . 5 ((𝑑𝐷𝑥𝐷) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
1413adantl 474 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15 fveqeq2 6420 . . . . . . . 8 (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
16 oveq1 6885 . . . . . . . . 9 (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
1716eqeq1d 2801 . . . . . . . 8 (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
18 fveq2 6411 . . . . . . . . . 10 (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
1918preq2d 4464 . . . . . . . . 9 (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
2019eleq1d 2863 . . . . . . . 8 (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
2115, 17, 203anbi123d 1561 . . . . . . 7 (𝑤 = 𝑑 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
2221, 3elrab2 3560 . . . . . 6 (𝑑𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
23 fveqeq2 6420 . . . . . . . 8 (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
24 oveq1 6885 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2524eqeq1d 2801 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
26 fveq2 6411 . . . . . . . . . 10 (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
2726preq2d 4464 . . . . . . . . 9 (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
2827eleq1d 2863 . . . . . . . 8 (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
2923, 25, 283anbi123d 1561 . . . . . . 7 (𝑤 = 𝑥 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
3029, 3elrab2 3560 . . . . . 6 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
31 eqtr3 2820 . . . . . . . . . . . . . . . . 17 (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
3231expcom 403 . . . . . . . . . . . . . . . 16 ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
33323ad2ant1 1164 . . . . . . . . . . . . . . 15 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3433adantl 474 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3534com12 32 . . . . . . . . . . . . 13 ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
36353ad2ant1 1164 . . . . . . . . . . . 12 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
3736adantl 474 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
3837imp 396 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (♯‘𝑑) = (♯‘𝑥))
3938adantr 473 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (♯‘𝑑) = (♯‘𝑥))
4039adantr 473 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (♯‘𝑑) = (♯‘𝑥))
41 simpr 478 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (lastS‘𝑑) = (lastS‘𝑥))
42 eqtr3 2820 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
43 1e2m1 11447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 − 1)
4443a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 = (2 − 1))
4544oveq2d 6894 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
46 nn0cn 11591 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
47 2cnd 11391 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
48 1cnd 10323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
4946, 47, 48addsubassd 10704 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
5045, 49eqtr4d 2836 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
5150adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
52 oveq1 6885 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
5352eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5453adantl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5551, 54mpbird 249 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
56 opeq2 4594 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 + 1) = ((♯‘𝑑) − 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, ((♯‘𝑑) − 1)⟩)
5756oveq2d 6894 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩))
5856oveq2d 6894 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))
5957, 58eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 + 1) = ((♯‘𝑑) − 1) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ↔ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))
6055, 59syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) ↔ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))
6160biimpd 221 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))
6261ex 402 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
6362com13 88 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
6442, 63syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
6564ex 402 . . . . . . . . . . . . . . . . . 18 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))))
6665com23 86 . . . . . . . . . . . . . . . . 17 ((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))))
6766impcom 397 . . . . . . . . . . . . . . . 16 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
6867com12 32 . . . . . . . . . . . . . . 15 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
69683ad2ant2 1165 . . . . . . . . . . . . . 14 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
7069adantl 474 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
7170com12 32 . . . . . . . . . . . 12 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
72713adant3 1163 . . . . . . . . . . 11 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
7372adantl 474 . . . . . . . . . 10 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
7473imp31 409 . . . . . . . . 9 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))
7574adantr 473 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))
76 simpl 475 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
77 simpl 475 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
7876, 77anim12i 607 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
7978adantr 473 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → (𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉))
80 nn0re 11590 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
81 2re 11387 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8281a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
83 nn0ge0 11607 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
84 2pos 11423 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8584a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
8680, 82, 83, 85addgegt0d 10893 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
8786adantl 474 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
88 breq2 4847 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
8988adantr 473 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
9087, 89mpbird 249 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑑))
91 hashgt0n0 13406 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
9290, 91sylan2 587 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
9392exp32 412 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9493com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
95943ad2ant1 1164 . . . . . . . . . . . . . 14 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9695impcom 397 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9796adantr 473 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9897imp 396 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑑 ≠ ∅)
9986adantl 474 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
100 breq2 4847 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
101100adantr 473 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
10299, 101mpbird 249 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑥))
103 hashgt0n0 13406 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
104102, 103sylan2 587 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
105104exp32 412 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝑉 → ((♯‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
106105com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
1071063ad2ant1 1164 . . . . . . . . . . . . . 14 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
108107impcom 397 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
109108adantl 474 . . . . . . . . . . . 12 (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
110109imp 396 . . . . . . . . . . 11 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → 𝑥 ≠ ∅)
11179, 98, 110jca32 512 . . . . . . . . . 10 ((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
112111adantr 473 . . . . . . . . 9 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)))
113 simpl 475 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
114113adantr 473 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
115 simpr 478 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
116115adantr 473 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
117 hashneq0 13405 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
118117biimprd 240 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
119118adantr 473 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
120119com12 32 . . . . . . . . . . . . 13 (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
121120adantr 473 . . . . . . . . . . . 12 ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
122121impcom 397 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
123 2swrd1eqwrdeqOLD 13708 . . . . . . . . . . 11 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
124114, 116, 122, 123syl3anc 1491 . . . . . . . . . 10 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩) ∧ (lastS‘𝑑) = (lastS‘𝑥)))))
125 ancom 453 . . . . . . . . . . . 12 (((𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩) ∧ (lastS‘𝑑) = (lastS‘𝑥)) ↔ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))
126125anbi2i 617 . . . . . . . . . . 11 (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
127 3anass 1117 . . . . . . . . . . 11 (((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ ((lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
128126, 127bitr4i 270 . . . . . . . . . 10 (((♯‘𝑑) = (♯‘𝑥) ∧ ((𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩) ∧ (lastS‘𝑑) = (lastS‘𝑥))) ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩)))
129124, 128syl6bb 279 . . . . . . . . 9 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
130112, 129syl 17 . . . . . . . 8 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → (𝑑 = 𝑥 ↔ ((♯‘𝑑) = (♯‘𝑥) ∧ (lastS‘𝑑) = (lastS‘𝑥) ∧ (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))))
13140, 41, 75, 130mpbir3and 1443 . . . . . . 7 (((((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) ∧ 𝑁 ∈ ℕ0) ∧ (lastS‘𝑑) = (lastS‘𝑥)) → 𝑑 = 𝑥)
132131exp31 411 . . . . . 6 (((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) ∧ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
13322, 30, 132syl2anb 592 . . . . 5 ((𝑑𝐷𝑥𝐷) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
134133impcom 397 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
13514, 134sylbid 232 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
136135ralrimivva 3152 . 2 (𝑁 ∈ ℕ0 → ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
137 dff13 6740 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥)))
1386, 136, 137sylanbrc 579 1 (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157   ≠ wne 2971  ∀wral 3089  {crab 3093  ∅c0 4115  {cpr 4370  ⟨cop 4374   class class class wbr 4843   ↦ cmpt 4922  ⟶wf 6097  –1-1→wf1 6098  ‘cfv 6101  (class class class)co 6878  ℝcr 10223  0cc0 10224  1c1 10225   + caddc 10227   < clt 10363   − cmin 10556  2c2 11368  ℕ0cn0 11580  ♯chash 13370  Word cword 13534  lastSclsw 13582   substr csubstr 13664  Vtxcvtx 26231  Edgcedg 26282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-lsw 13583  df-s1 13616  df-substr 13665 This theorem is referenced by:  wwlksnextbij0OLD  27178
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