MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wwlksnextinj Structured version   Visualization version   GIF version

Theorem wwlksnextinj 27043
Description: Lemma for wwlksnextbij 27046. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextbij0.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij.r 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij.f 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
Assertion
Ref Expression
wwlksnextinj (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 wwlksnextinj
Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef 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 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
4 wwlksnextbij.r . . 3 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
5 wwlksnextbij.f . . 3 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
61, 2, 3, 4, 5wwlksnextfun 27042 . 2 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
7 fveq2 6332 . . . . . . 7 (𝑡 = 𝑑 → (lastS‘𝑡) = (lastS‘𝑑))
8 fvex 6342 . . . . . . 7 (lastS‘𝑑) ∈ V
97, 5, 8fvmpt 6424 . . . . . 6 (𝑑𝐷 → (𝐹𝑑) = (lastS‘𝑑))
10 fveq2 6332 . . . . . . 7 (𝑡 = 𝑥 → (lastS‘𝑡) = (lastS‘𝑥))
11 fvex 6342 . . . . . . 7 (lastS‘𝑥) ∈ V
1210, 5, 11fvmpt 6424 . . . . . 6 (𝑥𝐷 → (𝐹𝑥) = (lastS‘𝑥))
139, 12eqeqan12d 2787 . . . . 5 ((𝑑𝐷𝑥𝐷) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
1413adantl 467 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) ↔ (lastS‘𝑑) = (lastS‘𝑥)))
15 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑑 → (♯‘𝑤) = (♯‘𝑑))
1615eqeq1d 2773 . . . . . . . 8 (𝑤 = 𝑑 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑑) = (𝑁 + 2)))
17 oveq1 6800 . . . . . . . . 9 (𝑤 = 𝑑 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, (𝑁 + 1)⟩))
1817eqeq1d 2773 . . . . . . . 8 (𝑤 = 𝑑 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
19 fveq2 6332 . . . . . . . . . 10 (𝑤 = 𝑑 → (lastS‘𝑤) = (lastS‘𝑑))
2019preq2d 4411 . . . . . . . . 9 (𝑤 = 𝑑 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑑)})
2120eleq1d 2835 . . . . . . . 8 (𝑤 = 𝑑 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸))
2216, 18, 213anbi123d 1547 . . . . . . 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)))
2322, 3elrab2 3518 . . . . . 6 (𝑑𝐷 ↔ (𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)))
24 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
2524eqeq1d 2773 . . . . . . . 8 (𝑤 = 𝑥 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑥) = (𝑁 + 2)))
26 oveq1 6800 . . . . . . . . 9 (𝑤 = 𝑥 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
2726eqeq1d 2773 . . . . . . . 8 (𝑤 = 𝑥 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
28 fveq2 6332 . . . . . . . . . 10 (𝑤 = 𝑥 → (lastS‘𝑤) = (lastS‘𝑥))
2928preq2d 4411 . . . . . . . . 9 (𝑤 = 𝑥 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑥)})
3029eleq1d 2835 . . . . . . . 8 (𝑤 = 𝑥 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸))
3125, 27, 303anbi123d 1547 . . . . . . 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)))
3231, 3elrab2 3518 . . . . . 6 (𝑥𝐷 ↔ (𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)))
33 eqtr3 2792 . . . . . . . . . . . . . . . . 17 (((♯‘𝑑) = (𝑁 + 2) ∧ (♯‘𝑥) = (𝑁 + 2)) → (♯‘𝑑) = (♯‘𝑥))
3433expcom 398 . . . . . . . . . . . . . . . 16 ((♯‘𝑥) = (𝑁 + 2) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
35343ad2ant1 1127 . . . . . . . . . . . . . . 15 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3635adantl 467 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → ((♯‘𝑑) = (𝑁 + 2) → (♯‘𝑑) = (♯‘𝑥)))
3736com12 32 . . . . . . . . . . . . 13 ((♯‘𝑑) = (𝑁 + 2) → ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (♯‘𝑑) = (♯‘𝑥)))
38373ad2ant1 1127 . . . . . . . . . . . 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)))
3938adantl 467 . . . . . . . . . . 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)))
4039imp 393 . . . . . . . . . 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))
4140adantr 466 . . . . . . . . 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))
4241adantr 466 . . . . . . . 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))
43 simpr 471 . . . . . . . 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))
44 eqtr3 2792 . . . . . . . . . . . . . . . . . . . 20 (((𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, (𝑁 + 1)⟩))
45 1e2m1 11338 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1 = (2 − 1)
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 = (2 − 1))
4746oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → (𝑁 + 1) = (𝑁 + (2 − 1)))
48 nn0cn 11504 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
49 2cnd 11295 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 2 ∈ ℂ)
50 1cnd 10258 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
5148, 49, 50addsubassd 10614 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → ((𝑁 + 2) − 1) = (𝑁 + (2 − 1)))
5247, 51eqtr4d 2808 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → (𝑁 + 1) = ((𝑁 + 2) − 1))
5352adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((𝑁 + 2) − 1))
54 oveq1 6800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((♯‘𝑑) = (𝑁 + 2) → ((♯‘𝑑) − 1) = ((𝑁 + 2) − 1))
5554eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((♯‘𝑑) = (𝑁 + 2) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5655adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → ((𝑁 + 1) = ((♯‘𝑑) − 1) ↔ (𝑁 + 1) = ((𝑁 + 2) − 1)))
5753, 56mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑁 ∈ ℕ0 ∧ (♯‘𝑑) = (𝑁 + 2)) → (𝑁 + 1) = ((♯‘𝑑) − 1))
58 opeq2 4540 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 + 1) = ((♯‘𝑑) − 1) → ⟨0, (𝑁 + 1)⟩ = ⟨0, ((♯‘𝑑) − 1)⟩)
5958oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑑 substr ⟨0, (𝑁 + 1)⟩) = (𝑑 substr ⟨0, ((♯‘𝑑) − 1)⟩))
6058oveq2d 6809 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 + 1) = ((♯‘𝑑) − 1) → (𝑥 substr ⟨0, (𝑁 + 1)⟩) = (𝑥 substr ⟨0, ((♯‘𝑑) − 1)⟩))
6159, 60eqeq12d 2786 . . . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
6257, 61syl 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)⟩)))
6362biimpd 219 . . . . . . . . . . . . . . . . . . . . . 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)⟩)))
6463ex 397 . . . . . . . . . . . . . . . . . . . . 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)⟩))))
6564com13 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)⟩))))
6644, 65syl 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)⟩))))
6766ex 397 . . . . . . . . . . . . . . . . . 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)⟩)))))
6867com23 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)⟩)))))
6968impcom 394 . . . . . . . . . . . . . . . 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)⟩))))
7069com12 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)⟩))))
71703ad2ant2 1128 . . . . . . . . . . . . . 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)⟩))))
7271adantl 467 . . . . . . . . . . . . 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)⟩))))
7372com12 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)⟩))))
74733adant3 1126 . . . . . . . . . . 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)⟩))))
7574adantl 467 . . . . . . . . . 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)⟩))))
7675imp31 404 . . . . . . . . 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)⟩))
7776adantr 466 . . . . . . . 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)⟩))
78 simpl 468 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → 𝑑 ∈ Word 𝑉)
79 simpl 468 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → 𝑥 ∈ Word 𝑉)
8078, 79anim12i 592 . . . . . . . . . . . 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))
8180adantr 466 . . . . . . . . . . 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))
82 nn0re 11503 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
83 2re 11292 . . . . . . . . . . . . . . . . . . . . . 22 2 ∈ ℝ
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
85 nn0ge0 11520 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
86 2pos 11314 . . . . . . . . . . . . . . . . . . . . . 22 0 < 2
8786a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ0 → 0 < 2)
8882, 84, 85, 87addgegt0d 10803 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
8988adantl 467 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
90 breq2 4790 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑑) = (𝑁 + 2) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
9190adantr 466 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑑) ↔ 0 < (𝑁 + 2)))
9289, 91mpbird 247 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑑))
93 hashgt0n0 13358 . . . . . . . . . . . . . . . . . 18 ((𝑑 ∈ Word 𝑉 ∧ 0 < (♯‘𝑑)) → 𝑑 ≠ ∅)
9492, 93sylan2 572 . . . . . . . . . . . . . . . . 17 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑑 ≠ ∅)
9594exp32 407 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → ((♯‘𝑑) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9695com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑑) = (𝑁 + 2) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
97963ad2ant1 1127 . . . . . . . . . . . . . 14 (((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸) → (𝑑 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑑 ≠ ∅)))
9897impcom 394 . . . . . . . . . . . . 13 ((𝑑 ∈ Word 𝑉 ∧ ((♯‘𝑑) = (𝑁 + 2) ∧ (𝑑 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑑)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑑 ≠ ∅))
9998adantr 466 . . . . . . . . . . . 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 ≠ ∅))
10099imp 393 . . . . . . . . . . 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 ≠ ∅)
10188adantl 467 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (𝑁 + 2))
102 breq2 4790 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑥) = (𝑁 + 2) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
103102adantr 466 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → (0 < (♯‘𝑥) ↔ 0 < (𝑁 + 2)))
104101, 103mpbird 247 . . . . . . . . . . . . . . . . . 18 (((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0) → 0 < (♯‘𝑥))
105 hashgt0n0 13358 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ Word 𝑉 ∧ 0 < (♯‘𝑥)) → 𝑥 ≠ ∅)
106104, 105sylan2 572 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ 𝑁 ∈ ℕ0)) → 𝑥 ≠ ∅)
107106exp32 407 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word 𝑉 → ((♯‘𝑥) = (𝑁 + 2) → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
108107com12 32 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = (𝑁 + 2) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
1091083ad2ant1 1127 . . . . . . . . . . . . . 14 (((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 ∈ Word 𝑉 → (𝑁 ∈ ℕ0𝑥 ≠ ∅)))
110109impcom 394 . . . . . . . . . . . . 13 ((𝑥 ∈ Word 𝑉 ∧ ((♯‘𝑥) = (𝑁 + 2) ∧ (𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑥)} ∈ 𝐸)) → (𝑁 ∈ ℕ0𝑥 ≠ ∅))
111110adantl 467 . . . . . . . . . . . 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 ≠ ∅))
112111imp 393 . . . . . . . . . . 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 ≠ ∅)
11381, 100, 112jca32 499 . . . . . . . . . 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 ≠ ∅)))
114113adantr 466 . . . . . . . . 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 ≠ ∅)))
115 simpl 468 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑑 ∈ Word 𝑉)
116115adantr 466 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑑 ∈ Word 𝑉)
117 simpr 471 . . . . . . . . . . . 12 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 𝑥 ∈ Word 𝑉)
118117adantr 466 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 𝑥 ∈ Word 𝑉)
119 hashneq0 13357 . . . . . . . . . . . . . . . 16 (𝑑 ∈ Word 𝑉 → (0 < (♯‘𝑑) ↔ 𝑑 ≠ ∅))
120119biimprd 238 . . . . . . . . . . . . . . 15 (𝑑 ∈ Word 𝑉 → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
121120adantr 466 . . . . . . . . . . . . . 14 ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → (𝑑 ≠ ∅ → 0 < (♯‘𝑑)))
122121com12 32 . . . . . . . . . . . . 13 (𝑑 ≠ ∅ → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
123122adantr 466 . . . . . . . . . . . 12 ((𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅) → ((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) → 0 < (♯‘𝑑)))
124123impcom 394 . . . . . . . . . . 11 (((𝑑 ∈ Word 𝑉𝑥 ∈ Word 𝑉) ∧ (𝑑 ≠ ∅ ∧ 𝑥 ≠ ∅)) → 0 < (♯‘𝑑))
125 2swrd1eqwrdeq 13663 . . . . . . . . . . 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)))))
126116, 118, 124, 125syl3anc 1476 . . . . . . . . . 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)))))
127 ancom 452 . . . . . . . . . . . 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)⟩)))
128127anbi2i 601 . . . . . . . . . . 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)⟩))))
129 3anass 1080 . . . . . . . . . . 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)⟩))))
130128, 129bitr4i 267 . . . . . . . . . 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)⟩)))
131126, 130syl6bb 276 . . . . . . . . 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)⟩))))
132114, 131syl 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)⟩))))
13342, 43, 77, 132mpbir3and 1427 . . . . . . 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)
134133exp31 406 . . . . . 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)))
13523, 32, 134syl2anb 577 . . . . 5 ((𝑑𝐷𝑥𝐷) → (𝑁 ∈ ℕ0 → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥)))
136135impcom 394 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((lastS‘𝑑) = (lastS‘𝑥) → 𝑑 = 𝑥))
13714, 136sylbid 230 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑑𝐷𝑥𝐷)) → ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
138137ralrimivva 3120 . 2 (𝑁 ∈ ℕ0 → ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥))
139 dff13 6655 . 2 (𝐹:𝐷1-1𝑅 ↔ (𝐹:𝐷𝑅 ∧ ∀𝑑𝐷𝑥𝐷 ((𝐹𝑑) = (𝐹𝑥) → 𝑑 = 𝑥)))
1406, 138, 139sylanbrc 564 1 (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  c0 4063  {cpr 4318  cop 4322   class class class wbr 4786  cmpt 4863  wf 6027  1-1wf1 6028  cfv 6031  (class class class)co 6793  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276  cmin 10468  2c2 11272  0cn0 11494  chash 13321  Word cword 13487  lastSclsw 13488   substr csubstr 13491  Vtxcvtx 26095  Edgcedg 26160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-s1 13498  df-substr 13499
This theorem is referenced by:  wwlksnextbij0  27045
  Copyright terms: Public domain W3C validator