numclwlk1lem2f1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > numclwlk1lem2f1		Structured version Visualization version GIF version

Theorem numclwlk1lem2f1 27076

Description: 𝑇 is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Revised by AV, 29-May-2021.)

**Hypotheses**
Ref	Expression
extwwlkfab.v	⊢ 𝑉 = (Vtx‘𝐺)
extwwlkfab.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
extwwlkfab.c	⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
numclwwlk.t	⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)

**Assertion**
Ref	Expression
numclwlk1lem2f1	⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐶𝑁)–1-1→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))

Distinct variable groups: 𝑛,𝐺,𝑣,𝑤 𝑛,𝑁,𝑣,𝑤 𝑛,𝑉,𝑣,𝑤 𝑛,𝑋,𝑣,𝑤 𝑤,𝐹 𝑤,𝐶 Allowed substitution hints: 𝐶(𝑣,𝑛) 𝑇(𝑤,𝑣,𝑛) 𝐹(𝑣,𝑛)

Proof of Theorem numclwlk1lem2f1 Dummy variables 𝑢 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		extwwlkfab.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		extwwlkfab.f	. . 3 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
3		extwwlkfab.c	. . 3 ⊢ 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})
4		numclwwlk.t	. . 3 ⊢ 𝑇 = (𝑤 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)
5	1, 2, 3, 4	numclwlk1lem2f 27074	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐶𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))
6	1, 2, 3, 4	numclwlk1lem2fv 27075	. . . . . 6 ⊢ (𝑝 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑝) = ⟨(𝑝 substr ⟨0, (𝑁 − 2)⟩), (𝑝‘(𝑁 − 1))⟩)
7	6	ad2antrl 763	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → (𝑇‘𝑝) = ⟨(𝑝 substr ⟨0, (𝑁 − 2)⟩), (𝑝‘(𝑁 − 1))⟩)
8	1, 2, 3, 4	numclwlk1lem2fv 27075	. . . . . 6 ⊢ (𝑢 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑢) = ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩)
9	8	ad2antll 764	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → (𝑇‘𝑢) = ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩)
10	7, 9	eqeq12d 2641	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → ((𝑇‘𝑝) = (𝑇‘𝑢) ↔ ⟨(𝑝 substr ⟨0, (𝑁 − 2)⟩), (𝑝‘(𝑁 − 1))⟩ = ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩))
11		ovex 6633	. . . . . 6 ⊢ (𝑝 substr ⟨0, (𝑁 − 2)⟩) ∈ V
12		fvex 6160	. . . . . 6 ⊢ (𝑝‘(𝑁 − 1)) ∈ V
13	11, 12	opth 4910	. . . . 5 ⊢ (⟨(𝑝 substr ⟨0, (𝑁 − 2)⟩), (𝑝‘(𝑁 − 1))⟩ = ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩ ↔ ((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))))
14		uzuzle23 11673	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ (ℤ_≥‘2))
15	3	numclwwlkovgel 27071	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑝 ∈ (𝑋𝐶𝑁) ↔ (𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))))
16	14, 15	sylan2 491	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑝 ∈ (𝑋𝐶𝑁) ↔ (𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))))
17	1	clwwlknbp 26746	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (𝑁 ClWWalksN 𝐺) → (𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁))
18	17	3ad2ant1 1080	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → (𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁))
19		3simpc 1058	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)))
20	18, 19	jca 554	. . . . . . . . 9 ⊢ ((𝑝 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))))
21	16, 20	syl6bi 243	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑝 ∈ (𝑋𝐶𝑁) → ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)))))
22	21	3adant1 1077	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑝 ∈ (𝑋𝐶𝑁) → ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)))))
23	3	numclwwlkovgel 27071	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑢 ∈ (𝑋𝐶𝑁) ↔ (𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))
24	14, 23	sylan2 491	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑢 ∈ (𝑋𝐶𝑁) ↔ (𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))
25	1	clwwlknbp 26746	. . . . . . . . . . 11 ⊢ (𝑢 ∈ (𝑁 ClWWalksN 𝐺) → (𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁))
26	25	3ad2ant1 1080	. . . . . . . . . 10 ⊢ ((𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)) → (𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁))
27		3simpc 1058	. . . . . . . . . 10 ⊢ ((𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)) → ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))
28	26, 27	jca 554	. . . . . . . . 9 ⊢ ((𝑢 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)) → ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))
29	24, 28	syl6bi 243	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑢 ∈ (𝑋𝐶𝑁) → ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))))
30	29	3adant1 1077	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑢 ∈ (𝑋𝐶𝑁) → ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))))
31		simpll 789	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) → 𝑝 ∈ Word 𝑉)
32	31	ad2antrl 763	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → 𝑝 ∈ Word 𝑉)
33		simprll 801	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → 𝑢 ∈ Word 𝑉)
34	33	adantl 482	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → 𝑢 ∈ Word 𝑉)
35		eleq1 2692	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (#‘𝑝) → (𝑁 ∈ (ℤ_≥‘3) ↔ (#‘𝑝) ∈ (ℤ_≥‘3)))
36	35	eqcoms 2634	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑝) = 𝑁 → (𝑁 ∈ (ℤ_≥‘3) ↔ (#‘𝑝) ∈ (ℤ_≥‘3)))
37		eluz2 11637	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑝) ∈ (ℤ_≥‘3) ↔ (3 ∈ ℤ ∧ (#‘𝑝) ∈ ℤ ∧ 3 ≤ (#‘𝑝)))
38		1red 10000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑝) ∈ ℤ → 1 ∈ ℝ)
39		3re 11039	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 3 ∈ ℝ
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑝) ∈ ℤ → 3 ∈ ℝ)
41		zre 11326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑝) ∈ ℤ → (#‘𝑝) ∈ ℝ)
42	38, 40, 41	3jca 1240	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑝) ∈ ℤ → (1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (#‘𝑝) ∈ ℝ))
43		1lt3 11141	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 < 3
44		ltletr 10074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (#‘𝑝) ∈ ℝ) → ((1 < 3 ∧ 3 ≤ (#‘𝑝)) → 1 < (#‘𝑝)))
45	44	expd 452	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ (#‘𝑝) ∈ ℝ) → (1 < 3 → (3 ≤ (#‘𝑝) → 1 < (#‘𝑝))))
46	42, 43, 45	mpisyl 21	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑝) ∈ ℤ → (3 ≤ (#‘𝑝) → 1 < (#‘𝑝)))
47	46	imp 445	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑝) ∈ ℤ ∧ 3 ≤ (#‘𝑝)) → 1 < (#‘𝑝))
48	47	3adant1 1077	. . . . . . . . . . . . . . . . 17 ⊢ ((3 ∈ ℤ ∧ (#‘𝑝) ∈ ℤ ∧ 3 ≤ (#‘𝑝)) → 1 < (#‘𝑝))
49	37, 48	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑝) ∈ (ℤ_≥‘3) → 1 < (#‘𝑝))
50	36, 49	syl6bi 243	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑝) = 𝑁 → (𝑁 ∈ (ℤ_≥‘3) → 1 < (#‘𝑝)))
51	50	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘3) → ((#‘𝑝) = 𝑁 → 1 < (#‘𝑝)))
52	51	3ad2ant3 1082	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((#‘𝑝) = 𝑁 → 1 < (#‘𝑝)))
53	52	com12 32	. . . . . . . . . . . 12 ⊢ ((#‘𝑝) = 𝑁 → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 1 < (#‘𝑝)))
54	53	ad3antlr 766	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 1 < (#‘𝑝)))
55	54	impcom 446	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → 1 < (#‘𝑝))
56		2swrd2eqwrdeq 13625	. . . . . . . . . 10 ⊢ ((𝑝 ∈ Word 𝑉 ∧ 𝑢 ∈ Word 𝑉 ∧ 1 < (#‘𝑝)) → (𝑝 = 𝑢 ↔ ((#‘𝑝) = (#‘𝑢) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)))))
57	32, 34, 55, 56	syl3anc 1323	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (𝑝 = 𝑢 ↔ ((#‘𝑝) = (#‘𝑢) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)))))
58		eqtr3 2647	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝑝) = 𝑁 ∧ (#‘𝑢) = 𝑁) → (#‘𝑝) = (#‘𝑢))
59	58	expcom 451	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑢) = 𝑁 → ((#‘𝑝) = 𝑁 → (#‘𝑝) = (#‘𝑢)))
60	59	ad2antlr 762	. . . . . . . . . . . . . 14 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))) → ((#‘𝑝) = 𝑁 → (#‘𝑝) = (#‘𝑢)))
61	60	com12 32	. . . . . . . . . . . . 13 ⊢ ((#‘𝑝) = 𝑁 → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))) → (#‘𝑝) = (#‘𝑢)))
62	61	ad2antlr 762	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) → (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))) → (#‘𝑝) = (#‘𝑢)))
63	62	imp 445	. . . . . . . . . . 11 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (#‘𝑝) = (#‘𝑢))
64	63	adantl 482	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (#‘𝑝) = (#‘𝑢))
65	64	biantrurd 529	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((#‘𝑝) = (#‘𝑢) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)))))
66		3anan12 1049	. . . . . . . . . . 11 ⊢ (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢))))
67	66	a1i 11	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)))))
68		eqeq2 2637	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘0) = 𝑋 → ((𝑝‘(𝑁 − 2)) = (𝑝‘0) ↔ (𝑝‘(𝑁 − 2)) = 𝑋))
69		oveq1 6612	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 = (#‘𝑝) → (𝑁 − 2) = ((#‘𝑝) − 2))
70	69	eqcoms 2634	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑝) = 𝑁 → (𝑁 − 2) = ((#‘𝑝) − 2))
71	70	fveq2d 6154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑝) = 𝑁 → (𝑝‘(𝑁 − 2)) = (𝑝‘((#‘𝑝) − 2)))
72	71	eqeq1d 2628	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑝) = 𝑁 → ((𝑝‘(𝑁 − 2)) = 𝑋 ↔ (𝑝‘((#‘𝑝) − 2)) = 𝑋))
73	72	biimpcd 239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘(𝑁 − 2)) = 𝑋 → ((#‘𝑝) = 𝑁 → (𝑝‘((#‘𝑝) − 2)) = 𝑋))
74	68, 73	syl6bi 243	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝‘0) = 𝑋 → ((𝑝‘(𝑁 − 2)) = (𝑝‘0) → ((#‘𝑝) = 𝑁 → (𝑝‘((#‘𝑝) − 2)) = 𝑋)))
75	74	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → ((#‘𝑝) = 𝑁 → (𝑝‘((#‘𝑝) − 2)) = 𝑋))
76	75	com12 32	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑝) = 𝑁 → (((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → (𝑝‘((#‘𝑝) − 2)) = 𝑋))
77	76	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) → (((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0)) → (𝑝‘((#‘𝑝) − 2)) = 𝑋))
78	77	imp 445	. . . . . . . . . . . . . 14 ⊢ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) → (𝑝‘((#‘𝑝) − 2)) = 𝑋)
79	78	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (𝑝‘((#‘𝑝) − 2)) = 𝑋)
80		oveq1 6612	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑝) = 𝑁 → ((#‘𝑝) − 2) = (𝑁 − 2))
81	80	fveq2d 6154	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑝) = 𝑁 → (𝑢‘((#‘𝑝) − 2)) = (𝑢‘(𝑁 − 2)))
82		eqeq1 2630	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢‘0) = (𝑢‘(𝑁 − 2)) → ((𝑢‘0) = 𝑋 ↔ (𝑢‘(𝑁 − 2)) = 𝑋))
83	82	eqcoms 2634	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢‘(𝑁 − 2)) = (𝑢‘0) → ((𝑢‘0) = 𝑋 ↔ (𝑢‘(𝑁 − 2)) = 𝑋))
84	83	biimpac 503	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)) → (𝑢‘(𝑁 − 2)) = 𝑋)
85	84	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))) → (𝑢‘(𝑁 − 2)) = 𝑋)
86	81, 85	sylan9eq 2680	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑝) = 𝑁 ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (𝑢‘((#‘𝑝) − 2)) = 𝑋)
87	86	ad4ant24 1295	. . . . . . . . . . . . 13 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (𝑢‘((#‘𝑝) − 2)) = 𝑋)
88	79, 87	eqtr4d 2663	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)))
89	88	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)))
90	89	biantrurd 529	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)))))
91	80	opeq2d 4382	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑝) = 𝑁 → ⟨0, ((#‘𝑝) − 2)⟩ = ⟨0, (𝑁 − 2)⟩)
92	91	oveq2d 6621	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑝) = 𝑁 → (𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑝 substr ⟨0, (𝑁 − 2)⟩))
93	91	oveq2d 6621	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑝) = 𝑁 → (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩))
94	92, 93	eqeq12d 2641	. . . . . . . . . . . . 13 ⊢ ((#‘𝑝) = 𝑁 → ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ↔ (𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩)))
95	94	ad3antlr 766	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ↔ (𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩)))
96	95	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → ((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ↔ (𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩)))
97		lsw 13285	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ Word 𝑉 → ( lastS ‘𝑝) = (𝑝‘((#‘𝑝) − 1)))
98		oveq1 6612	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑝) = 𝑁 → ((#‘𝑝) − 1) = (𝑁 − 1))
99	98	fveq2d 6154	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑝) = 𝑁 → (𝑝‘((#‘𝑝) − 1)) = (𝑝‘(𝑁 − 1)))
100	97, 99	sylan9eq 2680	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) → ( lastS ‘𝑝) = (𝑝‘(𝑁 − 1)))
101	100	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) → ( lastS ‘𝑝) = (𝑝‘(𝑁 − 1)))
102		lsw 13285	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 ∈ Word 𝑉 → ( lastS ‘𝑢) = (𝑢‘((#‘𝑢) − 1)))
103	102	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) → ( lastS ‘𝑢) = (𝑢‘((#‘𝑢) − 1)))
104		oveq1 6612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (#‘𝑢) → (𝑁 − 1) = ((#‘𝑢) − 1))
105	104	eqcoms 2634	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑢) = 𝑁 → (𝑁 − 1) = ((#‘𝑢) − 1))
106	105	fveq2d 6154	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑢) = 𝑁 → (𝑢‘(𝑁 − 1)) = (𝑢‘((#‘𝑢) − 1)))
107	106	eqeq2d 2636	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑢) = 𝑁 → (( lastS ‘𝑢) = (𝑢‘(𝑁 − 1)) ↔ ( lastS ‘𝑢) = (𝑢‘((#‘𝑢) − 1))))
108	107	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) → (( lastS ‘𝑢) = (𝑢‘(𝑁 − 1)) ↔ ( lastS ‘𝑢) = (𝑢‘((#‘𝑢) − 1))))
109	103, 108	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) → ( lastS ‘𝑢) = (𝑢‘(𝑁 − 1)))
110	109	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))) → ( lastS ‘𝑢) = (𝑢‘(𝑁 − 1)))
111	101, 110	eqeqan12d 2642	. . . . . . . . . . . 12 ⊢ ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (( lastS ‘𝑝) = ( lastS ‘𝑢) ↔ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))))
112	111	adantl 482	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (( lastS ‘𝑝) = ( lastS ‘𝑢) ↔ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))))
113	96, 112	anbi12d 746	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1)))))
114	67, 90, 113	3bitr2d 296	. . . . . . . . 9 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (((𝑝 substr ⟨0, ((#‘𝑝) − 2)⟩) = (𝑢 substr ⟨0, ((#‘𝑝) − 2)⟩) ∧ (𝑝‘((#‘𝑝) − 2)) = (𝑢‘((#‘𝑝) − 2)) ∧ ( lastS ‘𝑝) = ( lastS ‘𝑢)) ↔ ((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1)))))
115	57, 65, 114	3bitr2d 296	. . . . . . . 8 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0))))) → (𝑝 = 𝑢 ↔ ((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1)))))
116	115	exbiri 651	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((((𝑝 ∈ Word 𝑉 ∧ (#‘𝑝) = 𝑁) ∧ ((𝑝‘0) = 𝑋 ∧ (𝑝‘(𝑁 − 2)) = (𝑝‘0))) ∧ ((𝑢 ∈ Word 𝑉 ∧ (#‘𝑢) = 𝑁) ∧ ((𝑢‘0) = 𝑋 ∧ (𝑢‘(𝑁 − 2)) = (𝑢‘0)))) → (((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))) → 𝑝 = 𝑢)))
117	22, 30, 116	syl2and 500	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁)) → (((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))) → 𝑝 = 𝑢)))
118	117	imp 445	. . . . 5 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → (((𝑝 substr ⟨0, (𝑁 − 2)⟩) = (𝑢 substr ⟨0, (𝑁 − 2)⟩) ∧ (𝑝‘(𝑁 − 1)) = (𝑢‘(𝑁 − 1))) → 𝑝 = 𝑢))
119	13, 118	syl5bi 232	. . . 4 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → (⟨(𝑝 substr ⟨0, (𝑁 − 2)⟩), (𝑝‘(𝑁 − 1))⟩ = ⟨(𝑢 substr ⟨0, (𝑁 − 2)⟩), (𝑢‘(𝑁 − 1))⟩ → 𝑝 = 𝑢))
120	10, 119	sylbid 230	. . 3 ⊢ (((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) ∧ (𝑝 ∈ (𝑋𝐶𝑁) ∧ 𝑢 ∈ (𝑋𝐶𝑁))) → ((𝑇‘𝑝) = (𝑇‘𝑢) → 𝑝 = 𝑢))
121	120	ralrimivva 2970	. 2 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ∀𝑝 ∈ (𝑋𝐶𝑁)∀𝑢 ∈ (𝑋𝐶𝑁)((𝑇‘𝑝) = (𝑇‘𝑢) → 𝑝 = 𝑢))
122		dff13 6467	. 2 ⊢ (𝑇:(𝑋𝐶𝑁)–1-1→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)) ↔ (𝑇:(𝑋𝐶𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)) ∧ ∀𝑝 ∈ (𝑋𝐶𝑁)∀𝑢 ∈ (𝑋𝐶𝑁)((𝑇‘𝑝) = (𝑇‘𝑢) → 𝑝 = 𝑢)))
123	5, 121, 122	sylanbrc 697	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐶𝑁)–1-1→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1036 = wceq 1480 ∈ wcel 1992 ∀wral 2912 {crab 2916 ⟨cop 4159 class class class wbr 4618 ↦ cmpt 4678 × cxp 5077 ⟶wf 5846 –1-1→wf1 5847 ‘cfv 5850 (class class class)co 6605 ↦ cmpt2 6607 ℝcr 9880 0cc0 9881 1c1 9882 < clt 10019 ≤ cle 10020 − cmin 10211 ℕcn 10965 2c2 11015 3c3 11016 ℤcz 11322 ℤ_≥cuz 11631 #chash 13054 Word cword 13225 lastS clsw 13226 substr csubstr 13229 Vtxcvtx 25769 USGraph cusgr 25932 NeighbVtx cnbgr 26105 ClWWalksN cclwwlksn 26737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1841 ax-6 1890 ax-7 1937 ax-8 1994 ax-9 2001 ax-10 2021 ax-11 2036 ax-12 2049 ax-13 2250 ax-ext 2606 ax-rep 4736 ax-sep 4746 ax-nul 4754 ax-pow 4808 ax-pr 4872 ax-un 6903 ax-cnex 9937 ax-resscn 9938 ax-1cn 9939 ax-icn 9940 ax-addcl 9941 ax-addrcl 9942 ax-mulcl 9943 ax-mulrcl 9944 ax-mulcom 9945 ax-addass 9946 ax-mulass 9947 ax-distr 9948 ax-i2m1 9949 ax-1ne0 9950 ax-1rid 9951 ax-rnegex 9952 ax-rrecex 9953 ax-cnre 9954 ax-pre-lttri 9955 ax-pre-lttrn 9956 ax-pre-ltadd 9957 ax-pre-mulgt0 9958

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-fal 1486 df-ex 1702 df-nf 1707 df-sb 1883 df-eu 2478 df-mo 2479 df-clab 2613 df-cleq 2619 df-clel 2622 df-nfc 2756 df-ne 2797 df-nel 2900 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3193 df-sbc 3423 df-csb 3520 df-dif 3563 df-un 3565 df-in 3567 df-ss 3574 df-pss 3576 df-nul 3897 df-if 4064 df-pw 4137 df-sn 4154 df-pr 4156 df-tp 4158 df-op 4160 df-uni 4408 df-int 4446 df-iun 4492 df-br 4619 df-opab 4679 df-mpt 4680 df-tr 4718 df-eprel 4990 df-id 4994 df-po 5000 df-so 5001 df-fr 5038 df-we 5040 df-xp 5085 df-rel 5086 df-cnv 5087 df-co 5088 df-dm 5089 df-rn 5090 df-res 5091 df-ima 5092 df-pred 5642 df-ord 5688 df-on 5689 df-lim 5690 df-suc 5691 df-iota 5813 df-fun 5852 df-fn 5853 df-f 5854 df-f1 5855 df-fo 5856 df-f1o 5857 df-fv 5858 df-riota 6566 df-ov 6608 df-oprab 6609 df-mpt2 6610 df-om 7014 df-1st 7116 df-2nd 7117 df-wrecs 7353 df-recs 7414 df-rdg 7452 df-1o 7506 df-2o 7507 df-oadd 7510 df-er 7688 df-map 7805 df-pm 7806 df-en 7901 df-dom 7902 df-sdom 7903 df-fin 7904 df-card 8710 df-cda 8935 df-pnf 10021 df-mnf 10022 df-xr 10023 df-ltxr 10024 df-le 10025 df-sub 10213 df-neg 10214 df-nn 10966 df-2 11024 df-3 11025 df-n0 11238 df-z 11323 df-uz 11632 df-fz 12266 df-fzo 12404 df-hash 13055 df-word 13233 df-lsw 13234 df-concat 13235 df-s1 13236 df-substr 13237 df-s2 13525 df-edg 25835 df-upgr 25868 df-umgr 25869 df-usgr 25934 df-nbgr 26109 df-clwwlks 26738 df-clwwlksn 26739

This theorem is referenced by: numclwlk1lem2f1o 27078

W3C validator