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Theorem clwlkclwwlklem3 26902

Description: Lemma 3 for clwlkclwwlk 26903. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)

**Assertion**
Ref	Expression
clwlkclwwlklem3	⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))

Distinct variable groups: 𝑓,𝐸,𝑖 𝑃,𝑓,𝑖 𝑅,𝑓,𝑖 𝑓,𝑉,𝑖

**Proof of Theorem clwlkclwwlklem3**
Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → 𝐸:dom 𝐸–1-1→𝑅)
2		simp1 1061	. . . . . . . 8 ⊢ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → 𝑓 ∈ Word dom 𝐸)
3	2	adantr 481	. . . . . . 7 ⊢ (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → 𝑓 ∈ Word dom 𝐸)
4	1, 3	anim12i 590	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑓 ∈ Word dom 𝐸))
5		simp3 1063	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → 2 ≤ (#‘𝑃))
6		simpl2 1065	. . . . . . 7 ⊢ (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → 𝑃:(0...(#‘𝑓))⟶𝑉)
7	5, 6	anim12ci 591	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (𝑃:(0...(#‘𝑓))⟶𝑉 ∧ 2 ≤ (#‘𝑃)))
8		simp3 1063	. . . . . . . 8 ⊢ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
9	8	anim1i 592	. . . . . . 7 ⊢ (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → (∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))))
10	9	adantl 482	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))))
11		clwlkclwwlklem2 26901	. . . . . 6 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑓 ∈ Word dom 𝐸) ∧ (𝑃:(0...(#‘𝑓))⟶𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ (∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))
12	4, 7, 10, 11	syl3anc 1326	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))
13		lencl 13324	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ₀)
14		lencl 13324	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ Word dom 𝐸 → (#‘𝑓) ∈ ℕ₀)
15		ffz0hash 13231	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝑓) ∈ ℕ₀ ∧ 𝑃:(0...(#‘𝑓))⟶𝑉) → (#‘𝑃) = ((#‘𝑓) + 1))
16		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑃) = ((#‘𝑓) + 1) → ((#‘𝑃) − 1) = (((#‘𝑓) + 1) − 1))
17	16	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑃) = ((#‘𝑓) + 1) → (((#‘𝑃) − 1) − 0) = ((((#‘𝑓) + 1) − 1) − 0))
18		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑓) ∈ ℕ₀ → (#‘𝑓) ∈ ℂ)
19		peano2cn 10208	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((#‘𝑓) ∈ ℂ → ((#‘𝑓) + 1) ∈ ℂ)
20		peano2cnm 10347	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((#‘𝑓) + 1) ∈ ℂ → (((#‘𝑓) + 1) − 1) ∈ ℂ)
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑓) ∈ ℕ₀ → (((#‘𝑓) + 1) − 1) ∈ ℂ)
22	21	subid1d 10381	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑓) ∈ ℕ₀ → ((((#‘𝑓) + 1) − 1) − 0) = (((#‘𝑓) + 1) − 1))
23		1cnd 10056	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑓) ∈ ℕ₀ → 1 ∈ ℂ)
24	18, 23	pncand 10393	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑓) ∈ ℕ₀ → (((#‘𝑓) + 1) − 1) = (#‘𝑓))
25	22, 24	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑓) ∈ ℕ₀ → ((((#‘𝑓) + 1) − 1) − 0) = (#‘𝑓))
26	25	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) → ((((#‘𝑓) + 1) − 1) − 0) = (#‘𝑓))
27	17, 26	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → (((#‘𝑃) − 1) − 0) = (#‘𝑓))
28	27	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ((((#‘𝑃) − 1) − 0) − 1) = ((#‘𝑓) − 1))
29	28	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → (0..^((((#‘𝑃) − 1) − 0) − 1)) = (0..^((#‘𝑓) − 1)))
30	29	raleqdv 3144	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))
31		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((#‘𝑃) = ((#‘𝑓) + 1) → ((#‘𝑃) − 2) = (((#‘𝑓) + 1) − 2))
32		2cnd 11093	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑓) ∈ ℕ₀ → 2 ∈ ℂ)
33	18, 32, 23	subsub3d 10422	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑓) − (2 − 1)) = (((#‘𝑓) + 1) − 2))
34		2m1e1 11135	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (2 − 1) = 1
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((#‘𝑓) ∈ ℕ₀ → (2 − 1) = 1)
36	35	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑓) − (2 − 1)) = ((#‘𝑓) − 1))
37	33, 36	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((#‘𝑓) ∈ ℕ₀ → (((#‘𝑓) + 1) − 2) = ((#‘𝑓) − 1))
38	37	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) → (((#‘𝑓) + 1) − 2) = ((#‘𝑓) − 1))
39	31, 38	sylan9eqr 2678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ((#‘𝑃) − 2) = ((#‘𝑓) − 1))
40	39	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → (𝑃‘((#‘𝑃) − 2)) = (𝑃‘((#‘𝑓) − 1)))
41	40	preq1d 4274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} = {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)})
42	41	eleq1d 2686	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ({(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸 ↔ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))
43	30, 42	anbi12d 747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ((∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸) ↔ (∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))
44	43	anbi2d 740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
45		3anass 1042	. . . . . . . . . . . . . . . . . 18 ⊢ ((( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))
46	44, 45	syl6bbr 278	. . . . . . . . . . . . . . . . 17 ⊢ ((((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) ∧ (#‘𝑃) = ((#‘𝑓) + 1)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))
47	46	expcom 451	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑃) = ((#‘𝑓) + 1) → (((#‘𝑓) ∈ ℕ₀ ∧ (#‘𝑃) ∈ ℕ₀) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
48	47	expd 452	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑃) = ((#‘𝑓) + 1) → ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))))
49	15, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑓) ∈ ℕ₀ ∧ 𝑃:(0...(#‘𝑓))⟶𝑉) → ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))))
50	49	ex 450	. . . . . . . . . . . . 13 ⊢ ((#‘𝑓) ∈ ℕ₀ → (𝑃:(0...(#‘𝑓))⟶𝑉 → ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))))
51	50	com23 86	. . . . . . . . . . . 12 ⊢ ((#‘𝑓) ∈ ℕ₀ → ((#‘𝑓) ∈ ℕ₀ → (𝑃:(0...(#‘𝑓))⟶𝑉 → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))))
52	14, 14, 51	sylc 65	. . . . . . . . . . 11 ⊢ (𝑓 ∈ Word dom 𝐸 → (𝑃:(0...(#‘𝑓))⟶𝑉 → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))))
53	52	imp 445	. . . . . . . . . 10 ⊢ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉) → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
54	53	3adant3 1081	. . . . . . . . 9 ⊢ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
55	54	adantr 481	. . . . . . . 8 ⊢ (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → ((#‘𝑃) ∈ ℕ₀ → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
56	13, 55	syl5com 31	. . . . . . 7 ⊢ (𝑃 ∈ Word 𝑉 → (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
57	56	3ad2ant2 1083	. . . . . 6 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸))))
58	57	imp 445	. . . . 5 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝑓) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑓) − 1)), (𝑃‘0)} ∈ ran 𝐸)))
59	12, 58	mpbird 247	. . . 4 ⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ ((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)))
60	59	ex 450	. . 3 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))
61	60	exlimdv 1861	. 2 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))
62		clwlkclwwlklem1 26900	. 2 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))))
63	61, 62	impbid 202	1 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 {cpr 4179 class class class wbr 4653 dom cdm 5114 ran crn 5115 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 ≤ cle 10075 − cmin 10266 2c2 11070 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 lastS clsw 13292

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300

This theorem is referenced by: clwlkclwwlk 26903

W3C validator