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Theorem clwlkclwwlklem3 29522

Description: Lemma 3 for clwlkclwwlk 29523. (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 1135	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → 𝐸:dom 𝐸–1-1→𝑅)
2		simp1 1135	. . . . . . . 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 480	. . . . . . 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 612	. . . . . 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 1137	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → 2 ≤ (♯‘𝑃))
6		simpl2 1191	. . . . . . 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 613	. . . . . 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 1137	. . . . . . . 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 614	. . . . . . 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 481	. . . . . 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 29521	. . . . . 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 1370	. . . . 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 14488	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → (♯‘𝑃) ∈ ℕ₀)
14		lencl 14488	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ Word dom 𝐸 → (♯‘𝑓) ∈ ℕ₀)
15		ffz0hash 14411	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝑓) ∈ ℕ₀ ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉) → (♯‘𝑃) = ((♯‘𝑓) + 1))
16		oveq1 7419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑃) = ((♯‘𝑓) + 1) → ((♯‘𝑃) − 1) = (((♯‘𝑓) + 1) − 1))
17	16	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑃) = ((♯‘𝑓) + 1) → (((♯‘𝑃) − 1) − 0) = ((((♯‘𝑓) + 1) − 1) − 0))
18		nn0cn 12487	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑓) ∈ ℕ₀ → (♯‘𝑓) ∈ ℂ)
19		peano2cn 11391	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((♯‘𝑓) ∈ ℂ → ((♯‘𝑓) + 1) ∈ ℂ)
20		peano2cnm 11531	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘𝑓) + 1) ∈ ℂ → (((♯‘𝑓) + 1) − 1) ∈ ℂ)
21	18, 19, 20	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑓) ∈ ℕ₀ → (((♯‘𝑓) + 1) − 1) ∈ ℂ)
22	21	subid1d 11565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑓) ∈ ℕ₀ → ((((♯‘𝑓) + 1) − 1) − 0) = (((♯‘𝑓) + 1) − 1))
23		1cnd 11214	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑓) ∈ ℕ₀ → 1 ∈ ℂ)
24	18, 23	pncand 11577	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑓) ∈ ℕ₀ → (((♯‘𝑓) + 1) − 1) = (♯‘𝑓))
25	22, 24	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑓) ∈ ℕ₀ → ((((♯‘𝑓) + 1) − 1) − 0) = (♯‘𝑓))
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) → ((((♯‘𝑓) + 1) − 1) − 0) = (♯‘𝑓))
27	17, 26	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → (((♯‘𝑃) − 1) − 0) = (♯‘𝑓))
28	27	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → ((((♯‘𝑃) − 1) − 0) − 1) = ((♯‘𝑓) − 1))
29	28	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → (0..^((((♯‘𝑃) − 1) − 0) − 1)) = (0..^((♯‘𝑓) − 1)))
30	29	raleqdv 3324	. . . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((♯‘𝑃) = ((♯‘𝑓) + 1) → ((♯‘𝑃) − 2) = (((♯‘𝑓) + 1) − 2))
32		2cnd 12295	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑓) ∈ ℕ₀ → 2 ∈ ℂ)
33	18, 32, 23	subsub3d 11606	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑓) ∈ ℕ₀ → ((♯‘𝑓) − (2 − 1)) = (((♯‘𝑓) + 1) − 2))
34		2m1e1 12343	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (2 − 1) = 1
35	34	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((♯‘𝑓) ∈ ℕ₀ → (2 − 1) = 1)
36	35	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((♯‘𝑓) ∈ ℕ₀ → ((♯‘𝑓) − (2 − 1)) = ((♯‘𝑓) − 1))
37	33, 36	eqtr3d 2773	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((♯‘𝑓) ∈ ℕ₀ → (((♯‘𝑓) + 1) − 2) = ((♯‘𝑓) − 1))
38	37	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) → (((♯‘𝑓) + 1) − 2) = ((♯‘𝑓) − 1))
39	31, 38	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → ((♯‘𝑃) − 2) = ((♯‘𝑓) − 1))
40	39	fveq2d 6895	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → (𝑃‘((♯‘𝑃) − 2)) = (𝑃‘((♯‘𝑓) − 1)))
41	40	preq1d 4743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((♯‘𝑓) ∈ ℕ₀ ∧ (♯‘𝑃) ∈ ℕ₀) ∧ (♯‘𝑃) = ((♯‘𝑓) + 1)) → {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} = {(𝑃‘((♯‘𝑓) − 1)), (𝑃‘0)})
42	41	eleq1d 2817	. . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . 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 628	. . . . . . . . . . . . . . . . . 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 1094	. . . . . . . . . . . . . . . . . 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	bitr4di 289	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 406	. . . . . . . . . 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 1131	. . . . . . . . 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 480	. . . . . . . 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 1133	. . . . . 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 406	. . . . 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 257	. . . 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 412	. . 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 1935	. 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 29520	. 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 211	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 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∀wral 3060 {cpr 4630 class class class wbr 5148 dom cdm 5676 ran crn 5677 ⟶wf 6539 –1-1→wf1 6540 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 0cc0 11114 1c1 11115 + caddc 11117 ≤ cle 11254 − cmin 11449 2c2 12272 ℕ₀cn0 12477 ...cfz 13489 ..^cfzo 13632 ♯chash 14295 Word cword 14469 lastSclsw 14517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-hash 14296 df-word 14470 df-lsw 14518

This theorem is referenced by: clwlkclwwlk 29523

W3C validator