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Theorem swrdspsleq 14727

Description: Two words have a common subword (starting at the same position with the same length) iff they have the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Proof shortened by AV, 7-May-2020.)

**Assertion**
Ref	Expression
swrdspsleq	⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))

Distinct variable groups: 𝑖,𝑀 𝑖,𝑁 𝑈,𝑖 𝑖,𝑉 𝑖,𝑊

Proof of Theorem swrdspsleq Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdsb0eq 14725	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
2	1	3expa 1136	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
3	2	ancoms 464	. . . 4 ⊢ ((𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
4	3	3adantr3 1190	. . 3 ⊢ ((𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
5		ral0 4464	. . . . . . 7 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑈‘𝑖)
6		nn0z 12633	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
7		nn0z 12633	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
8		fzon 13728	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
9	6, 7, 8	syl2an 608	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
10	9	biimpa 482	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑀..^𝑁) = ∅)
11	10	raleqdv 3326	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑈‘𝑖)))
12	5, 11	mpbiri 261	. . . . . 6 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖))
13	12	ex 418	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
14	13	3ad2ant2 1152	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑁 ≤ 𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
15	14	impcom 413	. . 3 ⊢ ((𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖))
16	4, 15	2thd 268	. 2 ⊢ ((𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
17		swrdcl 14705	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
18		swrdcl 14705	. . . . . 6 ⊢ (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
19		eqwrd 14614	. . . . . 6 ⊢ (((𝑊 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ((♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)) ∧ ∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗))))
20	17, 18, 19	syl2an 608	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ((♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)) ∧ ∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗))))
21	20	3ad2ant1 1151	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ((♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)) ∧ ∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗))))
22	21	adantl 487	. . 3 ⊢ ((¬ 𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ((♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)) ∧ ∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗))))
23		swrdsbslen 14726	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
24	23	adantl 487	. . . 4 ⊢ ((¬ 𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
25	24	biantrurd 542	. . 3 ⊢ ((¬ 𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (♯‘(𝑈 substr ⟨𝑀, 𝑁⟩)) ∧ ∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗))))
26		nn0re 12531	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
27		nn0re 12531	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
28		ltnle 11307	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀))
29		ltle 11316	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 → 𝑀 ≤ 𝑁))
30	28, 29	sylbird 263	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
31	26, 27, 30	syl2an 608	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
32	31	3ad2ant2 1152	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
33		simpl1l 1243	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑊 ∈ Word 𝑉)
34		simpl2l 1245	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ₀)
35	6, 7	anim12i 625	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
36	35	3ad2ant2 1152	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
37	36	anim1i 627	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≤ 𝑁))
38		df-3an 1105	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≤ 𝑁))
39	37, 38	sylibr 237	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
40		eluz2 12886	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
41	39, 40	sylibr 237	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
42	34, 41	jca 521	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)))
43		simpl3l 1247	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ≤ (♯‘𝑊))
44		swrdlen2 14722	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑊)) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁 − 𝑀))
45	33, 42, 43, 44	syl3anc 1398	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁 − 𝑀))
46	45	oveq2d 7439	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩))) = (0..^(𝑁 − 𝑀)))
47	46	raleqdv 3326	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑗 ∈ (0..^(𝑁 − 𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
48		0zd 12621	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 0 ∈ ℤ)
49		zsubcl 12654	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ)
50	7, 6, 49	syl2anr 609	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 − 𝑀) ∈ ℤ)
51	50	3ad2ant2 1152	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑁 − 𝑀) ∈ ℤ)
52	6	adantr 486	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℤ)
53	52	3ad2ant2 1152	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ ℤ)
54		fzoshftral 13835	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 𝑀) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (∀𝑗 ∈ (0..^(𝑁 − 𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀))[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
55	48, 51, 53, 54	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (∀𝑗 ∈ (0..^(𝑁 − 𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀))[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
56	55	adantr 486	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑗 ∈ (0..^(𝑁 − 𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀))[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
57		nn0cn 12532	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
58		nn0cn 12532	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
59		addlid 11411	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℂ → (0 + 𝑀) = 𝑀)
60	59	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (0 + 𝑀) = 𝑀)
61		npcan 11484	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
62	60, 61	oveq12d 7441	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
63	57, 58, 62	syl2anr 609	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
64	63	3ad2ant2 1152	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
65	64	adantr 486	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
66	65	raleqdv 3326	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑖 ∈ ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀))[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
67		ovex 7456	. . . . . . . . . . 11 ⊢ (𝑖 − 𝑀) ∈ V
68		sbceqg 4380	. . . . . . . . . . . 12 ⊢ ((𝑖 − 𝑀) ∈ V → ([(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
69		csbfv2g 6934	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
70		csbvarg 4402	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌𝑗 = (𝑖 − 𝑀))
71	70	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
72	69, 71	eqtrd 2801	. . . . . . . . . . . . 13 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
73		csbfv2g 6934	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
74	70	fveq2d 6892	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
75	73, 74	eqtrd 2801	. . . . . . . . . . . . 13 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
76	72, 75	eqeq12d 2782	. . . . . . . . . . . 12 ⊢ ((𝑖 − 𝑀) ∈ V → (⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀))))
77	68, 76	bitrd 282	. . . . . . . . . . 11 ⊢ ((𝑖 − 𝑀) ∈ V → ([(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀))))
78	67, 77	mp1i 14	. . . . . . . . . 10 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ([(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀))))
79	33, 42, 43	3jca 1146	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑊)))
80		swrdfv2 14723	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑊)) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = (𝑊‘𝑖))
81	79, 80	sylan 592	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = (𝑊‘𝑖))
82		simpl1r 1244	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑈 ∈ Word 𝑉)
83		simpl3r 1248	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ≤ (♯‘𝑈))
84	82, 42, 83	3jca 1146	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (𝑈 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑈)))
85		swrdfv2 14723	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑈)) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = (𝑈‘𝑖))
86	84, 85	sylan 592	. . . . . . . . . . 11 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = (𝑈‘𝑖))
87	81, 86	eqeq12d 2782	. . . . . . . . . 10 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → (((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)) ↔ (𝑊‘𝑖) = (𝑈‘𝑖)))
88	78, 87	bitrd 282	. . . . . . . . 9 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ([(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ (𝑊‘𝑖) = (𝑈‘𝑖)))
89	88	ralbidva 3189	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑖 ∈ (𝑀..^𝑁)[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
90	66, 89	bitrd 282	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑖 ∈ ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀))[(𝑖 − 𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
91	47, 56, 90	3bitrd 308	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
92	91	ex 418	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑀 ≤ 𝑁 → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖))))
93	32, 92	syld 48	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (¬ 𝑁 ≤ 𝑀 → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖))))
94	93	impcom 413	. . 3 ⊢ ((¬ 𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → (∀𝑗 ∈ (0..^(♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
95	22, 25, 94	3bitr2d 310	. 2 ⊢ ((¬ 𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
96	16, 95	pm2.61ian 824	1 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3082 Vcvv 3458 [wsbc 3747 ⦋csb 3856 ∅c0 4289 ⟨cop 4600 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 ℝcr 11117 0cc0 11118 + caddc 11121 < clt 11261 ≤ cle 11262 − cmin 11459 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ..^cfzo 13701 ♯chash 14386 Word cword 14570 substr csubstr 14700

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-substr 14701

This theorem is used by: pfxsuffeqwrdeq 14759 clwwlkf1 30437

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