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Theorem swrdspsleq 14230

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 14228	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
2	1	3expa 1120	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
3	2	ancoms 462	. . . 4 ⊢ ((𝑁 ≤ 𝑀 ∧ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀))) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
4	3	3adantr3 1173	. . 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 4424	. . . . . . 7 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑈‘𝑖)
6		nn0z 12200	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
7		nn0z 12200	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
8		fzon 13263	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
9	6, 7, 8	syl2an 599	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
10	9	biimpa 480	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑀..^𝑁) = ∅)
11	10	raleqdv 3325	. . . . . . 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 416	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
14	13	3ad2ant2 1136	. . . 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 411	. . 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 14210	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
18		swrdcl 14210	. . . . . 6 ⊢ (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
19		eqwrd 14112	. . . . . 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 599	. . . . 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 1135	. . . 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 485	. . 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 14229	. . . . 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 485	. . . 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 536	. . 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 12099	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
27		nn0re 12099	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
28		ltnle 10912	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀))
29		ltle 10921	. . . . . . . 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 599	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
32	31	3ad2ant2 1136	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
33		simpl1l 1226	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑊 ∈ Word 𝑉)
34		simpl2l 1228	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑀 ∈ ℕ₀)
35	6, 7	anim12i 616	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
36	35	3ad2ant2 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
37	36	anim1i 618	. . . . . . . . . . . . 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 1091	. . . . . . . . . . . . 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 12444	. . . . . . . . . . . 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 515	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)))
43		simpl3l 1230	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ≤ (♯‘𝑊))
44		swrdlen2 14225	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑊)) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁 − 𝑀))
45	33, 42, 43, 44	syl3anc 1373	. . . . . . . . 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 7229	. . . . . . . 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 3325	. . . . . . 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 12188	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 0 ∈ ℤ)
49		zsubcl 12219	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ)
50	7, 6, 49	syl2anr 600	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 − 𝑀) ∈ ℤ)
51	50	3ad2ant2 1136	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑁 − 𝑀) ∈ ℤ)
52	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℤ)
53	52	3ad2ant2 1136	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ ℤ)
54		fzoshftral 13359	. . . . . . . . 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 1373	. . . . . . . 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 484	. . . . . . 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 12100	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
58		nn0cn 12100	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
59		addid2 11015	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℂ → (0 + 𝑀) = 𝑀)
60	59	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (0 + 𝑀) = 𝑀)
61		npcan 11087	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
62	60, 61	oveq12d 7231	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
63	57, 58, 62	syl2anr 600	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
64	63	3ad2ant2 1136	. . . . . . . . . 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 484	. . . . . . . . 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 3325	. . . . . . . 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 7246	. . . . . . . . . . 11 ⊢ (𝑖 − 𝑀) ∈ V
68		sbceqg 4324	. . . . . . . . . . . 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 6761	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
70		csbvarg 4346	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌𝑗 = (𝑖 − 𝑀))
71	70	fveq2d 6721	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
72	69, 71	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
73		csbfv2g 6761	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
74	70	fveq2d 6721	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
75	73, 74	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
76	72, 75	eqeq12d 2753	. . . . . . . . . . . 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 13	. . . . . . . . . 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 1130	. . . . . . . . . . . 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 14226	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 1227	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑈 ∈ Word 𝑉)
83		simpl3r 1231	. . . . . . . . . . . . 13 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ≤ (♯‘𝑈))
84	82, 42, 83	3jca 1130	. . . . . . . . . . . 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 14226	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 2753	. . . . . . . . . 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 3117	. . . . . . . 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 416	. . . . 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 47	. . . 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 411	. . 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 812	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 [wsbc 3694 ⦋csb 3811 ∅c0 4237 ⟨cop 4547 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 + caddc 10732 < clt 10867 ≤ cle 10868 − cmin 11062 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 ..^cfzo 13238 ♯chash 13896 Word cword 14069 substr csubstr 14205

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-fzo 13239 df-hash 13897 df-word 14070 df-substr 14206

This theorem is referenced by: pfxsuffeqwrdeq 14263 clwwlkf1 28132

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