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Theorem swrdspsleq 14601

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 14599	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
2	1	3expa 1119	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) ∧ 𝑁 ≤ 𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
3	2	ancoms 458	. . . 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 4453	. . . . . . 7 ⊢ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑈‘𝑖)
6		nn0z 12524	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
7		nn0z 12524	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
8		fzon 13608	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
9	6, 7, 8	syl2an 597	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 ↔ (𝑀..^𝑁) = ∅))
10	9	biimpa 476	. . . . . . . 8 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (𝑀..^𝑁) = ∅)
11	10	raleqdv 3298	. . . . . . 7 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → (∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝑊‘𝑖) = (𝑈‘𝑖)))
12	5, 11	mpbiri 258	. . . . . 6 ⊢ (((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑀) → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖))
13	12	ex 412	. . . . 5 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ 𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊‘𝑖) = (𝑈‘𝑖)))
14	13	3ad2ant2 1135	. . . 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 407	. . 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 265	. 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 14581	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
18		swrdcl 14581	. . . . . 6 ⊢ (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
19		eqwrd 14492	. . . . . 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 597	. . . . 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 1134	. . . 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 481	. . 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 14600	. . . . 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 481	. . . 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 532	. . 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 12422	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℝ)
27		nn0re 12422	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℝ)
28		ltnle 11224	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 ↔ ¬ 𝑁 ≤ 𝑀))
29		ltle 11233	. . . . . . . 8 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 → 𝑀 ≤ 𝑁))
30	28, 29	sylbird 260	. . . . . . 7 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
31	26, 27, 30	syl2an 597	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (¬ 𝑁 ≤ 𝑀 → 𝑀 ≤ 𝑁))
32	31	3ad2ant2 1135	. . . . 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 614	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
36	35	3ad2ant2 1135	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
37	36	anim1i 616	. . . . . . . . . . . . 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 1089	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 ≤ 𝑁))
39	37, 38	sylibr 234	. . . . . . . . . . . 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 12769	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁))
41	39, 40	sylibr 234	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
42	34, 41	jca 511	. . . . . . . . . 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 14596	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) ∧ 𝑁 ≤ (♯‘𝑊)) → (♯‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁 − 𝑀))
45	33, 42, 43, 44	syl3anc 1374	. . . . . . . . 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 7384	. . . . . . . 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 3298	. . . . . . 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 12512	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 0 ∈ ℤ)
49		zsubcl 12545	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ)
50	7, 6, 49	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 − 𝑀) ∈ ℤ)
51	50	3ad2ant2 1135	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (𝑁 − 𝑀) ∈ ℤ)
52	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℤ)
53	52	3ad2ant2 1135	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑁 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ ℤ)
54		fzoshftral 13715	. . . . . . . . 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 1374	. . . . . . . 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 480	. . . . . . 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 12423	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
58		nn0cn 12423	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
59		addlid 11328	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℂ → (0 + 𝑀) = 𝑀)
60	59	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (0 + 𝑀) = 𝑀)
61		npcan 11401	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
62	60, 61	oveq12d 7386	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
63	57, 58, 62	syl2anr 598	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((0 + 𝑀)..^((𝑁 − 𝑀) + 𝑀)) = (𝑀..^𝑁))
64	63	3ad2ant2 1135	. . . . . . . . . 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 480	. . . . . . . . 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 3298	. . . . . . . 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 7401	. . . . . . . . . . 11 ⊢ (𝑖 − 𝑀) ∈ V
68		sbceqg 4366	. . . . . . . . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
70		csbvarg 4388	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌𝑗 = (𝑖 − 𝑀))
71	70	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
72	69, 71	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
73		csbfv2g 6888	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ⦋(𝑖 − 𝑀) / 𝑗⦌((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗))
74	70	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ ((𝑖 − 𝑀) ∈ V → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘⦋(𝑖 − 𝑀) / 𝑗⦌𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖 − 𝑀)))
75	73, 74	eqtrd 2772	. . . . . . . . . . . . 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 279	. . . . . . . . . . 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 1129	. . . . . . . . . . . 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 14597	. . . . . . . . . . . 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 581	. . . . . . . . . . 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 1129	. . . . . . . . . . . 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 14597	. . . . . . . . . . . 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 581	. . . . . . . . . . 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 279	. . . . . . . . 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 3159	. . . . . . . 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 279	. . . . . . 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 305	. . . . . 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 412	. . . . 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 407	. . 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 307	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 [wsbc 3742 ⦋csb 3851 ∅c0 4287 ⟨cop 4588 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝcr 11037 0cc0 11038 + caddc 11041 < clt 11178 ≤ cle 11179 − cmin 11376 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ..^cfzo 13582 ♯chash 14265 Word cword 14448 substr csubstr 14576

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-substr 14577

This theorem is referenced by: pfxsuffeqwrdeq 14633 clwwlkf1 30136

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