swrdeq - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > swrdeq		Structured version Visualization version GIF version

Theorem swrdeq 13444

Description: Two subwords of words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.)

**Assertion**
Ref	Expression
swrdeq	⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))

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

**Proof of Theorem swrdeq**
Step	Hyp	Ref	Expression
1		swrdcl 13419	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉)
2		swrdcl 13419	. . . . 5 ⊢ (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
3	1, 2	anim12i 590	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → ((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉))
4	3	3ad2ant1 1082	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉))
5		eqwrd 13346	. . 3 ⊢ (((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ ((#‘(𝑊 substr ⟨0, 𝑀⟩)) = (#‘(𝑈 substr ⟨0, 𝑁⟩)) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖))))
6	4, 5	syl 17	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ ((#‘(𝑊 substr ⟨0, 𝑀⟩)) = (#‘(𝑈 substr ⟨0, 𝑁⟩)) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖))))
7		simpl 473	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
8	7	3ad2ant1 1082	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑊 ∈ Word 𝑉)
9		simpl 473	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
10	9	3ad2ant2 1083	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ ℕ₀)
11		lencl 13324	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ₀)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (#‘𝑊) ∈ ℕ₀)
13	12	3ad2ant1 1082	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘𝑊) ∈ ℕ₀)
14		simpl 473	. . . . . . 7 ⊢ ((𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)) → 𝑀 ≤ (#‘𝑊))
15	14	3ad2ant3 1084	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ≤ (#‘𝑊))
16		elfz2nn0 12431	. . . . . 6 ⊢ (𝑀 ∈ (0...(#‘𝑊)) ↔ (𝑀 ∈ ℕ₀ ∧ (#‘𝑊) ∈ ℕ₀ ∧ 𝑀 ≤ (#‘𝑊)))
17	10, 13, 15, 16	syl3anbrc 1246	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ (0...(#‘𝑊)))
18		swrd0len 13422	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
19	8, 17, 18	syl2anc 693	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
20		simpr 477	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑈 ∈ Word 𝑉)
21	20	3ad2ant1 1082	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑈 ∈ Word 𝑉)
22		simpr 477	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
23	22	3ad2ant2 1083	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ∈ ℕ₀)
24		lencl 13324	. . . . . . . 8 ⊢ (𝑈 ∈ Word 𝑉 → (#‘𝑈) ∈ ℕ₀)
25	24	adantl 482	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (#‘𝑈) ∈ ℕ₀)
26	25	3ad2ant1 1082	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘𝑈) ∈ ℕ₀)
27		simpr 477	. . . . . . 7 ⊢ ((𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)) → 𝑁 ≤ (#‘𝑈))
28	27	3ad2ant3 1084	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ≤ (#‘𝑈))
29		elfz2nn0 12431	. . . . . 6 ⊢ (𝑁 ∈ (0...(#‘𝑈)) ↔ (𝑁 ∈ ℕ₀ ∧ (#‘𝑈) ∈ ℕ₀ ∧ 𝑁 ≤ (#‘𝑈)))
30	23, 26, 28, 29	syl3anbrc 1246	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ∈ (0...(#‘𝑈)))
31		swrd0len 13422	. . . . 5 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈))) → (#‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
32	21, 30, 31	syl2anc 693	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
33	19, 32	eqeq12d 2637	. . 3 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((#‘(𝑊 substr ⟨0, 𝑀⟩)) = (#‘(𝑈 substr ⟨0, 𝑁⟩)) ↔ 𝑀 = 𝑁))
34	33	anbi1d 741	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (((#‘(𝑊 substr ⟨0, 𝑀⟩)) = (#‘(𝑈 substr ⟨0, 𝑁⟩)) ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖))))
35	8	adantr 481	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → 𝑊 ∈ Word 𝑉)
36	17	adantr 481	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → 𝑀 ∈ (0...(#‘𝑊)))
37	35, 36, 18	syl2anc 693	. . . . . 6 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
38	37	oveq2d 6666	. . . . 5 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩))) = (0..^𝑀))
39	38	raleqdv 3144	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖)))
40	35	adantr 481	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ Word 𝑉)
41	36	adantr 481	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ (0...(#‘𝑊)))
42		simpr 477	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
43		swrd0fv 13439	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(#‘𝑊)) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = (𝑊‘𝑖))
44	40, 41, 42, 43	syl3anc 1326	. . . . . 6 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = (𝑊‘𝑖))
45		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
46	45	eleq2d 2687	. . . . . . . . . 10 ⊢ (𝑀 = 𝑁 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
47	46	adantl 482	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
48	21	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑈 ∈ Word 𝑉)
49	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ (0...(#‘𝑈)))
50		simpr 477	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^𝑁))
51	48, 49, 50	3jca 1242	. . . . . . . . . . 11 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)))
52	51	ex 450	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑖 ∈ (0..^𝑁) → (𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
53	52	adantr 481	. . . . . . . . 9 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑁) → (𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
54	47, 53	sylbid 230	. . . . . . . 8 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑀) → (𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
55	54	imp 445	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)))
56		swrd0fv 13439	. . . . . . 7 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) = (𝑈‘𝑖))
57	55, 56	syl 17	. . . . . 6 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) = (𝑈‘𝑖))
58	44, 57	eqeq12d 2637	. . . . 5 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ (𝑊‘𝑖) = (𝑈‘𝑖)))
59	58	ralbidva 2985	. . . 4 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^𝑀)((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖)))
60	39, 59	bitrd 268	. . 3 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖)))
61	60	pm5.32da 673	. 2 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩)))((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖)) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))
62	6, 34, 61	3bitrd 294	1 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊‘𝑖) = (𝑈‘𝑖))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⟨cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ≤ cle 10075 ℕ₀cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 substr csubstr 13295

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-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-substr 13303

This theorem is referenced by: clwlksf1clwwlklem 26968

W3C validator