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Theorem swrdeqOLD 13767

Description: Obsolete version of pfxeq 13809 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

**Assertion**
Ref	Expression
swrdeqOLD	⊢ (((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 swrdeqOLD**
Step	Hyp	Ref	Expression
1		swrdcl 13739	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉)
2		swrdcl 13739	. . . . 5 ⊢ (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
3	1, 2	anim12i 606	. . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → ((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉))
4	3	3ad2ant1 1124	. . 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 13651	. . 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 476	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
8	7	3ad2ant1 1124	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑊 ∈ Word 𝑉)
9		simpl 476	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑀 ∈ ℕ₀)
10	9	3ad2ant2 1125	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ ℕ₀)
11		lencl 13625	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (♯‘𝑊) ∈ ℕ₀)
12	11	adantr 474	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑊) ∈ ℕ₀)
13	12	3ad2ant1 1124	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘𝑊) ∈ ℕ₀)
14		simpl 476	. . . . . . 7 ⊢ ((𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)) → 𝑀 ≤ (♯‘𝑊))
15	14	3ad2ant3 1126	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ≤ (♯‘𝑊))
16		elfz2nn0 12753	. . . . . 6 ⊢ (𝑀 ∈ (0...(♯‘𝑊)) ↔ (𝑀 ∈ ℕ₀ ∧ (♯‘𝑊) ∈ ℕ₀ ∧ 𝑀 ≤ (♯‘𝑊)))
17	10, 13, 15, 16	syl3anbrc 1400	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑀 ∈ (0...(♯‘𝑊)))
18		swrd0lenOLD 13742	. . . . 5 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
19	8, 17, 18	syl2anc 579	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
20		simpr 479	. . . . . 6 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → 𝑈 ∈ Word 𝑉)
21	20	3ad2ant1 1124	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑈 ∈ Word 𝑉)
22		simpr 479	. . . . . . 7 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → 𝑁 ∈ ℕ₀)
23	22	3ad2ant2 1125	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑁 ∈ ℕ₀)
24		lencl 13625	. . . . . . . 8 ⊢ (𝑈 ∈ Word 𝑉 → (♯‘𝑈) ∈ ℕ₀)
25	24	adantl 475	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) → (♯‘𝑈) ∈ ℕ₀)
26	25	3ad2ant1 1124	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘𝑈) ∈ ℕ₀)
27		simpr 479	. . . . . . 7 ⊢ ((𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈)) → 𝑁 ≤ (♯‘𝑈))
28	27	3ad2ant3 1126	. . . . . 6 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑁 ≤ (♯‘𝑈))
29		elfz2nn0 12753	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝑈)) ↔ (𝑁 ∈ ℕ₀ ∧ (♯‘𝑈) ∈ ℕ₀ ∧ 𝑁 ≤ (♯‘𝑈)))
30	23, 26, 28, 29	syl3anbrc 1400	. . . . 5 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → 𝑁 ∈ (0...(♯‘𝑈)))
31		swrd0lenOLD 13742	. . . . 5 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑁 ∈ (0...(♯‘𝑈))) → (♯‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
32	21, 30, 31	syl2anc 579	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) → (♯‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
33	19, 32	eqeq12d 2793	. . 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 623	. 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 474	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) → 𝑊 ∈ Word 𝑉)
36	17	adantr 474	. . . . . . 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 579	. . . . . 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 6940	. . . . 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 3340	. . . 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 474	. . . . . . 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 474	. . . . . . 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 479	. . . . . . 7 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
43		swrd0fvOLD 13762	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(♯‘𝑊)) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = (𝑊‘𝑖))
44	40, 41, 42, 43	syl3anc 1439	. . . . . 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 6932	. . . . . . . . . . 11 ⊢ (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
46	45	eleq2d 2845	. . . . . . . . . 10 ⊢ (𝑀 = 𝑁 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
47	46	adantl 475	. . . . . . . . 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 474	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑈 ∈ Word 𝑉)
49	30	adantr 474	. . . . . . . . . . . 12 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ 𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑀 ≤ (♯‘𝑊) ∧ 𝑁 ≤ (♯‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ (0...(♯‘𝑈)))
50		simpr 479	. . . . . . . . . . . 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 1119	. . . . . . . . . . 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 403	. . . . . . . . . 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 474	. . . . . . . . 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 232	. . . . . . . 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 397	. . . . . . 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		swrd0fvOLD 13762	. . . . . . 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 2793	. . . . 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 3167	. . . 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 271	. . 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 574	. 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 297	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 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⟨cop 4404 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 0cc0 10274 ≤ cle 10414 ℕ₀cn0 11646 ...cfz 12647 ..^cfzo 12788 ♯chash 13439 Word cword 13603 substr csubstr 13734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-fz 12648 df-fzo 12789 df-hash 13440 df-word 13604 df-substr 13735

This theorem is referenced by: clwlkclwwlkf1lem2OLD 27392

W3C validator