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Theorem swrdeq 13296
 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 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))
Distinct variable groups:   𝑖,𝑀   𝑖,𝑁   𝑈,𝑖   𝑖,𝑉   𝑖,𝑊

Proof of Theorem swrdeq
StepHypRef Expression
1 swrdcl 13271 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉)
2 swrdcl 13271 . . . . 5 (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉)
31, 2anim12i 588 . . . 4 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → ((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉))
433ad2ant1 1075 . . 3 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) ∈ Word 𝑉 ∧ (𝑈 substr ⟨0, 𝑁⟩) ∈ Word 𝑉))
5 eqwrd 13201 . . 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))))
64, 5syl 17 . 2 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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 472 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → 𝑊 ∈ Word 𝑉)
873ad2ant1 1075 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑊 ∈ Word 𝑉)
9 simpl 472 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℕ0)
1093ad2ant2 1076 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ ℕ0)
11 lencl 13179 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
1211adantr 480 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → (#‘𝑊) ∈ ℕ0)
13123ad2ant1 1075 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘𝑊) ∈ ℕ0)
14 simpl 472 . . . . . . 7 ((𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)) → 𝑀 ≤ (#‘𝑊))
15143ad2ant3 1077 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ≤ (#‘𝑊))
16 elfz2nn0 12300 . . . . . 6 (𝑀 ∈ (0...(#‘𝑊)) ↔ (𝑀 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0𝑀 ≤ (#‘𝑊)))
1710, 13, 15, 16syl3anbrc 1239 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ (0...(#‘𝑊)))
18 swrd0len 13274 . . . . 5 ((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
198, 17, 18syl2anc 691 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
20 simpr 476 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → 𝑈 ∈ Word 𝑉)
21203ad2ant1 1075 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑈 ∈ Word 𝑉)
22 simpr 476 . . . . . . 7 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
23223ad2ant2 1076 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ∈ ℕ0)
24 lencl 13179 . . . . . . . 8 (𝑈 ∈ Word 𝑉 → (#‘𝑈) ∈ ℕ0)
2524adantl 481 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) → (#‘𝑈) ∈ ℕ0)
26253ad2ant1 1075 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘𝑈) ∈ ℕ0)
27 simpr 476 . . . . . . 7 ((𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)) → 𝑁 ≤ (#‘𝑈))
28273ad2ant3 1077 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ≤ (#‘𝑈))
29 elfz2nn0 12300 . . . . . 6 (𝑁 ∈ (0...(#‘𝑈)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) ∈ ℕ0𝑁 ≤ (#‘𝑈)))
3023, 26, 28, 29syl3anbrc 1239 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ∈ (0...(#‘𝑈)))
31 swrd0len 13274 . . . . 5 ((𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈))) → (#‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
3221, 30, 31syl2anc 691 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑈 substr ⟨0, 𝑁⟩)) = 𝑁)
3319, 32eqeq12d 2625 . . 3 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((#‘(𝑊 substr ⟨0, 𝑀⟩)) = (#‘(𝑈 substr ⟨0, 𝑁⟩)) ↔ 𝑀 = 𝑁))
3433anbi1d 737 . 2 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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))))
358adantr 480 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → 𝑊 ∈ Word 𝑉)
3617adantr 480 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → 𝑀 ∈ (0...(#‘𝑊)))
3735, 36, 18syl2anc 691 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (#‘(𝑊 substr ⟨0, 𝑀⟩)) = 𝑀)
3837oveq2d 6565 . . . . 5 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (0..^(#‘(𝑊 substr ⟨0, 𝑀⟩))) = (0..^𝑀))
3938raleqdv 3121 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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)))
4035adantr 480 . . . . . . 7 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑊 ∈ Word 𝑉)
4136adantr 480 . . . . . . 7 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ (0...(#‘𝑊)))
42 simpr 476 . . . . . . 7 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
43 swrd0fv 13291 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊)) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = (𝑊𝑖))
4440, 41, 42, 43syl3anc 1318 . . . . . 6 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = (𝑊𝑖))
45 oveq2 6557 . . . . . . . . . . 11 (𝑀 = 𝑁 → (0..^𝑀) = (0..^𝑁))
4645eleq2d 2673 . . . . . . . . . 10 (𝑀 = 𝑁 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
4746adantl 481 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ (0..^𝑁)))
4821adantr 480 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑈 ∈ Word 𝑉)
4930adantr 480 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑁 ∈ (0...(#‘𝑈)))
50 simpr 476 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ (0..^𝑁))
5148, 49, 503jca 1235 . . . . . . . . . . 11 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑖 ∈ (0..^𝑁)) → (𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)))
5251ex 449 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑖 ∈ (0..^𝑁) → (𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
5352adantr 480 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑁) → (𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
5447, 53sylbid 229 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (𝑖 ∈ (0..^𝑀) → (𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁))))
5554imp 444 . . . . . . 7 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)))
56 swrd0fv 13291 . . . . . . 7 ((𝑈 ∈ Word 𝑉𝑁 ∈ (0...(#‘𝑈)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) = (𝑈𝑖))
5755, 56syl 17 . . . . . 6 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) = (𝑈𝑖))
5844, 57eqeq12d 2625 . . . . 5 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) ∧ 𝑖 ∈ (0..^𝑀)) → (((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ (𝑊𝑖) = (𝑈𝑖)))
5958ralbidva 2968 . . . 4 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀 = 𝑁) → (∀𝑖 ∈ (0..^𝑀)((𝑊 substr ⟨0, 𝑀⟩)‘𝑖) = ((𝑈 substr ⟨0, 𝑁⟩)‘𝑖) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖)))
6039, 59bitrd 267 . . 3 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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)(𝑊𝑖) = (𝑈𝑖)))
6160pm5.32da 671 . 2 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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)(𝑊𝑖) = (𝑈𝑖))))
626, 34, 613bitrd 293 1 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑀 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨0, 𝑀⟩) = (𝑈 substr ⟨0, 𝑁⟩) ↔ (𝑀 = 𝑁 ∧ ∀𝑖 ∈ (0..^𝑀)(𝑊𝑖) = (𝑈𝑖))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ≤ cle 9954  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-substr 13158 This theorem is referenced by:  clwlkf1clwwlklem  26376  clwlksf1clwwlklem  41275
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