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Theorem swrdspsleq 13301
 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 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
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.
StepHypRef Expression
1 simpr1 1060 . . . 4 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → (𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉))
2 simpr2 1061 . . . 4 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → (𝑀 ∈ ℕ0𝑁 ∈ ℕ0))
3 simpl 472 . . . 4 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → 𝑁𝑀)
4 swrdsb0eq 13299 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
51, 2, 3, 4syl3anc 1318 . . 3 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → (𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩))
6 ral0 4028 . . . . . . 7 𝑖 ∈ ∅ (𝑊𝑖) = (𝑈𝑖)
7 nn0z 11277 . . . . . . . . . 10 (𝑀 ∈ ℕ0𝑀 ∈ ℤ)
8 nn0z 11277 . . . . . . . . . 10 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
9 fzon 12358 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))
107, 8, 9syl2an 493 . . . . . . . . 9 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁𝑀 ↔ (𝑀..^𝑁) = ∅))
1110biimpa 500 . . . . . . . 8 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → (𝑀..^𝑁) = ∅)
1211raleqdv 3121 . . . . . . 7 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → (∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖) ↔ ∀𝑖 ∈ ∅ (𝑊𝑖) = (𝑈𝑖)))
136, 12mpbiri 247 . . . . . 6 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ 𝑁𝑀) → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖))
1413ex 449 . . . . 5 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
15143ad2ant2 1076 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑁𝑀 → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
1615impcom 445 . . 3 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖))
175, 162thd 254 . 2 ((𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
18 swrdcl 13271 . . . . . 6 (𝑊 ∈ Word 𝑉 → (𝑊 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
19 swrdcl 13271 . . . . . 6 (𝑈 ∈ Word 𝑉 → (𝑈 substr ⟨𝑀, 𝑁⟩) ∈ Word 𝑉)
20 eqwrd 13201 . . . . . 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))))
2118, 19, 20syl2an 493 . . . . 5 ((𝑊 ∈ Word 𝑉𝑈 ∈ 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))))
22213ad2ant1 1075 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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))))
2322adantl 481 . . 3 ((¬ 𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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))))
24 swrdsbslen 13300 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (#‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (#‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
2524adantl 481 . . . 4 ((¬ 𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → (#‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (#‘(𝑈 substr ⟨𝑀, 𝑁⟩)))
2625biantrurd 528 . . 3 ((¬ 𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ 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))))
27 nn0re 11178 . . . . . . 7 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
28 nn0re 11178 . . . . . . 7 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
29 ltnle 9996 . . . . . . . 8 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁 ↔ ¬ 𝑁𝑀))
30 ltle 10005 . . . . . . . 8 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 < 𝑁𝑀𝑁))
3129, 30sylbird 249 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁𝑀𝑀𝑁))
3227, 28, 31syl2an 493 . . . . . 6 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (¬ 𝑁𝑀𝑀𝑁))
33323ad2ant2 1076 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (¬ 𝑁𝑀𝑀𝑁))
34 simpl1l 1105 . . . . . . . . . . 11 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑊 ∈ Word 𝑉)
35 simpl 472 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℕ0)
36353ad2ant2 1076 . . . . . . . . . . . . 13 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ ℕ0)
3736adantr 480 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑀 ∈ ℕ0)
387, 8anim12i 588 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
39383ad2ant2 1076 . . . . . . . . . . . . . . 15 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
4039anim1i 590 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀𝑁))
41 df-3an 1033 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀𝑁))
4240, 41sylibr 223 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
43 eluz2 11569 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
4442, 43sylibr 223 . . . . . . . . . . . 12 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑁 ∈ (ℤ𝑀))
4537, 44jca 553 . . . . . . . . . . 11 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)))
46 simpl 472 . . . . . . . . . . . . 13 ((𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)) → 𝑁 ≤ (#‘𝑊))
47463ad2ant3 1077 . . . . . . . . . . . 12 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑁 ≤ (#‘𝑊))
4847adantr 480 . . . . . . . . . . 11 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑁 ≤ (#‘𝑊))
4934, 45, 483jca 1235 . . . . . . . . . 10 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ 𝑁 ≤ (#‘𝑊)))
50 swrdlen2 13297 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ 𝑁 ≤ (#‘𝑊)) → (#‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁𝑀))
5149, 50syl 17 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (#‘(𝑊 substr ⟨𝑀, 𝑁⟩)) = (𝑁𝑀))
5251oveq2d 6565 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩))) = (0..^(𝑁𝑀)))
5352raleqdv 3121 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑗 ∈ (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑗 ∈ (0..^(𝑁𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
54 0zd 11266 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 0 ∈ ℤ)
55 zsubcl 11296 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁𝑀) ∈ ℤ)
568, 7, 55syl2anr 494 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝑁𝑀) ∈ ℤ)
57563ad2ant2 1076 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑁𝑀) ∈ ℤ)
587adantr 480 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℤ)
59583ad2ant2 1076 . . . . . . . . . 10 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → 𝑀 ∈ ℤ)
60 fzoshftral 12447 . . . . . . . . . 10 ((0 ∈ ℤ ∧ (𝑁𝑀) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (∀𝑗 ∈ (0..^(𝑁𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁𝑀) + 𝑀))[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
6154, 57, 59, 60syl3anc 1318 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (∀𝑗 ∈ (0..^(𝑁𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁𝑀) + 𝑀))[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
6261adantr 480 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑗 ∈ (0..^(𝑁𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ ((0 + 𝑀)..^((𝑁𝑀) + 𝑀))[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
63 nn0cn 11179 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
64 nn0cn 11179 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
65 addid2 10098 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℂ → (0 + 𝑀) = 𝑀)
6665adantl 481 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (0 + 𝑀) = 𝑀)
67 npcan 10169 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((𝑁𝑀) + 𝑀) = 𝑁)
6866, 67oveq12d 6567 . . . . . . . . . . . . 13 ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → ((0 + 𝑀)..^((𝑁𝑀) + 𝑀)) = (𝑀..^𝑁))
6963, 64, 68syl2anr 494 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → ((0 + 𝑀)..^((𝑁𝑀) + 𝑀)) = (𝑀..^𝑁))
70693ad2ant2 1076 . . . . . . . . . . 11 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((0 + 𝑀)..^((𝑁𝑀) + 𝑀)) = (𝑀..^𝑁))
7170adantr 480 . . . . . . . . . 10 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → ((0 + 𝑀)..^((𝑁𝑀) + 𝑀)) = (𝑀..^𝑁))
7271raleqdv 3121 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑖 ∈ ((0 + 𝑀)..^((𝑁𝑀) + 𝑀))[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
73 ovex 6577 . . . . . . . . . . . 12 (𝑖𝑀) ∈ V
74 sbceqg 3936 . . . . . . . . . . . . 13 ((𝑖𝑀) ∈ V → ([(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ (𝑖𝑀) / 𝑗((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = (𝑖𝑀) / 𝑗((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗)))
75 csbfv2g 6142 . . . . . . . . . . . . . . 15 ((𝑖𝑀) ∈ V → (𝑖𝑀) / 𝑗((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀) / 𝑗𝑗))
76 csbvarg 3955 . . . . . . . . . . . . . . . 16 ((𝑖𝑀) ∈ V → (𝑖𝑀) / 𝑗𝑗 = (𝑖𝑀))
7776fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑖𝑀) ∈ V → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀) / 𝑗𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)))
7875, 77eqtrd 2644 . . . . . . . . . . . . . 14 ((𝑖𝑀) ∈ V → (𝑖𝑀) / 𝑗((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)))
79 csbfv2g 6142 . . . . . . . . . . . . . . 15 ((𝑖𝑀) ∈ V → (𝑖𝑀) / 𝑗((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀) / 𝑗𝑗))
8076fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑖𝑀) ∈ V → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀) / 𝑗𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)))
8179, 80eqtrd 2644 . . . . . . . . . . . . . 14 ((𝑖𝑀) ∈ V → (𝑖𝑀) / 𝑗((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)))
8278, 81eqeq12d 2625 . . . . . . . . . . . . 13 ((𝑖𝑀) ∈ V → ((𝑖𝑀) / 𝑗((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = (𝑖𝑀) / 𝑗((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀))))
8374, 82bitrd 267 . . . . . . . . . . . 12 ((𝑖𝑀) ∈ V → ([(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀))))
8473, 83mp1i 13 . . . . . . . . . . 11 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ([(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀))))
85 swrdfv2 13298 . . . . . . . . . . . . 13 (((𝑊 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ 𝑁 ≤ (#‘𝑊)) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = (𝑊𝑖))
8649, 85sylan 487 . . . . . . . . . . . 12 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = (𝑊𝑖))
87 simpl1r 1106 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑈 ∈ Word 𝑉)
88 simpl3r 1110 . . . . . . . . . . . . . 14 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → 𝑁 ≤ (#‘𝑈))
8987, 45, 883jca 1235 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (𝑈 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ 𝑁 ≤ (#‘𝑈)))
90 swrdfv2 13298 . . . . . . . . . . . . 13 (((𝑈 ∈ Word 𝑉 ∧ (𝑀 ∈ ℕ0𝑁 ∈ (ℤ𝑀)) ∧ 𝑁 ≤ (#‘𝑈)) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = (𝑈𝑖))
9189, 90sylan 487 . . . . . . . . . . . 12 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = (𝑈𝑖))
9286, 91eqeq12d 2625 . . . . . . . . . . 11 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → (((𝑊 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘(𝑖𝑀)) ↔ (𝑊𝑖) = (𝑈𝑖)))
9384, 92bitrd 267 . . . . . . . . . 10 (((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) ∧ 𝑖 ∈ (𝑀..^𝑁)) → ([(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ (𝑊𝑖) = (𝑈𝑖)))
9493ralbidva 2968 . . . . . . . . 9 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑖 ∈ (𝑀..^𝑁)[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
9572, 94bitrd 267 . . . . . . . 8 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑖 ∈ ((0 + 𝑀)..^((𝑁𝑀) + 𝑀))[(𝑖𝑀) / 𝑗]((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
9662, 95bitrd 267 . . . . . . 7 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑗 ∈ (0..^(𝑁𝑀))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
9753, 96bitrd 267 . . . . . 6 ((((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) ∧ 𝑀𝑁) → (∀𝑗 ∈ (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
9897ex 449 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (𝑀𝑁 → (∀𝑗 ∈ (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖))))
9933, 98syld 46 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → (¬ 𝑁𝑀 → (∀𝑗 ∈ (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖))))
10099impcom 445 . . 3 ((¬ 𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → (∀𝑗 ∈ (0..^(#‘(𝑊 substr ⟨𝑀, 𝑁⟩)))((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑗) = ((𝑈 substr ⟨𝑀, 𝑁⟩)‘𝑗) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
10123, 26, 1003bitr2d 295 . 2 ((¬ 𝑁𝑀 ∧ ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈)))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
10217, 101pm2.61ian 827 1 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑁 ≤ (#‘𝑊) ∧ 𝑁 ≤ (#‘𝑈))) → ((𝑊 substr ⟨𝑀, 𝑁⟩) = (𝑈 substr ⟨𝑀, 𝑁⟩) ↔ ∀𝑖 ∈ (𝑀..^𝑁)(𝑊𝑖) = (𝑈𝑖)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402  ⦋csb 3499  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ..^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-fal 1481  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:  2swrdeqwrdeq  13305  clwwlkf1  26324  pfxsuffeqwrdeq  40269  clwwlksf1  41224
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