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Theorem pfxccat3a 14088
Description: A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.) (Revised by AV, 10-May-2020.)
Hypotheses
Ref Expression
swrdccatin2.l 𝐿 = (♯‘𝐴)
pfxccatpfx2.m 𝑀 = (♯‘𝐵)
Assertion
Ref Expression
pfxccat3a ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿))))))

Proof of Theorem pfxccat3a
StepHypRef Expression
1 simprl 767 . . . . . 6 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 elfznn0 12988 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
32adantl 482 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
43adantl 482 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
5 swrdccatin2.l . . . . . . . . . . 11 𝐿 = (♯‘𝐴)
6 lencl 13871 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
75, 6eqeltrid 2914 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ ℕ0)
87adantr 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
98adantr 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
109adantl 482 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
11 simpl 483 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁𝐿)
12 elfz2nn0 12986 . . . . . . 7 (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0𝐿 ∈ ℕ0𝑁𝐿))
134, 10, 11, 12syl3anbrc 1335 . . . . . 6 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14 df-3an 1081 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
151, 13, 14sylanbrc 583 . . . . 5 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)))
165pfxccatpfx1 14086 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
1715, 16syl 17 . . . 4 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 prefix 𝑁))
18 iftrue 4469 . . . . 5 (𝑁𝐿 → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 prefix 𝑁))
1918adantr 481 . . . 4 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 prefix 𝑁))
2017, 19eqtr4d 2856 . . 3 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))))
21 simprl 767 . . . . . 6 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
22 elfz2nn0 12986 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)))
235eleq1i 2900 . . . . . . . . . . . 12 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24 nn0ltp1le 12028 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25 nn0re 11894 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
26 nn0re 11894 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
27 ltnle 10708 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
2825, 26, 27syl2an 595 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
2924, 28bitr3d 282 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁𝐿))
30293ad2antr1 1180 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁𝐿))
31 simpr3 1188 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
3231anim1ci 615 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀)))
33 nn0z 11993 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
34333ad2ant1 1125 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
3534adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
3635adantr 481 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
37 peano2nn0 11925 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
3837nn0zd 12073 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
3938adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
4039adantr 481 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
41 nn0z 11993 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
42413ad2ant2 1126 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
4342adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
4443adantr 481 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
45 elfz 12886 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℤ ∧ (𝐿 + 1) ∈ ℤ ∧ (𝐿 + 𝑀) ∈ ℤ) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀))))
4636, 40, 44, 45syl3anc 1363 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀))))
4732, 46mpbird 258 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
4847ex 413 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
4930, 48sylbird 261 . . . . . . . . . . . . 13 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5049ex 413 . . . . . . . . . . . 12 (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5123, 50sylbir 236 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
526, 51syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5352adantr 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5422, 53syl5bi 243 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5554imp 407 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5655impcom 408 . . . . . 6 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
57 df-3an 1081 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5821, 56, 57sylanbrc 583 . . . . 5 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
59 pfxccatpfx2.m . . . . . 6 𝑀 = (♯‘𝐵)
605, 59pfxccatpfx2 14087 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6158, 60syl 17 . . . 4 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
62 iffalse 4472 . . . . 5 𝑁𝐿 → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6362adantr 481 . . . 4 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6461, 63eqtr4d 2856 . . 3 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))))
6520, 64pm2.61ian 808 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))))
6665ex 413 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  ifcif 4463   class class class wbr 5057  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cle 10664  cmin 10858  0cn0 11885  cz 11969  ...cfz 12880  chash 13678  Word cword 13849   ++ cconcat 13910   prefix cpfx 14020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-substr 13991  df-pfx 14021
This theorem is referenced by: (None)
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