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Theorem pfxccat3a 14095
 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 12995 . . . . . . . . 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 13878 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
75, 6eqeltrid 2922 . . . . . . . . . 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 12993 . . . . . . 7 (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0𝐿 ∈ ℕ0𝑁𝐿))
134, 10, 11, 12syl3anbrc 1337 . . . . . 6 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14 df-3an 1083 . . . . . 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 14093 . . . . 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 4476 . . . . 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 2864 . . 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 12993 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)))
235eleq1i 2908 . . . . . . . . . . . 12 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
24 nn0ltp1le 12034 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25 nn0re 11900 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
26 nn0re 11900 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
27 ltnle 10714 . . . . . . . . . . . . . . . . 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 1182 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁𝐿))
31 simpr3 1190 . . . . . . . . . . . . . . . . 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 11999 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
34333ad2ant1 1127 . . . . . . . . . . . . . . . . . . 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 11931 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
3837nn0zd 12079 . . . . . . . . . . . . . . . . . . 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 11999 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
42413ad2ant2 1128 . . . . . . . . . . . . . . . . . . 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 12893 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℤ ∧ (𝐿 + 1) ∈ ℤ ∧ (𝐿 + 𝑀) ∈ ℤ) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀))))
4636, 40, 44, 45syl3anc 1365 . . . . . . . . . . . . . . . 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 1083 . . . . . 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 14094 . . . . 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 4479 . . . . 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 2864 . . 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 1081   = wceq 1530   ∈ wcel 2107  ifcif 4470   class class class wbr 5063  ‘cfv 6354  (class class class)co 7150  ℝcr 10530  0cc0 10531  1c1 10532   + caddc 10534   < clt 10669   ≤ cle 10670   − cmin 10864  ℕ0cn0 11891  ℤcz 11975  ...cfz 12887  ♯chash 13685  Word cword 13856   ++ cconcat 13917   prefix cpfx 14027 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-hash 13686  df-word 13857  df-concat 13918  df-substr 13998  df-pfx 14028 This theorem is referenced by: (None)
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