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Theorem pfxccat3a 40090
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. Could replace swrdccat3a 13291. (Contributed by AV, 10-May-2020.)
Hypotheses
Ref Expression
pfxccatin12.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 789 . . . . . 6 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 elfznn0 12257 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + 𝑀)) → 𝑁 ∈ ℕ0)
32adantl 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝑁 ∈ ℕ0)
43adantl 480 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ℕ0)
5 pfxccatin12.l . . . . . . . . . . 11 𝐿 = (#‘𝐴)
6 lencl 13125 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
75, 6syl5eqel 2691 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ ℕ0)
87adantr 479 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐿 ∈ ℕ0)
98adantr 479 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → 𝐿 ∈ ℕ0)
109adantl 480 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝐿 ∈ ℕ0)
11 simpl 471 . . . . . . 7 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁𝐿)
12 elfz2nn0 12255 . . . . . . 7 (𝑁 ∈ (0...𝐿) ↔ (𝑁 ∈ ℕ0𝐿 ∈ ℕ0𝑁𝐿))
134, 10, 11, 12syl3anbrc 1238 . . . . . 6 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ (0...𝐿))
14 df-3an 1032 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...𝐿)))
151, 13, 14sylanbrc 694 . . . . 5 ((𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ (0...𝐿)))
165pfxccatpfx1 40088 . . . . 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 4041 . . . . 5 (𝑁𝐿 → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 prefix 𝑁))
1918adantr 479 . . . 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 2646 . . 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 789 . . . . . 6 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
22 elfz2nn0 12255 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + 𝑀)) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)))
235eleq1i 2678 . . . . . . . . . . . 12 (𝐿 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0)
24 nn0ltp1le 11268 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ (𝐿 + 1) ≤ 𝑁))
25 nn0re 11148 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
26 nn0re 11148 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
27 ltnle 9968 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
2825, 26, 27syl2an 492 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
2924, 28bitr3d 268 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0𝑁 ∈ ℕ0) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁𝐿))
30293ad2antr1 1218 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁 ↔ ¬ 𝑁𝐿))
31 simpr3 1061 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ≤ (𝐿 + 𝑀))
3231anim1i 589 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝑁 ≤ (𝐿 + 𝑀) ∧ (𝐿 + 1) ≤ 𝑁))
3332ancomd 465 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀)))
34 nn0z 11233 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
35343ad2ant1 1074 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → 𝑁 ∈ ℤ)
3635adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → 𝑁 ∈ ℤ)
3736adantr 479 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ℤ)
38 peano2nn0 11180 . . . . . . . . . . . . . . . . . . . 20 (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℕ0)
3938nn0zd 11312 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0 → (𝐿 + 1) ∈ ℤ)
4039adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 1) ∈ ℤ)
4140adantr 479 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 1) ∈ ℤ)
42 nn0z 11233 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 + 𝑀) ∈ ℕ0 → (𝐿 + 𝑀) ∈ ℤ)
43423ad2ant2 1075 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (𝐿 + 𝑀) ∈ ℤ)
4443adantl 480 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (𝐿 + 𝑀) ∈ ℤ)
4544adantr 479 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝐿 + 𝑀) ∈ ℤ)
46 elfz 12158 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℤ ∧ (𝐿 + 1) ∈ ℤ ∧ (𝐿 + 𝑀) ∈ ℤ) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀))))
4737, 41, 45, 46syl3anc 1317 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → (𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)) ↔ ((𝐿 + 1) ≤ 𝑁𝑁 ≤ (𝐿 + 𝑀))))
4833, 47mpbird 245 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) ∧ (𝐿 + 1) ≤ 𝑁) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
4948ex 448 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → ((𝐿 + 1) ≤ 𝑁𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5030, 49sylbird 248 . . . . . . . . . . . . 13 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀))) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5150ex 448 . . . . . . . . . . . 12 (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5223, 51sylbir 223 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
536, 52syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5453adantr 479 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + 𝑀) ∈ ℕ0𝑁 ≤ (𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5522, 54syl5bi 230 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(𝐿 + 𝑀)) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))))
5655imp 443 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → (¬ 𝑁𝐿𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5756impcom 444 . . . . . 6 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀)))
58 df-3an 1032 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
5921, 57, 58sylanbrc 694 . . . . 5 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))))
60 pfxccatpfx2.m . . . . . 6 𝑀 = (#‘𝐵)
615, 60pfxccatpfx2 40089 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉𝑁 ∈ ((𝐿 + 1)...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6259, 61syl 17 . . . 4 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
63 iffalse 4044 . . . . 5 𝑁𝐿 → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6463adantr 479 . . . 4 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))) = (𝐴 ++ (𝐵 prefix (𝑁𝐿))))
6562, 64eqtr4d 2646 . . 3 ((¬ 𝑁𝐿 ∧ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀)))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))))
6620, 65pm2.61ian 826 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (0...(𝐿 + 𝑀))) → ((𝐴 ++ 𝐵) prefix 𝑁) = if(𝑁𝐿, (𝐴 prefix 𝑁), (𝐴 ++ (𝐵 prefix (𝑁𝐿)))))
6766ex 448 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  ifcif 4035   class class class wbr 4577  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  0cn0 11139  cz 11210  ...cfz 12152  #chash 12934  Word cword 13092   ++ cconcat 13094   prefix cpfx 40042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-substr 13104  df-pfx 40043
This theorem is referenced by: (None)
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