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Theorem pfxccat3 40090
Description: The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. Could replace swrdccat3 13285. (Contributed by AV, 10-May-2020.)
Hypothesis
Ref Expression
pfxccatin12.l 𝐿 = (#‘𝐴)
Assertion
Ref Expression
pfxccat3 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))))

Proof of Theorem pfxccat3
StepHypRef Expression
1 simpll 785 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simplrl 795 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → 𝑀 ∈ (0...𝑁))
3 lencl 13121 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
4 elfznn0 12253 . . . . . . . . . . . . . 14 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℕ0)
54adantr 479 . . . . . . . . . . . . 13 ((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
65adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ ℕ0)
7 simplr 787 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → (#‘𝐴) ∈ ℕ0)
8 pfxccatin12.l . . . . . . . . . . . . . . 15 𝐿 = (#‘𝐴)
98breq2i 4581 . . . . . . . . . . . . . 14 (𝑁𝐿𝑁 ≤ (#‘𝐴))
109biimpi 204 . . . . . . . . . . . . 13 (𝑁𝐿𝑁 ≤ (#‘𝐴))
1110adantl 480 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ≤ (#‘𝐴))
12 elfz2nn0 12251 . . . . . . . . . . . 12 (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)))
136, 7, 11, 12syl3anbrc 1238 . . . . . . . . . . 11 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(#‘𝐴)))
1413exp31 627 . . . . . . . . . 10 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
1514adantl 480 . . . . . . . . 9 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((#‘𝐴) ∈ ℕ0 → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
163, 15syl5com 31 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
1716adantr 479 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
1817imp 443 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴))))
1918imp 443 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(#‘𝐴)))
202, 19jca 552 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))))
21 swrdccatin1 13276 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
221, 20, 21sylc 62 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩))
23 simp1l 1077 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
248eleq1i 2674 . . . . . . . . . . 11 (𝐿 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0)
25 elfz2nn0 12251 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
26 nn0z 11229 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
2726adantl 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
28 nn0z 11229 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
29283ad2ant2 1075 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑁 ∈ ℤ)
3029adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
31 nn0z 11229 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℤ)
32313ad2ant1 1074 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℤ)
3332adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
3427, 30, 333jca 1234 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
3534adantr 479 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
36 simpl3 1058 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀𝑁)
3736anim1i 589 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝑀𝑁𝐿𝑀))
3837ancomd 465 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿𝑀𝑀𝑁))
39 elfz2 12155 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
4035, 38, 39sylanbrc 694 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
4140exp31 627 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4225, 41sylbi 205 . . . . . . . . . . . . 13 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4342adantr 479 . . . . . . . . . . . 12 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4443com12 32 . . . . . . . . . . 11 (𝐿 ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4524, 44sylbir 223 . . . . . . . . . 10 ((#‘𝐴) ∈ ℕ0 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
463, 45syl 17 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4746adantr 479 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4847imp 443 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁)))
4948a1d 25 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
50493imp 1248 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
51 elfz2nn0 12251 . . . . . . . . . . . 12 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))))
52 nn0z 11229 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℤ)
538, 52syl5eqel 2687 . . . . . . . . . . . . . . . . 17 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℤ)
5453adantr 479 . . . . . . . . . . . . . . . 16 (((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿) → 𝐿 ∈ ℤ)
5554adantl 480 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿 ∈ ℤ)
56 nn0z 11229 . . . . . . . . . . . . . . . . 17 ((𝐿 + (#‘𝐵)) ∈ ℕ0 → (𝐿 + (#‘𝐵)) ∈ ℤ)
57563ad2ant2 1075 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (𝐿 + (#‘𝐵)) ∈ ℤ)
5857adantr 479 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 + (#‘𝐵)) ∈ ℤ)
59283ad2ant1 1074 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
6059adantr 479 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ ℤ)
6155, 58, 603jca 1234 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
628eqcomi 2614 . . . . . . . . . . . . . . . . . . 19 (#‘𝐴) = 𝐿
6362eleq1i 2674 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
64 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
65 nn0re 11144 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
66 ltnle 9964 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6764, 65, 66syl2anr 493 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6867bicomd 211 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿 < 𝑁))
69 ltle 9973 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁𝐿𝑁))
7064, 65, 69syl2anr 493 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁𝐿𝑁))
7168, 70sylbid 228 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿𝑁))
7271ex 448 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7363, 72syl5bi 230 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
74733ad2ant1 1074 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7574imp32 447 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿𝑁)
76 simpl3 1058 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ≤ (𝐿 + (#‘𝐵)))
7775, 76jca 552 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵))))
78 elfz2 12155 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵)))))
7961, 77, 78sylanbrc 694 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
8079exp32 628 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8151, 80sylbi 205 . . . . . . . . . . 11 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8281adantl 480 . . . . . . . . . 10 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
833, 82syl5com 31 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8483adantr 479 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8584imp 443 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
8685a1dd 47 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
87863imp 1248 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
8850, 87jca 552 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
898swrdccatin2 13280 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)))
9023, 88, 89sylc 62 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))
91 simp1l 1077 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
92 nn0re 11144 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
9392adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
94 ltnle 9964 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9593, 64, 94syl2anr 493 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9695bicomd 211 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 < 𝐿))
97 simpll 785 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
98 simplr 787 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
99 ltle 9973 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿𝑀𝐿))
10092, 64, 99syl2an 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑀 < 𝐿𝑀𝐿))
101100imp 443 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀𝐿)
102 elfz2nn0 12251 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
10397, 98, 101, 102syl3anbrc 1238 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
104103exp31 627 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
105104adantr 479 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
106105impcom 444 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿𝑀 ∈ (0...𝐿)))
10796, 106sylbid 228 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿)))
108107expcom 449 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1091083adant3 1073 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11025, 109sylbi 205 . . . . . . . . . . . 12 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11163, 110syl5bi 230 . . . . . . . . . . 11 (𝑀 ∈ (0...𝑁) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
112111adantr 479 . . . . . . . . . 10 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1133, 112syl5com 31 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
114113adantr 479 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
115114imp 443 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿)))
116115a1d 25 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1171163imp 1248 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑀 ∈ (0...𝐿))
118653ad2ant1 1074 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → 𝑁 ∈ ℝ)
11966bicomd 211 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁𝐿𝐿 < 𝑁))
12064, 118, 119syl2an 492 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝐿 < 𝑁))
12126adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → 𝐿 ∈ ℤ)
12257adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 + (#‘𝐵)) ∈ ℤ)
12359adantl 480 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → 𝑁 ∈ ℤ)
124121, 122, 1233jca 1234 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
125124adantr 479 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
12664, 118, 69syl2an 492 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 < 𝑁𝐿𝑁))
127126imp 443 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿𝑁)
128 simplr3 1097 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (#‘𝐵)))
129127, 128jca 552 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵))))
130125, 129, 78sylanbrc 694 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
131130ex 448 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 < 𝑁𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
132120, 131sylbid 228 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
133132ex 448 . . . . . . . . . . . . . 14 (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
13463, 133sylbi 205 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
1353, 134syl 17 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
136135adantr 479 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
137136com12 32 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
13851, 137sylbi 205 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
139138adantl 480 . . . . . . . 8 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
140139impcom 444 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
141140a1dd 47 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (¬ 𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
1421413imp 1248 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
143117, 142jca 552 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
1448pfxccatin12 40089 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))
14591, 143, 144sylc 62 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))))
14622, 90, 1452if2 4081 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))))))
147146ex 448 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = if(𝑁𝐿, (𝐴 substr ⟨𝑀, 𝑁⟩), if(𝐿𝑀, (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩), ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  ifcif 4031  cop 4126   class class class wbr 4573  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788   + caddc 9791   < clt 9926  cle 9927  cmin 10113  0cn0 11135  cz 11206  ...cfz 12148  #chash 12930  Word cword 13088   ++ cconcat 13090   substr csubstr 13092   prefix cpfx 40045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
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 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-fz 12149  df-fzo 12286  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-pfx 40046
This theorem is referenced by: (None)
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