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Theorem pfxccat3 41751
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 13538. (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 805 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simplrl 817 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → 𝑀 ∈ (0...𝑁))
3 lencl 13356 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
4 elfznn0 12471 . . . . . . . . . . . . . 14 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℕ0)
54adantr 480 . . . . . . . . . . . . 13 ((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
65adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ ℕ0)
7 simplr 807 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → (#‘𝐴) ∈ ℕ0)
8 pfxccatin12.l . . . . . . . . . . . . . . 15 𝐿 = (#‘𝐴)
98breq2i 4693 . . . . . . . . . . . . . 14 (𝑁𝐿𝑁 ≤ (#‘𝐴))
109biimpi 206 . . . . . . . . . . . . 13 (𝑁𝐿𝑁 ≤ (#‘𝐴))
1110adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ≤ (#‘𝐴))
12 elfz2nn0 12469 . . . . . . . . . . . 12 (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)))
136, 7, 11, 12syl3anbrc 1265 . . . . . . . . . . 11 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(#‘𝐴)))
1413exp31 629 . . . . . . . . . 10 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
1514adantl 481 . . . . . . . . 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 480 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴)))))
1817imp 444 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (𝑁𝐿𝑁 ∈ (0...(#‘𝐴))))
1918imp 444 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(#‘𝐴)))
202, 19jca 553 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))))
21 swrdccatin1 13529 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
221, 20, 21sylc 65 . . 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 1105 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
248eleq1i 2721 . . . . . . . . . . 11 (𝐿 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0)
25 elfz2nn0 12469 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
26 nn0z 11438 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
2726adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
28 nn0z 11438 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
29283ad2ant2 1103 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑁 ∈ ℤ)
3029adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
31 nn0z 11438 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℤ)
32313ad2ant1 1102 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℤ)
3332adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
3427, 30, 333jca 1261 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
3534adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
36 simpl3 1086 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀𝑁)
3736anim1i 591 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝑀𝑁𝐿𝑀))
3837ancomd 466 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿𝑀𝑀𝑁))
39 elfz2 12371 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
4035, 38, 39sylanbrc 699 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
4140exp31 629 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4225, 41sylbi 207 . . . . . . . . . . . . 13 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4342adantr 480 . . . . . . . . . . . 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 225 . . . . . . . . . 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 480 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4847imp 444 . . . . . . 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 1275 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
51 elfz2nn0 12469 . . . . . . . . . . . 12 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))))
52 nn0z 11438 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℤ)
538, 52syl5eqel 2734 . . . . . . . . . . . . . . . . 17 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℤ)
5453adantr 480 . . . . . . . . . . . . . . . 16 (((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿) → 𝐿 ∈ ℤ)
5554adantl 481 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿 ∈ ℤ)
56 nn0z 11438 . . . . . . . . . . . . . . . . 17 ((𝐿 + (#‘𝐵)) ∈ ℕ0 → (𝐿 + (#‘𝐵)) ∈ ℤ)
57563ad2ant2 1103 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (𝐿 + (#‘𝐵)) ∈ ℤ)
5857adantr 480 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 + (#‘𝐵)) ∈ ℤ)
59283ad2ant1 1102 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
6059adantr 480 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ ℤ)
6155, 58, 603jca 1261 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
628eqcomi 2660 . . . . . . . . . . . . . . . . . . 19 (#‘𝐴) = 𝐿
6362eleq1i 2721 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
64 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
65 nn0re 11339 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
66 ltnle 10155 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6764, 65, 66syl2anr 494 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6867bicomd 213 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿 < 𝑁))
69 ltle 10164 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁𝐿𝑁))
7064, 65, 69syl2anr 494 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁𝐿𝑁))
7168, 70sylbid 230 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿𝑁))
7271ex 449 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7363, 72syl5bi 232 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
74733ad2ant1 1102 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7574imp32 448 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿𝑁)
76 simpl3 1086 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ≤ (𝐿 + (#‘𝐵)))
7775, 76jca 553 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵))))
78 elfz2 12371 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵)))))
7961, 77, 78sylanbrc 699 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
8079exp32 630 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8151, 80sylbi 207 . . . . . . . . . . 11 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8281adantl 481 . . . . . . . . . 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 480 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8584imp 444 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
8685a1dd 50 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
87863imp 1275 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
8850, 87jca 553 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
898swrdccatin2 13533 . . . 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 65 . . 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 1105 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
92 nn0re 11339 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
9392adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
94 ltnle 10155 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9593, 64, 94syl2anr 494 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9695bicomd 213 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 < 𝐿))
97 simpll 805 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
98 simplr 807 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
99 ltle 10164 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿𝑀𝐿))
10092, 64, 99syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑀 < 𝐿𝑀𝐿))
101100imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀𝐿)
102 elfz2nn0 12469 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
10397, 98, 101, 102syl3anbrc 1265 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
104103exp31 629 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
105104adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
106105impcom 445 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿𝑀 ∈ (0...𝐿)))
10796, 106sylbid 230 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿)))
108107expcom 450 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1091083adant3 1101 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11025, 109sylbi 207 . . . . . . . . . . . 12 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11163, 110syl5bi 232 . . . . . . . . . . 11 (𝑀 ∈ (0...𝑁) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
112111adantr 480 . . . . . . . . . 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 480 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
115114imp 444 . . . . . . 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 1275 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑀 ∈ (0...𝐿))
118653ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → 𝑁 ∈ ℝ)
11966bicomd 213 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁𝐿𝐿 < 𝑁))
12064, 118, 119syl2an 493 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝐿 < 𝑁))
12126adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → 𝐿 ∈ ℤ)
12257adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 + (#‘𝐵)) ∈ ℤ)
12359adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → 𝑁 ∈ ℤ)
124121, 122, 1233jca 1261 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
125124adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
12664, 118, 69syl2an 493 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 < 𝑁𝐿𝑁))
127126imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿𝑁)
128 simplr3 1125 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (#‘𝐵)))
129127, 128jca 553 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵))))
130125, 129, 78sylanbrc 699 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
131130ex 449 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (𝐿 < 𝑁𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
132120, 131sylbid 230 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
133132ex 449 . . . . . . . . . . . . . 14 (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
13463, 133sylbi 207 . . . . . . . . . . . . 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 480 . . . . . . . . . . 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 207 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
139138adantl 481 . . . . . . . 8 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
140139impcom 445 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
141140a1dd 50 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (¬ 𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
1421413imp 1275 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
143117, 142jca 553 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
1448pfxccatin12 41750 . . . 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 65 . . 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 4169 . 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 449 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  ifcif 4119  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974   + caddc 9977   < clt 10112  cle 10113  cmin 10304  0cn0 11330  cz 11415  ...cfz 12364  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327   prefix cpfx 41706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335  df-pfx 41707
This theorem is referenced by: (None)
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