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Theorem swrdccat3 13343
 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. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 28-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l 𝐿 = (#‘𝐴)
Assertion
Ref Expression
swrdccat3 ((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 substr ⟨0, (𝑁𝐿)⟩))))))

Proof of Theorem swrdccat3
StepHypRef Expression
1 simpll 786 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
2 simplrl 796 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ 𝑁𝐿) → 𝑀 ∈ (0...𝑁))
3 lencl 13179 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
4 elfznn0 12302 . . . . . . . . . . . . . 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 788 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → (#‘𝐴) ∈ ℕ0)
8 swrdccatin12.l . . . . . . . . . . . . . . 15 𝐿 = (#‘𝐴)
98breq2i 4591 . . . . . . . . . . . . . 14 (𝑁𝐿𝑁 ≤ (#‘𝐴))
109biimpi 205 . . . . . . . . . . . . 13 (𝑁𝐿𝑁 ≤ (#‘𝐴))
1110adantl 481 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ≤ (#‘𝐴))
12 elfz2nn0 12300 . . . . . . . . . . . 12 (𝑁 ∈ (0...(#‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (#‘𝐴) ∈ ℕ0𝑁 ≤ (#‘𝐴)))
136, 7, 11, 12syl3anbrc 1239 . . . . . . . . . . 11 (((𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ∧ (#‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(#‘𝐴)))
1413exp31 628 . . . . . . . . . 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 13334 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝐴))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐴 substr ⟨𝑀, 𝑁⟩)))
221, 20, 21sylc 63 . . 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 1078 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
248eleq1i 2679 . . . . . . . . . . 11 (𝐿 ∈ ℕ0 ↔ (#‘𝐴) ∈ ℕ0)
25 elfz2nn0 12300 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
26 nn0z 11277 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
2726adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
28 nn0z 11277 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
29283ad2ant2 1076 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑁 ∈ ℤ)
3029adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
31 nn0z 11277 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℤ)
32313ad2ant1 1075 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℤ)
3332adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
3427, 30, 333jca 1235 . . . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀𝑁)
3736anim1i 590 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝑀𝑁𝐿𝑀))
3837ancomd 466 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿𝑀𝑀𝑁))
39 elfz2 12204 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
4035, 38, 39sylanbrc 695 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
4140exp31 628 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4225, 41sylbi 206 . . . . . . . . . . . . 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 224 . . . . . . . . . 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 1249 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
51 elfz2nn0 12300 . . . . . . . . . . . 12 (𝑁 ∈ (0...(𝐿 + (#‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))))
52 nn0z 11277 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℤ)
538, 52syl5eqel 2692 . . . . . . . . . . . . . . . . 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 11277 . . . . . . . . . . . . . . . . 17 ((𝐿 + (#‘𝐵)) ∈ ℕ0 → (𝐿 + (#‘𝐵)) ∈ ℤ)
57563ad2ant2 1076 . . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
628eqcomi 2619 . . . . . . . . . . . . . . . . . . 19 (#‘𝐴) = 𝐿
6362eleq1i 2679 . . . . . . . . . . . . . . . . . 18 ((#‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
64 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
65 nn0re 11178 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
66 ltnle 9996 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6764, 65, 66syl2anr 494 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6867bicomd 212 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿 < 𝑁))
69 ltle 10005 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁𝐿𝑁))
7064, 65, 69syl2anr 494 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁𝐿𝑁))
7168, 70sylbid 229 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿𝑁))
7271ex 449 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7363, 72syl5bi 231 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
74733ad2ant1 1075 . . . . . . . . . . . . . . . 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 1059 . . . . . . . . . . . . . . 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 12204 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (#‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿𝑁𝑁 ≤ (𝐿 + (#‘𝐵)))))
7961, 77, 78sylanbrc 695 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) ∧ ((#‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
8079exp32 629 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → ((#‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
8151, 80sylbi 206 . . . . . . . . . . 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 48 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
87863imp 1249 . . . . 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 13338 . . . 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 63 . . 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 1078 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
92 nn0re 11178 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
9392adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
94 ltnle 9996 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9593, 64, 94syl2anr 494 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9695bicomd 212 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 < 𝐿))
97 simpll 786 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
98 simplr 788 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
99 ltle 10005 . . . . . . . . . . . . . . . . . . . . . 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 12300 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
10397, 98, 101, 102syl3anbrc 1239 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
104103exp31 628 . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿)))
108107expcom 450 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1091083adant3 1074 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11025, 109sylbi 206 . . . . . . . . . . . 12 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11163, 110syl5bi 231 . . . . . . . . . . 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 1249 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑀 ∈ (0...𝐿))
118653ad2ant1 1075 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (#‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (#‘𝐵))) → 𝑁 ∈ ℝ)
11966bicomd 212 . . . . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . . . . . . . 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 1098 . . . . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . 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 206 . . . . . . . . . . . . 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 206 . . . . . . . . 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 48 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (#‘𝐵))))) → (¬ 𝑁𝐿 → (¬ 𝐿𝑀𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))))
1421413imp 1249 . . . . 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)))))
1448swrdccatin12 13342 . . . 4 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))
14591, 143, 144sylc 63 . . 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 substr ⟨0, (𝑁𝐿)⟩)))
14622, 90, 1452if2 4086 . 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 substr ⟨0, (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 substr ⟨0, (𝑁𝐿)⟩))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ifcif 4036  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158 This theorem is referenced by:  swrdccat  13344  swrdccat3a  13345  swrdccat3b  13347
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