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Theorem pfxccat3 13741
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 AV, 10-May-2020.)
Hypothesis
Ref Expression
swrdccatin2.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 783 . . . 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 13505 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
4 elfznn0 12640 . . . . . . . . . . . . . 14 (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℕ0)
54adantr 472 . . . . . . . . . . . . 13 ((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) → 𝑁 ∈ ℕ0)
65adantr 472 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ ℕ0)
7 simplr 785 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → (♯‘𝐴) ∈ ℕ0)
8 swrdccatin2.l . . . . . . . . . . . . . . 15 𝐿 = (♯‘𝐴)
98breq2i 4817 . . . . . . . . . . . . . 14 (𝑁𝐿𝑁 ≤ (♯‘𝐴))
109biimpi 207 . . . . . . . . . . . . 13 (𝑁𝐿𝑁 ≤ (♯‘𝐴))
1110adantl 473 . . . . . . . . . . . 12 (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ≤ (♯‘𝐴))
12 elfz2nn0 12638 . . . . . . . . . . . 12 (𝑁 ∈ (0...(♯‘𝐴)) ↔ (𝑁 ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0𝑁 ≤ (♯‘𝐴)))
136, 7, 11, 12syl3anbrc 1443 . . . . . . . . . . 11 (((𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
1413exp31 410 . . . . . . . . . 10 (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (𝑁𝐿𝑁 ∈ (0...(♯‘𝐴)))))
1514adantl 473 . . . . . . . . 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 472 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿𝑁 ∈ (0...(♯‘𝐴)))))
1817imp 395 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿𝑁 ∈ (0...(♯‘𝐴))))
1918imp 395 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁𝐿) → 𝑁 ∈ (0...(♯‘𝐴)))
202, 19jca 507 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ 𝑁𝐿) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘𝐴))))
21 swrdccatin1 13729 . . . 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 1254 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
248eleq1i 2835 . . . . . . . . . . 11 (𝐿 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0)
25 elfz2nn0 12638 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝑁) ↔ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁))
26 nn0z 11647 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℕ0𝐿 ∈ ℤ)
2726adantl 473 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝐿 ∈ ℤ)
28 nn0z 11647 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
29283ad2ant2 1164 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑁 ∈ ℤ)
3029adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑁 ∈ ℤ)
31 nn0z 11647 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℕ0𝑀 ∈ ℤ)
32313ad2ant1 1163 . . . . . . . . . . . . . . . . . . 19 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → 𝑀 ∈ ℤ)
3332adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀 ∈ ℤ)
3427, 30, 333jca 1158 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
3534adantr 472 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
36 simpl3 1246 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) → 𝑀𝑁)
3736anim1i 608 . . . . . . . . . . . . . . . . 17 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝑀𝑁𝐿𝑀))
3837ancomd 453 . . . . . . . . . . . . . . . 16 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → (𝐿𝑀𝑀𝑁))
39 elfz2 12540 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
4035, 38, 39sylanbrc 578 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) ∧ 𝐿 ∈ ℕ0) ∧ 𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
4140exp31 410 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4225, 41sylbi 208 . . . . . . . . . . . . 13 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4342adantr 472 . . . . . . . . . . . 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 226 . . . . . . . . . 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 472 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (𝐿𝑀𝑀 ∈ (𝐿...𝑁))))
4847imp 395 . . . . . . 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 1137 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑀 ∈ (𝐿...𝑁))
51 elfz2nn0 12638 . . . . . . . . . . . 12 (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))))
52 nn0z 11647 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℤ)
538, 52syl5eqel 2848 . . . . . . . . . . . . . . . . 17 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℤ)
5453adantr 472 . . . . . . . . . . . . . . . 16 (((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿) → 𝐿 ∈ ℤ)
5554adantl 473 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿 ∈ ℤ)
56 nn0z 11647 . . . . . . . . . . . . . . . . 17 ((𝐿 + (♯‘𝐵)) ∈ ℕ0 → (𝐿 + (♯‘𝐵)) ∈ ℤ)
57563ad2ant2 1164 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
5857adantr 472 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
59283ad2ant1 1163 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
6059adantr 472 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ ℤ)
6155, 58, 603jca 1158 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
628eqcomi 2774 . . . . . . . . . . . . . . . . . . 19 (♯‘𝐴) = 𝐿
6362eleq1i 2835 . . . . . . . . . . . . . . . . . 18 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ ℕ0)
64 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . 22 (𝐿 ∈ ℕ0𝐿 ∈ ℝ)
65 nn0re 11548 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
66 ltnle 10371 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6764, 65, 66syl2anr 590 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁 ↔ ¬ 𝑁𝐿))
6867bicomd 214 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿 < 𝑁))
69 ltle 10380 . . . . . . . . . . . . . . . . . . . . 21 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝐿 < 𝑁𝐿𝑁))
7064, 65, 69syl2anr 590 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 < 𝑁𝐿𝑁))
7168, 70sylbid 231 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ0𝐿 ∈ ℕ0) → (¬ 𝑁𝐿𝐿𝑁))
7271ex 401 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7363, 72syl5bi 233 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ0 → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
74733ad2ant1 1163 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝐿𝑁)))
7574imp32 409 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝐿𝑁)
76 simpl3 1246 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
7775, 76jca 507 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → (𝐿𝑁𝑁 ≤ (𝐿 + (♯‘𝐵))))
78 elfz2 12540 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ ((𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐿𝑁𝑁 ≤ (𝐿 + (♯‘𝐵)))))
7961, 77, 78sylanbrc 578 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ ¬ 𝑁𝐿)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
8079exp32 411 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
8151, 80sylbi 208 . . . . . . . . . . 11 (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
8281adantl 473 . . . . . . . . . 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 472 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
8584imp 395 . . . . . . 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 1137 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
8850, 87jca 507 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿𝐿𝑀) → (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
898swrdccatin2 13734 . . . 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 1254 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
92 nn0re 11548 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℕ0𝑀 ∈ ℝ)
9392adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → 𝑀 ∈ ℝ)
94 ltnle 10371 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9593, 64, 94syl2anr 590 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿 ↔ ¬ 𝐿𝑀))
9695bicomd 214 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 < 𝐿))
97 simpll 783 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ ℕ0)
98 simplr 785 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝐿 ∈ ℕ0)
99 ltle 10380 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (𝑀 < 𝐿𝑀𝐿))
10092, 64, 99syl2an 589 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑀 < 𝐿𝑀𝐿))
101100imp 395 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀𝐿)
102 elfz2nn0 12638 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
10397, 98, 101, 102syl3anbrc 1443 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) ∧ 𝑀 < 𝐿) → 𝑀 ∈ (0...𝐿))
104103exp31 410 . . . . . . . . . . . . . . . . . 18 (𝑀 ∈ ℕ0 → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
105104adantr 472 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (𝑀 < 𝐿𝑀 ∈ (0...𝐿))))
106105impcom 396 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (𝑀 < 𝐿𝑀 ∈ (0...𝐿)))
10796, 106sylbid 231 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿)))
108107expcom 402 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
1091083adant3 1162 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ0𝑁 ∈ ℕ0𝑀𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11025, 109sylbi 208 . . . . . . . . . . . 12 (𝑀 ∈ (0...𝑁) → (𝐿 ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
11163, 110syl5bi 233 . . . . . . . . . . 11 (𝑀 ∈ (0...𝑁) → ((♯‘𝐴) ∈ ℕ0 → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
112111adantr 472 . . . . . . . . . 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 472 . . . . . . . 8 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → (¬ 𝐿𝑀𝑀 ∈ (0...𝐿))))
115114imp 395 . . . . . . 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 1137 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑀 ∈ (0...𝐿))
118653ad2ant1 1163 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℝ)
11966bicomd 214 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (¬ 𝑁𝐿𝐿 < 𝑁))
12064, 118, 119syl2an 589 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁𝐿𝐿 < 𝑁))
12126adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝐿 ∈ ℤ)
12257adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 + (♯‘𝐵)) ∈ ℤ)
12359adantl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → 𝑁 ∈ ℤ)
124121, 122, 1233jca 1158 . . . . . . . . . . . . . . . . . . 19 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
125124adantr 472 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿 ∈ ℤ ∧ (𝐿 + (♯‘𝐵)) ∈ ℤ ∧ 𝑁 ∈ ℤ))
12664, 118, 69syl2an 589 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁𝐿𝑁))
127126imp 395 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝐿𝑁)
128 simplr3 1279 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ≤ (𝐿 + (♯‘𝐵)))
129127, 128jca 507 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → (𝐿𝑁𝑁 ≤ (𝐿 + (♯‘𝐵))))
130125, 129, 78sylanbrc 578 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) ∧ 𝐿 < 𝑁) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
131130ex 401 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (𝐿 < 𝑁𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
132120, 131sylbid 231 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℕ0 ∧ (𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵)))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
133132ex 401 . . . . . . . . . . . . . 14 (𝐿 ∈ ℕ0 → ((𝑁 ∈ ℕ0 ∧ (𝐿 + (♯‘𝐵)) ∈ ℕ0𝑁 ≤ (𝐿 + (♯‘𝐵))) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
13463, 133sylbi 208 . . . . . . . . . . . . 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 472 . . . . . . . . . . 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 208 . . . . . . . . 9 (𝑁 ∈ (0...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
139138adantl 473 . . . . . . . 8 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (¬ 𝑁𝐿𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
140139impcom 396 . . . . . . 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 1137 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
143117, 142jca 507 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(𝐿 + (♯‘𝐵))))) ∧ ¬ 𝑁𝐿 ∧ ¬ 𝐿𝑀) → (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
1448pfxccatin12 13739 . . . 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 4296 . 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 401 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  ifcif 4243  cop 4340   class class class wbr 4809  cfv 6068  (class class class)co 6842  cr 10188  0cc0 10189   + caddc 10192   < clt 10328  cle 10329  cmin 10520  0cn0 11538  cz 11624  ...cfz 12533  chash 13321  Word cword 13486   ++ cconcat 13541   substr csubstr 13616   prefix cpfx 13661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-concat 13542  df-substr 13617  df-pfx 13662
This theorem is referenced by:  swrdccat  13743  swrdccat3b  13750
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