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Theorem pfxccatin12 41852
Description: The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13612. (Contributed by AV, 9-May-2020.)
Hypothesis
Ref Expression
pfxccatin12.l 𝐿 = (♯‘𝐴)
Assertion
Ref Expression
pfxccatin12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))

Proof of Theorem pfxccatin12
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 ccatcl 13467 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
21adantr 472 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3 elfz0fzfz0 12559 . . . . 5 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑀 ∈ (0...𝑁))
43adantl 473 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝑁))
5 elfzuz2 12460 . . . . . . . . 9 (𝑀 ∈ (0...𝐿) → 𝐿 ∈ (ℤ‘0))
65adantl 473 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈ (ℤ‘0))
7 fzss1 12494 . . . . . . . 8 (𝐿 ∈ (ℤ‘0) → (𝐿...(𝐿 + (♯‘𝐵))) ⊆ (0...(𝐿 + (♯‘𝐵))))
86, 7syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (♯‘𝐵))) ⊆ (0...(𝐿 + (♯‘𝐵))))
9 ccatlen 13468 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
10 pfxccatin12.l . . . . . . . . . . . 12 𝐿 = (♯‘𝐴)
1110eqcomi 2733 . . . . . . . . . . 11 (♯‘𝐴) = 𝐿
1211oveq1i 6775 . . . . . . . . . 10 ((♯‘𝐴) + (♯‘𝐵)) = (𝐿 + (♯‘𝐵))
139, 12syl6eq 2774 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = (𝐿 + (♯‘𝐵)))
1413adantr 472 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (♯‘(𝐴 ++ 𝐵)) = (𝐿 + (♯‘𝐵)))
1514oveq2d 6781 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (♯‘𝐵))))
168, 15sseqtr4d 3748 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (♯‘𝐵))) ⊆ (0...(♯‘(𝐴 ++ 𝐵))))
1716sseld 3708 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
1817impr 650 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
19 swrdvalfn 13547 . . . 4 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
202, 4, 18, 19syl3anc 1439 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
21 swrdcl 13539 . . . . . . 7 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
22 pfxcl 41813 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉)
2321, 22anim12i 591 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
2423adantr 472 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
25 ccatvalfn 13474 . . . . 5 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
2624, 25syl 17 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
27 simpll 807 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐴 ∈ Word 𝑉)
28 simprl 811 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝐿))
29 lencl 13431 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
30 elnn0uz 11839 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
31 eluzfz2 12463 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ (ℤ‘0) → (♯‘𝐴) ∈ (0...(♯‘𝐴)))
3230, 31sylbi 207 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (0...(♯‘𝐴)))
3310, 32syl5eqel 2807 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ (0...(♯‘𝐴)))
3429, 33syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝐴)))
3534ad2antrr 764 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐿 ∈ (0...(♯‘𝐴)))
36 swrdlen 13543 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐴))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
3727, 28, 35, 36syl3anc 1439 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
38 simplr 809 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
39 lencl 13431 . . . . . . . . . . . . 13 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
4039nn0zd 11593 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
4140adantl 473 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
42 simpr 479 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
4341, 42anim12i 591 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))))
44 elfzmlbp 12565 . . . . . . . . . 10 (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
4543, 44syl 17 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
46 pfxlen 41818 . . . . . . . . 9 ((𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4738, 45, 46syl2anc 696 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4837, 47oveq12d 6783 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
49 elfz2nn0 12545 . . . . . . . . . . 11 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
50 nn0cn 11415 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
5150adantl 473 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → 𝐿 ∈ ℂ)
5251adantl 473 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
53 nn0cn 11415 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
5453ad2antrl 766 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
55 zcn 11495 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
5655adantr 472 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
5752, 54, 563jca 1379 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
5857ex 449 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
59 elfzelz 12456 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
6058, 59syl11 33 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
61603adant3 1124 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6249, 61sylbi 207 . . . . . . . . . 10 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6362imp 444 . . . . . . . . 9 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
6463adantl 473 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
65 npncan3 10432 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6664, 65syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6748, 66eqtr2d 2759 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝑀) = ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))
6867oveq2d 6781 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (0..^(𝑁𝑀)) = (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
6968fneq2d 6095 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^(𝑁𝑀)) ↔ ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))))
7026, 69mpbird 247 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^(𝑁𝑀)))
71 simprl 811 . . . . . 6 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
72 simpr 479 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
7372anim2i 594 . . . . . . 7 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝑘 ∈ (0..^(𝐿𝑀)) ∧ 𝑘 ∈ (0..^(𝑁𝑀))))
7473ancomd 466 . . . . . 6 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝑘 ∈ (0..^(𝑁𝑀)) ∧ 𝑘 ∈ (0..^(𝐿𝑀))))
7510swrdccatin12lem3 13611 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ 𝑘 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘)))
7671, 74, 75sylc 65 . . . . 5 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
7724ad2antrl 766 . . . . . . 7 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
78 simpl 474 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(𝐿𝑀)))
79 nn0fz0 12552 . . . . . . . . . . . . . . 15 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (0...(♯‘𝐴)))
8029, 79sylib 208 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ (0...(♯‘𝐴)))
8110, 80syl5eqel 2807 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝐴)))
8281ad2antrr 764 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐿 ∈ (0...(♯‘𝐴)))
8327, 28, 823jca 1379 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐴))))
8483ad2antrl 766 . . . . . . . . . 10 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐴))))
8584, 36syl 17 . . . . . . . . 9 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
8685oveq2d 6781 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))) = (0..^(𝐿𝑀)))
8778, 86eleqtrrd 2806 . . . . . . 7 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))))
88 df-3an 1074 . . . . . . 7 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉) ∧ 𝑘 ∈ (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
8977, 87, 88sylanbrc 701 . . . . . 6 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
90 ccatval1 13470 . . . . . 6 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
9189, 90syl 17 . . . . 5 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
9276, 91eqtr4d 2761 . . . 4 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘))
93 simprl 811 . . . . . 6 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))))
9472anim2i 594 . . . . . . 7 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ 𝑘 ∈ (0..^(𝑁𝑀))))
9594ancomd 466 . . . . . 6 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))))
9610pfxccatin12lem2 41851 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 prefix (𝑁𝐿))‘(𝑘 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))
9793, 95, 96sylc 65 . . . . 5 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 prefix (𝑁𝐿))‘(𝑘 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
9824ad2antrl 766 . . . . . . 7 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
99 elfzuz 12452 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ (ℤ𝐿))
100 eluzelz 11810 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝐿) → 𝑁 ∈ ℤ)
101 simpll 807 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈ ℕ0)
102 simplr 809 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℕ0)
103 simpr 479 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
104101, 102, 1033jca 1379 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
105104ex 449 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
106105ancoms 468 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
1071063adant3 1124 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
10849, 107sylbi 207 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
109100, 108syl5com 31 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
11099, 109syl 17 . . . . . . . . . . . 12 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
111110impcom 445 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
112111adantl 473 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
113112ad2antrl 766 . . . . . . . . 9 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
114 swrdccatin12lem1 13605 . . . . . . . . 9 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
115113, 95, 114sylc 65 . . . . . . . 8 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
11627, 28, 82, 36syl3anc 1439 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
117 simpr 479 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
118117adantl 473 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
11941adantl 473 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (♯‘𝐵) ∈ ℤ)
120 simpl 474 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))
121119, 120, 44syl2anc 696 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
122118, 121jca 555 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
123122ex 449 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵)))))
124123adantl 473 . . . . . . . . . . . . 13 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵)))))
125124impcom 445 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
126125, 46syl 17 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
127116, 126oveq12d 6783 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
128116, 127oveq12d 6783 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))) = ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
129128ad2antrl 766 . . . . . . . 8 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))) = ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
130115, 129eleqtrrd 2806 . . . . . . 7 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
131 df-3an 1074 . . . . . . 7 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉) ∧ 𝑘 ∈ ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))))
13298, 130, 131sylanbrc 701 . . . . . 6 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))))
133 ccatval2 13471 . . . . . 6 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉𝑘 ∈ ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘) = ((𝐵 prefix (𝑁𝐿))‘(𝑘 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
134132, 133syl 17 . . . . 5 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘) = ((𝐵 prefix (𝑁𝐿))‘(𝑘 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
13597, 134eqtr4d 2761 . . . 4 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘))
13692, 135pm2.61ian 866 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))‘𝑘))
13720, 70, 136eqfnfvd 6429 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))))
138137ex 449 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1596  wcel 2103  wss 3680  cop 4291   class class class wbr 4760   Fn wfn 5996  cfv 6001  (class class class)co 6765  cc 10047  0cc0 10049   + caddc 10052  cle 10188  cmin 10379  0cn0 11405  cz 11490  cuz 11800  ...cfz 12440  ..^cfzo 12580  chash 13232  Word cword 13398   ++ cconcat 13400   substr csubstr 13402   prefix cpfx 41808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-hash 13233  df-word 13406  df-concat 13408  df-substr 13410  df-pfx 41809
This theorem is referenced by:  pfxccat3  41853  pfxccatpfx2  41855  pfxccatin12d  41859
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