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Theorem pfxccatin12 40089
Description: The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13284. (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 13154 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
21adantr 479 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3 elfz0fzfz0 12264 . . . . 5 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...𝑁))
43adantl 480 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝑁))
5 elfzuz2 12168 . . . . . . . . 9 (𝑀 ∈ (0...𝐿) → 𝐿 ∈ (ℤ‘0))
65adantl 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈ (ℤ‘0))
7 fzss1 12202 . . . . . . . 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 13155 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
10 pfxccatin12.l . . . . . . . . . . . 12 𝐿 = (#‘𝐴)
1110eqcomi 2614 . . . . . . . . . . 11 (#‘𝐴) = 𝐿
1211oveq1i 6533 . . . . . . . . . 10 ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))
139, 12syl6eq 2655 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
1413adantr 479 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
1514oveq2d 6539 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(#‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (#‘𝐵))))
168, 15sseqtr4d 3600 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(#‘(𝐴 ++ 𝐵))))
1716sseld 3562 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
1817impr 646 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
19 swrdvalfn 13220 . . . 4 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
202, 4, 18, 19syl3anc 1317 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
21 swrdcl 13213 . . . . . . 7 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
22 pfxcl 40050 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉)
2321, 22anim12i 587 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
2423adantr 479 . . . . 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 13160 . . . . 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 785 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐴 ∈ Word 𝑉)
28 simprl 789 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝐿))
29 lencl 13121 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
30 elnn0uz 11553 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (ℤ‘0))
31 eluzfz2 12171 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ (ℤ‘0) → (#‘𝐴) ∈ (0...(#‘𝐴)))
3230, 31sylbi 205 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...(#‘𝐴)))
3310, 32syl5eqel 2687 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0𝐿 ∈ (0...(#‘𝐴)))
3429, 33syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
3534ad2antrr 757 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
36 swrdlen 13217 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
3727, 28, 35, 36syl3anc 1317 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
38 simplr 787 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉)
39 lencl 13121 . . . . . . . . . . . . 13 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4039nn0zd 11308 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
4140adantl 480 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ)
42 simpr 475 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
4341, 42anim12i 587 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
44 elfzmlbp 12270 . . . . . . . . . 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 40055 . . . . . . . . 9 ((𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))) → (#‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4738, 45, 46syl2anc 690 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4837, 47oveq12d 6541 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
49 elfz2nn0 12251 . . . . . . . . . . 11 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
50 elfzelz 12164 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
51 nn0cn 11145 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
5251adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → 𝐿 ∈ ℂ)
5352adantl 480 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
54 nn0cn 11145 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
5554ad2antrl 759 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
56 zcn 11211 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
5756adantr 479 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
5853, 55, 573jca 1234 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
5958ex 448 . . . . . . . . . . . . . 14 (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6050, 59syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6160com12 32 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
62613adant3 1073 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6349, 62sylbi 205 . . . . . . . . . 10 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6463imp 443 . . . . . . . . 9 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
6564adantl 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
66 npncan3 10166 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6765, 66syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6848, 67eqtr2d 2640 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁𝑀) = ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))))
6968oveq2d 6539 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^(𝑁𝑀)) = (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿))))))
7069fneq2d 5878 . . . 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)))))))
7126, 70mpbird 245 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^(𝑁𝑀)))
72 simprl 789 . . . . . 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))))))
73 simpr 475 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
7473anim2i 590 . . . . . . 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))))
7574ancomd 465 . . . . . 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))))
7610swrdccatin12lem3 13283 . . . . . 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)))
7772, 75, 76sylc 62 . . . . 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))
7824ad2antrl 759 . . . . . . 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))
79 simpl 471 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(𝐿𝑀)))
80 nn0fz0 12257 . . . . . . . . . . . . . . 15 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴)))
8129, 80sylib 206 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (0...(#‘𝐴)))
8210, 81syl5eqel 2687 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
8382ad2antrr 757 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
8427, 28, 833jca 1234 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))))
8584ad2antrl 759 . . . . . . . . . 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))))
8685, 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))
8786oveq2d 6539 . . . . . . . 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)))
8879, 87eleqtrrd 2686 . . . . . . 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⟩))))
89 df-3an 1032 . . . . . . 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⟩)))))
9078, 88, 89sylanbrc 694 . . . . . 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⟩)))))
91 ccatval1 13156 . . . . . 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))
9290, 91syl 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))
9377, 92eqtr4d 2642 . . . 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))
94 simprl 789 . . . . . 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))))))
9573anim2i 590 . . . . . . 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))))
9695ancomd 465 . . . . . 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))))
9710pfxccatin12lem2 40088 . . . . . 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⟩))))))
9894, 96, 97sylc 62 . . . . 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⟩)))))
9924ad2antrl 759 . . . . . . 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))
100 elfzuz 12160 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (ℤ𝐿))
101 eluzelz 11525 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝐿) → 𝑁 ∈ ℤ)
102 simpll 785 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈ ℕ0)
103 simplr 787 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℕ0)
104 simpr 475 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
105102, 103, 1043jca 1234 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
106105ex 448 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
107106ancoms 467 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
1081073adant3 1073 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
10949, 108sylbi 205 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
110101, 109syl5com 31 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
111100, 110syl 17 . . . . . . . . . . . 12 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
112111impcom 444 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
113112adantl 480 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
114113ad2antrl 759 . . . . . . . . 9 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
115 swrdccatin12lem1 13277 . . . . . . . . 9 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
116114, 96, 115sylc 62 . . . . . . . 8 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
11727, 28, 83, 36syl3anc 1317 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
118 simpr 475 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
119118adantl 480 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
12041adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (#‘𝐵) ∈ ℤ)
121 simpl 471 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
122120, 121, 44syl2anc 690 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁𝐿) ∈ (0...(#‘𝐵)))
123119, 122jca 552 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))))
124123ex 448 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
125124adantl 480 . . . . . . . . . . . . 13 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
126125impcom 444 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))))
127126, 46syl 17 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
128117, 127oveq12d 6541 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
129117, 128oveq12d 6541 . . . . . . . . 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))))
130129ad2antrl 759 . . . . . . . 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))))
131116, 130eleqtrrd 2686 . . . . . . 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))))))
132 df-3an 1032 . . . . . . 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)))))))
13399, 131, 132sylanbrc 694 . . . . . 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)))))))
134 ccatval2 13157 . . . . . 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⟩)))))
135133, 134syl 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⟩)))))
13698, 135eqtr4d 2642 . . . 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))
13793, 136pm2.61ian 826 . . 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))
13820, 71, 137eqfnfvd 6203 . 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))))
139138ex 448 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 382  w3a 1030   = wceq 1474  wcel 1975  wss 3535  cop 4126   class class class wbr 4573   Fn wfn 5781  cfv 5786  (class class class)co 6523  cc 9786  0cc0 9788   + caddc 9791  cle 9927  cmin 10113  0cn0 11135  cz 11206  cuz 11515  ...cfz 12148  ..^cfzo 12285  #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:  pfxccat3  40090  pfxccatpfx2  40092  pfxccatin12d  40096
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