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Theorem pfxccatin12 40288
 Description: The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13342. (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 13212 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
21adantr 480 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
3 elfz0fzfz0 12313 . . . . 5 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑀 ∈ (0...𝑁))
43adantl 481 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝑁))
5 elfzuz2 12217 . . . . . . . . 9 (𝑀 ∈ (0...𝐿) → 𝐿 ∈ (ℤ‘0))
65adantl 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈ (ℤ‘0))
7 fzss1 12251 . . . . . . . 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 13213 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
10 pfxccatin12.l . . . . . . . . . . . 12 𝐿 = (#‘𝐴)
1110eqcomi 2619 . . . . . . . . . . 11 (#‘𝐴) = 𝐿
1211oveq1i 6559 . . . . . . . . . 10 ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))
139, 12syl6eq 2660 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
1413adantr 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (#‘(𝐴 ++ 𝐵)) = (𝐿 + (#‘𝐵)))
1514oveq2d 6565 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(#‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (#‘𝐵))))
168, 15sseqtr4d 3605 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(#‘(𝐴 ++ 𝐵))))
1716sseld 3567 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
1817impr 647 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
19 swrdvalfn 13278 . . . 4 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
202, 4, 18, 19syl3anc 1318 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
21 swrdcl 13271 . . . . . . 7 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
22 pfxcl 40249 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉)
2321, 22anim12i 588 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
2423adantr 480 . . . . 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 13218 . . . . 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 786 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐴 ∈ Word 𝑉)
28 simprl 790 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝐿))
29 lencl 13179 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
30 elnn0uz 11601 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (ℤ‘0))
31 eluzfz2 12220 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ (ℤ‘0) → (#‘𝐴) ∈ (0...(#‘𝐴)))
3230, 31sylbi 206 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...(#‘𝐴)))
3310, 32syl5eqel 2692 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0𝐿 ∈ (0...(#‘𝐴)))
3429, 33syl 17 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
3534ad2antrr 758 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
36 swrdlen 13275 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
3727, 28, 35, 36syl3anc 1318 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
38 simplr 788 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉)
39 lencl 13179 . . . . . . . . . . . . 13 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4039nn0zd 11356 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
4140adantl 481 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ)
42 simpr 476 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
4341, 42anim12i 588 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
44 elfzmlbp 12319 . . . . . . . . . 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 40254 . . . . . . . . 9 ((𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))) → (#‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4738, 45, 46syl2anc 691 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
4837, 47oveq12d 6567 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
49 elfz2nn0 12300 . . . . . . . . . . 11 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
50 elfzelz 12213 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
51 nn0cn 11179 . . . . . . . . . . . . . . . . . 18 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
5251adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → 𝐿 ∈ ℂ)
5352adantl 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
54 nn0cn 11179 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
5554ad2antrl 760 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
56 zcn 11259 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
5756adantr 480 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
5853, 55, 573jca 1235 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
5958ex 449 . . . . . . . . . . . . . 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 1074 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6349, 62sylbi 206 . . . . . . . . . 10 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6463imp 444 . . . . . . . . 9 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
6564adantl 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
66 npncan3 10198 . . . . . . . 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 2645 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁𝑀) = ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))))
6968oveq2d 6565 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^(𝑁𝑀)) = (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿))))))
7069fneq2d 5896 . . . 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 246 . . 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 790 . . . . . 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 476 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
7473anim2i 591 . . . . . . 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 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))))
7610swrdccatin12lem3 13341 . . . . . 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 63 . . . . 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 760 . . . . . . 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 472 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(𝐿𝑀)))
80 nn0fz0 12306 . . . . . . . . . . . . . . 15 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴)))
8129, 80sylib 207 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (0...(#‘𝐴)))
8210, 81syl5eqel 2692 . . . . . . . . . . . . 13 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
8382ad2antrr 758 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
8427, 28, 833jca 1235 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))))
8584ad2antrl 760 . . . . . . . . . 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 6565 . . . . . . . 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 2691 . . . . . . 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 1033 . . . . . . 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 695 . . . . . 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 13214 . . . . . 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 2647 . . . 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 790 . . . . . 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 591 . . . . . . 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 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))))
9710pfxccatin12lem2 40287 . . . . . 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 63 . . . . 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 760 . . . . . . 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 12209 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (ℤ𝐿))
101 eluzelz 11573 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝐿) → 𝑁 ∈ ℤ)
102 simpll 786 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈ ℕ0)
103 simplr 788 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℕ0)
104 simpr 476 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
105102, 103, 1043jca 1235 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
106105ex 449 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
107106ancoms 468 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
1081073adant3 1074 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
10949, 108sylbi 206 . . . . . . . . . . . . . 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 445 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
113112adantl 481 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
114113ad2antrl 760 . . . . . . . . 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 13335 . . . . . . . . 9 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
116114, 96, 115sylc 63 . . . . . . . 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 1318 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
118 simpr 476 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
119118adantl 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
12041adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (#‘𝐵) ∈ ℤ)
121 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
122120, 121, 44syl2anc 691 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁𝐿) ∈ (0...(#‘𝐵)))
123119, 122jca 553 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))))
124123ex 449 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
125124adantl 481 . . . . . . . . . . . . 13 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
126125impcom 445 . . . . . . . . . . . 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 6567 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
129117, 128oveq12d 6567 . . . . . . . . 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 760 . . . . . . . 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 2691 . . . . . . 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 1033 . . . . . . 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 695 . . . . . 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 13215 . . . . . 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 2647 . . . 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 827 . . 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 6222 . 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 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 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   ≤ cle 9954   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   prefix cpfx 40244 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  df-pfx 40245 This theorem is referenced by:  pfxccat3  40289  pfxccatpfx2  40291  pfxccatin12d  40295
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