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Theorem pfxccatin12 14374
Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by AV, 9-May-2020.)
Hypothesis
Ref Expression
swrdccatin2.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 swrdccatin2.l . . . . 5 𝐿 = (♯‘𝐴)
21pfxccatin12lem2c 14371 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
3 swrdvalfn 14292 . . . 4 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
42, 3syl 17 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
5 swrdcl 14286 . . . . . 6 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
6 pfxcl 14318 . . . . . 6 (𝐵 ∈ Word 𝑉 → (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉)
7 ccatvalfn 14214 . . . . . 6 (((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))))))
85, 6, 7syl2an 595 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
98adantr 480 . . . 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))))))
10 simpll 763 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐴 ∈ Word 𝑉)
11 simprl 767 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝐿))
12 lencl 14164 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
13 nn0fz0 13283 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (0...(♯‘𝐴)))
1412, 13sylib 217 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ (0...(♯‘𝐴)))
151, 14eqeltrid 2843 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝐴)))
1615ad2antrr 722 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐿 ∈ (0...(♯‘𝐴)))
17 swrdlen 14288 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝐴))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
1810, 11, 16, 17syl3anc 1369 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
19 lencl 14164 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
2019nn0zd 12353 . . . . . . . . . . 11 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
21 elfzmlbp 13296 . . . . . . . . . . 11 (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
2220, 21sylan 579 . . . . . . . . . 10 ((𝐵 ∈ Word 𝑉𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
23 pfxlen 14324 . . . . . . . . . 10 ((𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
2422, 23syldan 590 . . . . . . . . 9 ((𝐵 ∈ Word 𝑉𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
2524ad2ant2l 742 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (♯‘(𝐵 prefix (𝑁𝐿))) = (𝑁𝐿))
2618, 25oveq12d 7273 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))) = ((𝐿𝑀) + (𝑁𝐿)))
27 elfz2nn0 13276 . . . . . . . . . . 11 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
28 nn0cn 12173 . . . . . . . . . . . . . . . 16 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
2928ad2antll 725 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
30 nn0cn 12173 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
3130ad2antrl 724 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
32 zcn 12254 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3332adantr 480 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
3429, 31, 333jca 1126 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
3534ex 412 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
36 elfzelz 13185 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ ℤ)
3735, 36syl11 33 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
38373adant3 1130 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
3927, 38sylbi 216 . . . . . . . . . 10 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
4039imp 406 . . . . . . . . 9 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
41 npncan3 11189 . . . . . . . . 9 ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
4240, 41syl 17 . . . . . . . 8 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
4342adantl 481 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
4426, 43eqtr2d 2779 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝑀) = ((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿)))))
4544oveq2d 7271 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (0..^(𝑁𝑀)) = (0..^((♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (♯‘(𝐵 prefix (𝑁𝐿))))))
4645fneq2d 6511 . . . 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)))))))
479, 46mpbird 256 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 prefix (𝑁𝐿))) Fn (0..^(𝑁𝑀)))
48 simprl 767 . . . . . 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))))))
49 simpr 484 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
5049anim2i 616 . . . . . . 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))))
5150ancomd 461 . . . . . 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))))
521pfxccatin12lem3 14373 . . . . . 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)))
5348, 51, 52sylc 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))
545, 6anim12i 612 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
5554adantr 480 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 prefix (𝑁𝐿)) ∈ Word 𝑉))
5655ad2antrl 724 . . . . . . 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))
57 simpl 482 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(𝐿𝑀)))
5818oveq2d 7271 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (0..^(♯‘(𝐴 substr ⟨𝑀, 𝐿⟩))) = (0..^(𝐿𝑀)))
5958ad2antrl 724 . . . . . . . 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)))
6057, 59eleqtrrd 2842 . . . . . . 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⟩))))
61 df-3an 1087 . . . . . . 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⟩)))))
6256, 60, 61sylanbrc 582 . . . . . 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⟩)))))
63 ccatval1 14209 . . . . . 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))
6462, 63syl 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))
6553, 64eqtr4d 2781 . . . 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))
66 simprl 767 . . . . . 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))))))
6749anim2i 616 . . . . . . 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))))
6867ancomd 461 . . . . . 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))))
691pfxccatin12lem2 14372 . . . . . 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⟩))))))
7066, 68, 69sylc 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⟩)))))
7155ad2antrl 724 . . . . . . 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))
72 elfzuz 13181 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → 𝑁 ∈ (ℤ𝐿))
73 eluzelz 12521 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝐿) → 𝑁 ∈ ℤ)
74 id 22 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
75743expia 1119 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
7675ancoms 458 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
77763adant3 1130 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
7827, 77sylbi 216 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
7973, 78syl5com 31 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
8072, 79syl 17 . . . . . . . . . . . 12 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
8180impcom 407 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
8281adantl 481 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
8382ad2antrl 724 . . . . . . . . 9 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
84 pfxccatin12lem4 14367 . . . . . . . . 9 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
8583, 68, 84sylc 65 . . . . . . . 8 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
8618, 26oveq12d 7273 . . . . . . . . 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))))
8786ad2antrl 724 . . . . . . . 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))))
8885, 87eleqtrrd 2842 . . . . . . 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))))))
89 df-3an 1087 . . . . . . 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)))))))
9071, 88, 89sylanbrc 582 . . . . . 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)))))))
91 ccatval2 14211 . . . . . 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⟩)))))
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) = ((𝐵 prefix (𝑁𝐿))‘(𝑘 − (♯‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
9370, 92eqtr4d 2781 . . . 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))
9465, 93pm2.61ian 808 . . 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))
954, 47, 94eqfnfvd 6894 . 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))))
9695ex 412 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 395  w3a 1085   = wceq 1539  wcel 2108  cop 4564   class class class wbr 5070   Fn wfn 6413  cfv 6418  (class class class)co 7255  cc 10800  0cc0 10802   + caddc 10805  cle 10941  cmin 11135  0cn0 12163  cz 12249  cuz 12511  ...cfz 13168  ..^cfzo 13311  chash 13972  Word cword 14145   ++ cconcat 14201   substr csubstr 14281   prefix cpfx 14311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-substr 14282  df-pfx 14312
This theorem is referenced by:  pfxccat3  14375  pfxccatpfx2  14378  pfxccatin12d  14386
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