MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swrdccatin12 Structured version   Visualization version   GIF version

Theorem swrdccatin12 13537
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 Alexander van der Vekens, 27-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l 𝐿 = (#‘𝐴)
Assertion
Ref Expression
swrdccatin12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))

Proof of Theorem swrdccatin12
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 ccatcl 13392 . . . . 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 12483 . . . . 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 12384 . . . . . . . . 9 (𝑀 ∈ (0...𝐿) → 𝐿 ∈ (ℤ‘0))
65adantl 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → 𝐿 ∈ (ℤ‘0))
7 fzss1 12418 . . . . . . . 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 13393 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
10 swrdccatin12.l . . . . . . . . . . . 12 𝐿 = (#‘𝐴)
1110eqcomi 2660 . . . . . . . . . . 11 (#‘𝐴) = 𝐿
1211oveq1i 6700 . . . . . . . . . 10 ((#‘𝐴) + (#‘𝐵)) = (𝐿 + (#‘𝐵))
139, 12syl6eq 2701 . . . . . . . . 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 6706 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (0...(#‘(𝐴 ++ 𝐵))) = (0...(𝐿 + (#‘𝐵))))
168, 15sseqtr4d 3675 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝐿...(𝐿 + (#‘𝐵))) ⊆ (0...(#‘(𝐴 ++ 𝐵))))
1716sseld 3635 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑀 ∈ (0...𝐿)) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))))
1817impr 648 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵))))
19 swrdvalfn 13472 . . . 4 (((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐴 ++ 𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
202, 4, 18, 19syl3anc 1366 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
21 swrdcl 13464 . . . . . . 7 (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉)
22 swrdcl 13464 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉)
2321, 22anim12i 589 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (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 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉))
25 ccatvalfn 13399 . . . . 5 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)))))
2624, 25syl 17 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)))))
27 simpll 805 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐴 ∈ Word 𝑉)
28 simprl 809 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝑀 ∈ (0...𝐿))
29 lencl 13356 . . . . . . . . . . . 12 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ0)
30 elnn0uz 11763 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (ℤ‘0))
31 eluzfz2 12387 . . . . . . . . . . . . . 14 ((#‘𝐴) ∈ (ℤ‘0) → (#‘𝐴) ∈ (0...(#‘𝐴)))
3230, 31sylbi 207 . . . . . . . . . . . . 13 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...(#‘𝐴)))
3310, 32syl5eqel 2734 . . . . . . . . . . . 12 ((#‘𝐴) ∈ ℕ0𝐿 ∈ (0...(#‘𝐴)))
3429, 33syl 17 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
3534adantr 480 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴)))
3635adantr 480 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
37 swrdlen 13468 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
3827, 28, 36, 37syl3anc 1366 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
39 simpr 476 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
4039adantr 480 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐵 ∈ Word 𝑉)
41 lencl 13356 . . . . . . . . . . . . 13 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ0)
4241nn0zd 11518 . . . . . . . . . . . 12 (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
4342adantl 481 . . . . . . . . . . 11 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (#‘𝐵) ∈ ℤ)
44 simpr 476 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
4543, 44anim12i 589 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))))
46 elfzmlbp 12489 . . . . . . . . . 10 (((#‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝑁𝐿) ∈ (0...(#‘𝐵)))
4745, 46syl 17 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁𝐿) ∈ (0...(#‘𝐵)))
48 swrd0len 13467 . . . . . . . . 9 ((𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))) → (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)) = (𝑁𝐿))
4940, 47, 48syl2anc 694 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)) = (𝑁𝐿))
5038, 49oveq12d 6708 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))) = ((𝐿𝑀) + (𝑁𝐿)))
51 elfz2nn0 12469 . . . . . . . . . . 11 (𝑀 ∈ (0...𝐿) ↔ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿))
52 nn0cn 11340 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℕ0𝐿 ∈ ℂ)
5352adantl 481 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → 𝐿 ∈ ℂ)
5453adantl 481 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝐿 ∈ ℂ)
55 nn0cn 11340 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ0𝑀 ∈ ℂ)
5655ad2antrl 764 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑀 ∈ ℂ)
57 zcn 11420 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
5857adantr 480 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → 𝑁 ∈ ℂ)
5954, 56, 583jca 1261 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℤ ∧ (𝑀 ∈ ℕ0𝐿 ∈ ℕ0)) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
6059ex 449 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
61 elfzelz 12380 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ ℤ)
6260, 61syl11 33 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
63623adant3 1101 . . . . . . . . . . 11 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6451, 63sylbi 207 . . . . . . . . . 10 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ)))
6564imp 444 . . . . . . . . 9 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
6665adantl 481 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ))
67 npncan3 10357 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6866, 67syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐿𝑀) + (𝑁𝐿)) = (𝑁𝑀))
6950, 68eqtr2d 2686 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝑁𝑀) = ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))
7069oveq2d 6706 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (0..^(𝑁𝑀)) = (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)))))
7170fneq2d 6020 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)) Fn (0..^(𝑁𝑀)) ↔ ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)) Fn (0..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))))
7226, 71mpbird 247 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)) Fn (0..^(𝑁𝑀)))
73 simprl 809 . . . . . 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))))))
74 simpr 476 . . . . . . . 8 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → 𝑘 ∈ (0..^(𝑁𝑀)))
7574anim2i 592 . . . . . . 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))))
7675ancomd 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))))
7710swrdccatin12lem3 13536 . . . . . 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)))
7873, 76, 77sylc 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))
7924ad2antrl 764 . . . . . . 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 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉))
80 simpl 472 . . . . . . . 8 ((𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ (0..^(𝐿𝑀)))
81 nn0fz0 12476 . . . . . . . . . . . . . . . 16 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴)))
8229, 81sylib 208 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ (0...(#‘𝐴)))
8310, 82syl5eqel 2734 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉𝐿 ∈ (0...(#‘𝐴)))
8483adantr 480 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐿 ∈ (0...(#‘𝐴)))
8584adantr 480 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → 𝐿 ∈ (0...(#‘𝐴)))
8627, 28, 853jca 1261 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐴 ∈ Word 𝑉𝑀 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(#‘𝐴))))
8786ad2antrl 764 . . . . . . . . . 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))))
8887, 37syl 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))
8988oveq2d 6706 . . . . . . . 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)))
9080, 89eleqtrrd 2733 . . . . . . 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⟩))))
91 df-3an 1056 . . . . . . 7 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉) ∧ 𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
9279, 90, 91sylanbrc 699 . . . . . 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 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
93 ccatval1 13395 . . . . . 6 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ (0..^(#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
9492, 93syl 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 substr ⟨0, (𝑁𝐿)⟩))‘𝑘) = ((𝐴 substr ⟨𝑀, 𝐿⟩)‘𝑘))
9578, 94eqtr4d 2688 . . . 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 substr ⟨0, (𝑁𝐿)⟩))‘𝑘))
96 simprl 809 . . . . . 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))))))
9774anim2i 592 . . . . . . 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))))
9897ancomd 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))))
9910swrdccatin12lem2 13535 . . . . . 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 substr ⟨0, (𝑁𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩))))))
10096, 98, 99sylc 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 substr ⟨0, (𝑁𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
10124ad2antrl 764 . . . . . . 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 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉))
102 elfzuz 12376 . . . . . . . . . . . . 13 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → 𝑁 ∈ (ℤ𝐿))
103 eluzelz 11735 . . . . . . . . . . . . . 14 (𝑁 ∈ (ℤ𝐿) → 𝑁 ∈ ℤ)
104 simpll 805 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝐿 ∈ ℕ0)
105 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → 𝑀 ∈ ℕ0)
106105adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℕ0)
107 simpr 476 . . . . . . . . . . . . . . . . . . 19 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
108104, 106, 1073jca 1261 . . . . . . . . . . . . . . . . . 18 (((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) ∧ 𝑁 ∈ ℤ) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
109108ex 449 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
110109ancoms 468 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
1111103adant3 1101 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ0𝐿 ∈ ℕ0𝑀𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
11251, 111sylbi 207 . . . . . . . . . . . . . 14 (𝑀 ∈ (0...𝐿) → (𝑁 ∈ ℤ → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
113103, 112syl5com 31 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ𝐿) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
114102, 113syl 17 . . . . . . . . . . . 12 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → (𝑀 ∈ (0...𝐿) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ)))
115114impcom 445 . . . . . . . . . . 11 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
116115adantl 481 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
117116ad2antrl 764 . . . . . . . . 9 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → (𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ))
118 swrdccatin12lem1 13530 . . . . . . . . 9 ((𝐿 ∈ ℕ0𝑀 ∈ ℕ0𝑁 ∈ ℤ) → ((𝑘 ∈ (0..^(𝑁𝑀)) ∧ ¬ 𝑘 ∈ (0..^(𝐿𝑀))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿)))))
119117, 98, 118sylc 65 . . . . . . . 8 ((¬ 𝑘 ∈ (0..^(𝐿𝑀)) ∧ (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) ∧ 𝑘 ∈ (0..^(𝑁𝑀)))) → 𝑘 ∈ ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
12027, 28, 85, 37syl3anc 1366 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) = (𝐿𝑀))
12139adantl 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝐵 ∈ Word 𝑉)
12243adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (#‘𝐵) ∈ ℤ)
123 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))
124122, 123, 46syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁𝐿) ∈ (0...(#‘𝐵)))
125121, 124jca 553 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))))
126125ex 449 . . . . . . . . . . . . . 14 (𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
127126adantl 481 . . . . . . . . . . . . 13 ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵)))))
128127impcom 445 . . . . . . . . . . . 12 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑁𝐿) ∈ (0...(#‘𝐵))))
129128, 48syl 17 . . . . . . . . . . 11 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)) = (𝑁𝐿))
130120, 129oveq12d 6708 . . . . . . . . . 10 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))) = ((𝐿𝑀) + (𝑁𝐿)))
131120, 130oveq12d 6708 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩)))) = ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
132131ad2antrl 764 . . . . . . . 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 substr ⟨0, (𝑁𝐿)⟩)))) = ((𝐿𝑀)..^((𝐿𝑀) + (𝑁𝐿))))
133119, 132eleqtrrd 2733 . . . . . . 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 substr ⟨0, (𝑁𝐿)⟩)))))
134 df-3an 1056 . . . . . . 7 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))) ↔ (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉) ∧ 𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))))
135101, 133, 134sylanbrc 699 . . . . . 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 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))))
136 ccatval2 13396 . . . . . 6 (((𝐴 substr ⟨𝑀, 𝐿⟩) ∈ Word 𝑉 ∧ (𝐵 substr ⟨0, (𝑁𝐿)⟩) ∈ Word 𝑉𝑘 ∈ ((#‘(𝐴 substr ⟨𝑀, 𝐿⟩))..^((#‘(𝐴 substr ⟨𝑀, 𝐿⟩)) + (#‘(𝐵 substr ⟨0, (𝑁𝐿)⟩))))) → (((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))‘𝑘) = ((𝐵 substr ⟨0, (𝑁𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
137135, 136syl 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 substr ⟨0, (𝑁𝐿)⟩))‘𝑘) = ((𝐵 substr ⟨0, (𝑁𝐿)⟩)‘(𝑘 − (#‘(𝐴 substr ⟨𝑀, 𝐿⟩)))))
138100, 137eqtr4d 2688 . . . 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 substr ⟨0, (𝑁𝐿)⟩))‘𝑘))
13995, 138pm2.61ian 848 . . 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 substr ⟨0, (𝑁𝐿)⟩))‘𝑘))
14020, 72, 139eqfnfvd 6354 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩)))
141140ex 449 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (0...𝐿) ∧ 𝑁 ∈ (𝐿...(𝐿 + (#‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = ((𝐴 substr ⟨𝑀, 𝐿⟩) ++ (𝐵 substr ⟨0, (𝑁𝐿)⟩))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  cop 4216   class class class wbr 4685   Fn wfn 5921  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974   + caddc 9977  cle 10113  cmin 10304  0cn0 11330  cz 11415  cuz 11725  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335
This theorem is referenced by:  swrdccat3  13538  swrdccatin12d  13547
  Copyright terms: Public domain W3C validator