MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swrdccatin2 Structured version   Visualization version   GIF version

Theorem swrdccatin2 13707
Description: The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)
Hypothesis
Ref Expression
swrdccatin12.l 𝐿 = (♯‘𝐴)
Assertion
Ref Expression
swrdccatin2 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)))

Proof of Theorem swrdccatin2
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 swrdccatin12.l . . . . . . . 8 𝐿 = (♯‘𝐴)
2 oveq1 6821 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...𝑁) = ((♯‘𝐴)...𝑁))
32eleq2d 2825 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((♯‘𝐴)...𝑁)))
4 id 22 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → 𝐿 = (♯‘𝐴))
5 oveq1 6821 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
64, 5oveq12d 6832 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
76eleq2d 2825 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
83, 7anbi12d 749 . . . . . . . 8 (𝐿 = (♯‘𝐴) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) ↔ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))))
91, 8ax-mp 5 . . . . . . 7 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) ↔ (𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
10 lencl 13530 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 elnn0uz 11938 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
1211biimpi 206 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (ℤ‘0))
13 fzss1 12593 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1412, 13syl 17 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1514sseld 3743 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
16 fzss1 12593 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1712, 16syl 17 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1817sseld 3743 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
1915, 18anim12d 587 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ0 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2010, 19syl 17 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2120adantr 472 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
229, 21syl5bi 232 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2322imp 444 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
24 swrdccatfn 13702 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
2523, 24syldan 488 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
26 elfz2 12546 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
27 zcn 11594 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
28 zcn 11594 . . . . . . . . . . . . 13 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
29 zcn 11594 . . . . . . . . . . . . 13 (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
3027, 28, 293anim123i 1155 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
31303comr 1120 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
3231adantr 472 . . . . . . . . . 10 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
3326, 32sylbi 207 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
3433adantr 472 . . . . . . . 8 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ))
35 nnncan2 10530 . . . . . . . 8 ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝐿 ∈ ℂ) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
3634, 35syl 17 . . . . . . 7 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
3736adantl 473 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
3837oveq2d 6830 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (0..^((𝑁𝐿) − (𝑀𝐿))) = (0..^(𝑁𝑀)))
3938fneq2d 6143 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀))))
4025, 39mpbird 247 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
41 simpr 479 . . . . 5 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝐵 ∈ Word 𝑉)
4241adantr 472 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
43 elfzmlbm 12663 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
4443ad2antrl 766 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
45 elfzmlbp 12664 . . . . . . . 8 (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
4645ex 449 . . . . . . 7 ((♯‘𝐵) ∈ ℤ → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
47 lencl 13530 . . . . . . . . 9 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
4847nn0zd 11692 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
4948adantl 473 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
5046, 49syl11 33 . . . . . 6 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
5150adantl 473 . . . . 5 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
5251impcom 445 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
53 swrdvalfn 13646 . . . 4 ((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
5442, 44, 52, 53syl3anc 1477 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
55 simpl 474 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
5655adantr 472 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
57 elfzoelz 12684 . . . . . . . . 9 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℤ)
58 elfzelz 12555 . . . . . . . . . . 11 (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
59 zaddcl 11629 . . . . . . . . . . . 12 ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
6059expcom 450 . . . . . . . . . . 11 (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
6158, 60syl 17 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
6261ad2antrl 766 . . . . . . . . 9 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
6357, 62syl5com 31 . . . . . . . 8 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 + 𝑀) ∈ ℤ))
6463impcom 445 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝑘 + 𝑀) ∈ ℤ)
65 df-3an 1074 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
6656, 64, 65sylanbrc 701 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ))
67 ccatsymb 13574 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = if((𝑘 + 𝑀) < (♯‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴)))))
6866, 67syl 17 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = if((𝑘 + 𝑀) < (♯‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴)))))
69 elfzonn0 12727 . . . . . . . 8 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℕ0)
70 zre 11593 . . . . . . . . . . . . . . . . . . 19 (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
71 zre 11593 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
7270, 71anim12i 591 . . . . . . . . . . . . . . . . . 18 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
73 elnn0z 11602 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
74 zre 11593 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
75 0red 10253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑀 ∈ ℝ → 0 ∈ ℝ)
7675anim1i 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑀 ∈ ℝ ∧ 𝐿 ∈ ℝ) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
7776ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
7877adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
79 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ∈ ℝ)
8079anim2i 594 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
81 le2add 10722 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
8278, 80, 81syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
83 recn 10238 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
8483addid2d 10449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
8584ad2antrl 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
8685breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
8782, 86sylibd 229 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
88 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝐿 ∈ ℝ)
8988adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
90 readdcl 10231 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
9180, 90syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
9289, 91lenltd 10395 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
9387, 92sylibd 229 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
9493expd 451 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ≤ 𝑘 → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
9594com12 32 . . . . . . . . . . . . . . . . . . . . 21 (0 ≤ 𝑘 → ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
9695expd 451 . . . . . . . . . . . . . . . . . . . 20 (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
9774, 96mpan9 487 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
9873, 97sylbi 207 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ0 → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
9972, 98mpan9 487 . . . . . . . . . . . . . . . . 17 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
1001eqcomi 2769 . . . . . . . . . . . . . . . . . . 19 (♯‘𝐴) = 𝐿
101100breq2i 4812 . . . . . . . . . . . . . . . . . 18 ((𝑘 + 𝑀) < (♯‘𝐴) ↔ (𝑘 + 𝑀) < 𝐿)
102101notbii 309 . . . . . . . . . . . . . . . . 17 (¬ (𝑘 + 𝑀) < (♯‘𝐴) ↔ ¬ (𝑘 + 𝑀) < 𝐿)
10399, 102syl6ibr 242 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
104103ex 449 . . . . . . . . . . . . . . 15 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ0 → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
105104com23 86 . . . . . . . . . . . . . 14 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿𝑀 → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
1061053adant2 1126 . . . . . . . . . . . . 13 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿𝑀 → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
107106com12 32 . . . . . . . . . . . 12 (𝐿𝑀 → ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
108107adantr 472 . . . . . . . . . . 11 ((𝐿𝑀𝑀𝑁) → ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
109108impcom 445 . . . . . . . . . 10 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
11026, 109sylbi 207 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
111110ad2antrl 766 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
11269, 111syl5com 31 . . . . . . 7 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
113112impcom 445 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ¬ (𝑘 + 𝑀) < (♯‘𝐴))
114113iffalsed 4241 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → if((𝑘 + 𝑀) < (♯‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴)))) = (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴))))
115 zcn 11594 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
116115adantl 473 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
11728adantl 473 . . . . . . . . . . . . . . . . 17 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ)
118117adantr 472 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
11929ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
120116, 118, 119addsubassd 10624 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)))
121 oveq2 6822 . . . . . . . . . . . . . . . 16 (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
122121eqeq1d 2762 . . . . . . . . . . . . . . 15 (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
123120, 122syl5ib 234 . . . . . . . . . . . . . 14 (𝐿 = (♯‘𝐴) → (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
1241, 123ax-mp 5 . . . . . . . . . . . . 13 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿)))
125124ex 449 . . . . . . . . . . . 12 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
1261253adant2 1126 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
127126adantr 472 . . . . . . . . . 10 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
12826, 127sylbi 207 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
129128ad2antrl 766 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
13057, 129syl5com 31 . . . . . . 7 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
131130impcom 445 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿)))
132131fveq2d 6357 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴))) = (𝐵‘(𝑘 + (𝑀𝐿))))
13368, 114, 1323eqtrd 2798 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐵‘(𝑘 + (𝑀𝐿))))
134 ccatcl 13566 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
135134ad2antrr 764 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
1361, 12syl5eqel 2843 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ (ℤ‘0))
137 fzss1 12593 . . . . . . . . . . . 12 (𝐿 ∈ (ℤ‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
13810, 136, 1373syl 18 . . . . . . . . . . 11 (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
139138sseld 3743 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 → (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ (0...𝑁)))
140139adantr 472 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ (0...𝑁)))
141140com12 32 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑀 ∈ (0...𝑁)))
142141adantr 472 . . . . . . 7 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑀 ∈ (0...𝑁)))
143142impcom 445 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝑁))
144143adantr 472 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → 𝑀 ∈ (0...𝑁))
1451, 7ax-mp 5 . . . . . . . . 9 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
14610, 12syl 17 . . . . . . . . . . . . . . 15 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ (ℤ‘0))
147146adantr 472 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐴) ∈ (ℤ‘0))
148147, 16syl 17 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
149148sseld 3743 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
150149impcom 445 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
151 ccatlen 13567 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
152151oveq2d 6830 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...((♯‘𝐴) + (♯‘𝐵))))
153152eleq2d 2825 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
154153adantl 473 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
155150, 154mpbird 247 . . . . . . . . . 10 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
156155ex 449 . . . . . . . . 9 (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
157145, 156sylbi 207 . . . . . . . 8 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
158157adantl 473 . . . . . . 7 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
159158impcom 445 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
160159adantr 472 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
161 fzmmmeqm 12587 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
162161oveq2d 6830 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁𝐿) − (𝑀𝐿))) = (0..^(𝑁𝑀)))
163162eleq2d 2825 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ 𝑘 ∈ (0..^(𝑁𝑀))))
164163biimpd 219 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ (0..^(𝑁𝑀))))
165164ad2antrl 766 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ (0..^(𝑁𝑀))))
166165imp 444 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → 𝑘 ∈ (0..^(𝑁𝑀)))
167 swrdfv 13643 . . . . 5 ((((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
168135, 144, 160, 166, 167syl31anc 1480 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
16948, 46syl 17 . . . . . . . . . 10 (𝐵 ∈ Word 𝑉 → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
170169adantl 473 . . . . . . . . 9 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
171170com12 32 . . . . . . . 8 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
172171adantl 473 . . . . . . 7 ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁𝐿) ∈ (0...(♯‘𝐵))))
173172impcom 445 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
17442, 44, 1733jca 1123 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
175 swrdfv 13643 . . . . 5 (((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀𝐿))))
176174, 175sylan 489 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀𝐿))))
177133, 168, 1763eqtr4d 2804 . . 3 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)‘𝑘))
17840, 54, 177eqfnfvd 6478 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))
179178ex 449 1 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wss 3715  ifcif 4230  cop 4327   class class class wbr 4804   Fn wfn 6044  cfv 6049  (class class class)co 6814  cc 10146  cr 10147  0cc0 10148   + caddc 10151   < clt 10286  cle 10287  cmin 10478  0cn0 11504  cz 11589  cuz 11899  ...cfz 12539  ..^cfzo 12679  chash 13331  Word cword 13497   ++ cconcat 13499   substr csubstr 13501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-fzo 12680  df-hash 13332  df-word 13505  df-concat 13507  df-substr 13509
This theorem is referenced by:  swrdccat3  13712  swrdccatin2d  13720  pfxccat3  41954
  Copyright terms: Public domain W3C validator