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

Theorem swrdccatin2 14294
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
swrdccatin2.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 swrdccatin2.l . . . . . . . 8 𝐿 = (♯‘𝐴)
2 oveq1 7220 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...𝑁) = ((♯‘𝐴)...𝑁))
32eleq2d 2823 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((♯‘𝐴)...𝑁)))
4 id 22 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → 𝐿 = (♯‘𝐴))
5 oveq1 7220 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
64, 5oveq12d 7231 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
76eleq2d 2823 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
83, 7anbi12d 634 . . . . . . . 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 14088 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 elnn0uz 12479 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
12 fzss1 13151 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1311, 12sylbi 220 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1413sseld 3900 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
15 fzss1 13151 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1611, 15sylbi 220 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1716sseld 3900 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
1814, 17anim12d 612 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ0 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
1910, 18syl 17 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2019adantr 484 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
219, 20syl5bi 245 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2221imp 410 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
23 swrdccatfn 14289 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
2422, 23syldan 594 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
25 fzmmmeqm 13145 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
2625oveq2d 7229 . . . . . 6 (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁𝐿) − (𝑀𝐿))) = (0..^(𝑁𝑀)))
2726fneq2d 6473 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀))))
2827ad2antrl 728 . . . 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))))
2924, 28mpbird 260 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
30 simplr 769 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
31 elfzmlbm 13222 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
3231ad2antrl 728 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
33 lencl 14088 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
3433nn0zd 12280 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
3534adantl 485 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
36 elfzmlbp 13223 . . . . . 6 (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
3735, 36sylan 583 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
3837adantrl 716 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
39 swrdvalfn 14216 . . . 4 ((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
4030, 32, 38, 39syl3anc 1373 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
41 simpll 767 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
42 elfzelz 13112 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
43 zaddcl 12217 . . . . . . . . . . 11 ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
4443expcom 417 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
4542, 44syl 17 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
4645ad2antrl 728 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
47 elfzoelz 13243 . . . . . . . 8 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℤ)
4846, 47impel 509 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝑘 + 𝑀) ∈ ℤ)
49 df-3an 1091 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
5041, 48, 49sylanbrc 586 . . . . . 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))
51 ccatsymb 14139 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = if((𝑘 + 𝑀) < (♯‘𝐴), (𝐴‘(𝑘 + 𝑀)), (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴)))))
5250, 51syl 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)))))
53 elfz2 13102 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
54 zre 12180 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
55 zre 12180 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
5654, 55anim12i 616 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
57 elnn0z 12189 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
58 zre 12180 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
59 0re 10835 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ ℝ
6059jctl 527 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
6160ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
62 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
6362adantrl 716 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
64 le2add 11314 . . . . . . . . . . . . . . . . . . . . . . . 24 (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
6561, 63, 64syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
66 recn 10819 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
6766addid2d 11033 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
6867ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
6968breq1d 5063 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
7065, 69sylibd 242 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
71 simprl 771 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
72 readdcl 10812 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
7372adantrl 716 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
7471, 73lenltd 10978 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
7570, 74sylibd 242 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
7675expd 419 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ≤ 𝑘 → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
7776com12 32 . . . . . . . . . . . . . . . . . . 19 (0 ≤ 𝑘 → ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
7877expd 419 . . . . . . . . . . . . . . . . . 18 (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
7958, 78mpan9 510 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
8057, 79sylbi 220 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
8156, 80mpan9 510 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
821breq2i 5061 . . . . . . . . . . . . . . . 16 ((𝑘 + 𝑀) < 𝐿 ↔ (𝑘 + 𝑀) < (♯‘𝐴))
8382notbii 323 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 𝑀) < 𝐿 ↔ ¬ (𝑘 + 𝑀) < (♯‘𝐴))
8481, 83syl6ib 254 . . . . . . . . . . . . . 14 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
8584ex 416 . . . . . . . . . . . . 13 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℕ0 → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
8685com23 86 . . . . . . . . . . . 12 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿𝑀 → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
87863adant2 1133 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿𝑀 → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
8887imp 410 . . . . . . . . . 10 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝐿𝑀) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
8988adantrr 717 . . . . . . . . 9 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9053, 89sylbi 220 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9190ad2antrl 728 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
92 elfzonn0 13287 . . . . . . 7 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℕ0)
9391, 92impel 509 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ¬ (𝑘 + 𝑀) < (♯‘𝐴))
9493iffalsed 4450 . . . . 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))))
95 zcn 12181 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
9695adantl 485 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
97 zcn 12181 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
9897ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
99 zcn 12181 . . . . . . . . . . . . . . . 16 (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
10099ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
10196, 98, 100addsubassd 11209 . . . . . . . . . . . . . 14 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)))
102 oveq2 7221 . . . . . . . . . . . . . . 15 (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
103102eqeq1d 2739 . . . . . . . . . . . . . 14 (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
104101, 103syl5ib 247 . . . . . . . . . . . . 13 (𝐿 = (♯‘𝐴) → (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
1051, 104ax-mp 5 . . . . . . . . . . . 12 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿)))
106105ex 416 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
1071063adant2 1133 . . . . . . . . . 10 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
108107adantr 484 . . . . . . . . 9 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
10953, 108sylbi 220 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
110109ad2antrl 728 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
111110, 47impel 509 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿)))
112111fveq2d 6721 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐵‘((𝑘 + 𝑀) − (♯‘𝐴))) = (𝐵‘(𝑘 + (𝑀𝐿))))
11352, 94, 1123eqtrd 2781 . . . 4 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)) = (𝐵‘(𝑘 + (𝑀𝐿))))
114 ccatcl 14129 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
115114adantr 484 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
11611biimpi 219 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (ℤ‘0))
1171, 116eqeltrid 2842 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ (ℤ‘0))
118 fzss1 13151 . . . . . . . . 9 (𝐿 ∈ (ℤ‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
11910, 117, 1183syl 18 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
120119sselda 3901 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝑀 ∈ (𝐿...𝑁)) → 𝑀 ∈ (0...𝑁))
121120ad2ant2r 747 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑀 ∈ (0...𝑁))
1221, 7ax-mp 5 . . . . . . . . 9 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
12310, 116, 153syl 18 . . . . . . . . . . . . . 14 (𝐴 ∈ Word 𝑉 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
124123adantr 484 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
125124sseld 3900 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
126125impcom 411 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
127 ccatlen 14130 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
128127oveq2d 7229 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...((♯‘𝐴) + (♯‘𝐵))))
129128eleq2d 2823 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
130129adantl 485 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
131126, 130mpbird 260 . . . . . . . . . 10 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
132131ex 416 . . . . . . . . 9 (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
133122, 132sylbi 220 . . . . . . . 8 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
134133impcom 411 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
135134adantrl 716 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
136115, 121, 1353jca 1130 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
13726eleq2d 2823 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ 𝑘 ∈ (0..^(𝑁𝑀))))
138137ad2antrl 728 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ 𝑘 ∈ (0..^(𝑁𝑀))))
139138biimpa 480 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → 𝑘 ∈ (0..^(𝑁𝑀)))
140 swrdfv 14213 . . . . 5 ((((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))) ∧ 𝑘 ∈ (0..^(𝑁𝑀))) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩)‘𝑘) = ((𝐴 ++ 𝐵)‘(𝑘 + 𝑀)))
141136, 139, 140syl2an2r 685 . . . 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)))
14234, 36sylan 583 . . . . . . 7 ((𝐵 ∈ Word 𝑉𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
143142ad2ant2l 746 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
14430, 32, 1433jca 1130 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
145 swrdfv 14213 . . . . 5 (((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀𝐿))))
146144, 145sylan 583 . . . 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))))
147113, 141, 1463eqtr4d 2787 . . 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))
14829, 40, 147eqfnfvd 6855 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))
149148ex 416 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wss 3866  ifcif 4439  cop 4547   class class class wbr 5053   Fn wfn 6375  cfv 6380  (class class class)co 7213  cc 10727  cr 10728  0cc0 10729   + caddc 10732   < clt 10867  cle 10868  cmin 11062  0cn0 12090  cz 12176  cuz 12438  ...cfz 13095  ..^cfzo 13238  chash 13896  Word cword 14069   ++ cconcat 14125   substr csubstr 14205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-hash 13897  df-word 14070  df-concat 14126  df-substr 14206
This theorem is referenced by:  pfxccat3  14299  swrdccatin2d  14309
  Copyright terms: Public domain W3C validator