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Theorem swrdccatin2 14086
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 7157 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...𝑁) = ((♯‘𝐴)...𝑁))
32eleq2d 2903 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑀 ∈ (𝐿...𝑁) ↔ 𝑀 ∈ ((♯‘𝐴)...𝑁)))
4 id 22 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → 𝐿 = (♯‘𝐴))
5 oveq1 7157 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
64, 5oveq12d 7168 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
76eleq2d 2903 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
83, 7anbi12d 630 . . . . . . . 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 13878 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 elnn0uz 12277 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
12 fzss1 12941 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1311, 12sylbi 218 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1413sseld 3970 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
15 fzss1 12941 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1611, 15sylbi 218 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1716sseld 3970 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
1814, 17anim12d 608 . . . . . . . . 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 481 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ ((♯‘𝐴)...𝑁) ∧ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
219, 20syl5bi 243 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
2221imp 407 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
23 swrdccatfn 14081 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
2422, 23syldan 591 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
25 fzmmmeqm 12935 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
2625oveq2d 7166 . . . . . 6 (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁𝐿) − (𝑀𝐿))) = (0..^(𝑁𝑀)))
2726fneq2d 6446 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀))))
2827ad2antrl 724 . . . 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 258 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
30 simplr 765 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
31 elfzmlbm 13012 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
3231ad2antrl 724 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
33 lencl 13878 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
3433nn0zd 12079 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
3534adantl 482 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
36 elfzmlbp 13013 . . . . . 6 (((♯‘𝐵) ∈ ℤ ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
3735, 36sylan 580 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
3837adantrl 712 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
39 swrdvalfn 14008 . . . 4 ((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
4030, 32, 38, 39syl3anc 1365 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
41 simpll 763 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
42 elfzelz 12903 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
43 zaddcl 12016 . . . . . . . . . . 11 ((𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 + 𝑀) ∈ ℤ)
4443expcom 414 . . . . . . . . . 10 (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
4542, 44syl 17 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
4645ad2antrl 724 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
47 elfzoelz 13033 . . . . . . . 8 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℤ)
4846, 47impel 506 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝑘 + 𝑀) ∈ ℤ)
49 df-3an 1083 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉 ∧ (𝑘 + 𝑀) ∈ ℤ) ↔ ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑘 + 𝑀) ∈ ℤ))
5041, 48, 49sylanbrc 583 . . . . . 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 13931 . . . . . 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 12894 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
54 zre 11979 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
55 zre 11979 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
5654, 55anim12i 612 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
57 elnn0z 11988 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
58 zre 11979 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
59 0re 10637 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ ℝ
6059jctl 524 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
6160ad2antrl 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
62 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
6362adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
64 le2add 11116 . . . . . . . . . . . . . . . . . . . . . . . 24 (((0 ∈ ℝ ∧ 𝐿 ∈ ℝ) ∧ (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
6561, 63, 64syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → (0 + 𝐿) ≤ (𝑘 + 𝑀)))
66 recn 10621 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
6766addid2d 10835 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
6867ad2antrl 724 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
6968breq1d 5073 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
7065, 69sylibd 240 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
71 simprl 767 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
72 readdcl 10614 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
7372adantrl 712 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
7471, 73lenltd 10780 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
7570, 74sylibd 240 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → ¬ (𝑘 + 𝑀) < 𝐿))
7675expd 416 . . . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . . . 18 (0 ≤ 𝑘 → (𝑘 ∈ ℝ → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))))
7958, 78mpan9 507 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℤ ∧ 0 ≤ 𝑘) → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
8057, 79sylbi 218 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ0 → ((𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿)))
8156, 80mpan9 507 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < 𝐿))
821breq2i 5071 . . . . . . . . . . . . . . . 16 ((𝑘 + 𝑀) < 𝐿 ↔ (𝑘 + 𝑀) < (♯‘𝐴))
8382notbii 321 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 𝑀) < 𝐿 ↔ ¬ (𝑘 + 𝑀) < (♯‘𝐴))
8481, 83syl6ib 252 . . . . . . . . . . . . . 14 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐿𝑀 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
8584ex 413 . . . . . . . . . . . . 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 1125 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿𝑀 → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴))))
8887imp 407 . . . . . . . . . 10 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝐿𝑀) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
8988adantrr 713 . . . . . . . . 9 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9053, 89sylbi 218 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9190ad2antrl 724 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
92 elfzonn0 13077 . . . . . . 7 (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) → 𝑘 ∈ ℕ0)
9391, 92impel 506 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ¬ (𝑘 + 𝑀) < (♯‘𝐴))
9493iffalsed 4481 . . . . 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 11980 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
9695adantl 482 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
97 zcn 11980 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
9897ad2antlr 723 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
99 zcn 11980 . . . . . . . . . . . . . . . 16 (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
10099ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
10196, 98, 100addsubassd 11011 . . . . . . . . . . . . . 14 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)))
102 oveq2 7158 . . . . . . . . . . . . . . 15 (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
103102eqeq1d 2828 . . . . . . . . . . . . . 14 (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
104101, 103syl5ib 245 . . . . . . . . . . . . 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 413 . . . . . . . . . . 11 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
1071063adant2 1125 . . . . . . . . . 10 ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
108107adantr 481 . . . . . . . . 9 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
10953, 108sylbi 218 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
110109ad2antrl 724 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
111110, 47impel 506 . . . . . 6 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿)))
112111fveq2d 6673 . . . . 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 2865 . . . 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 13921 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
115114adantr 481 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
11611biimpi 217 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (ℤ‘0))
1171, 116eqeltrid 2922 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ (ℤ‘0))
118 fzss1 12941 . . . . . . . . 9 (𝐿 ∈ (ℤ‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
11910, 117, 1183syl 18 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
120119sselda 3971 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝑀 ∈ (𝐿...𝑁)) → 𝑀 ∈ (0...𝑁))
121120ad2ant2r 743 . . . . . 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 481 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
125124sseld 3970 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
126125impcom 408 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
127 ccatlen 13922 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
128127oveq2d 7166 . . . . . . . . . . . . 13 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (0...(♯‘(𝐴 ++ 𝐵))) = (0...((♯‘𝐴) + (♯‘𝐵))))
129128eleq2d 2903 . . . . . . . . . . . 12 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
130129adantl 482 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
131126, 130mpbird 258 . . . . . . . . . 10 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
132131ex 413 . . . . . . . . 9 (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
133122, 132sylbi 218 . . . . . . . 8 (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) → ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
134133impcom 408 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
135134adantrl 712 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
136115, 121, 1353jca 1122 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) ∈ Word 𝑉𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵)))))
13726eleq2d 2903 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ 𝑘 ∈ (0..^(𝑁𝑀))))
138137ad2antrl 724 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ 𝑘 ∈ (0..^(𝑁𝑀))))
139138biimpa 477 . . . . 5 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → 𝑘 ∈ (0..^(𝑁𝑀)))
140 swrdfv 14005 . . . . 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 681 . . . 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 580 . . . . . . 7 ((𝐵 ∈ Word 𝑉𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵)))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
143142ad2ant2l 742 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
14430, 32, 1433jca 1122 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
145 swrdfv 14005 . . . . 5 (((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → ((𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩)‘𝑘) = (𝐵‘(𝑘 + (𝑀𝐿))))
146144, 145sylan 580 . . . 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 2871 . . 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 6803 . 2 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) = (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩))
149148ex 413 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wss 3940  ifcif 4470  cop 4570   class class class wbr 5063   Fn wfn 6349  cfv 6354  (class class class)co 7150  cc 10529  cr 10530  0cc0 10531   + caddc 10534   < clt 10669  cle 10670  cmin 10864  0cn0 11891  cz 11975  cuz 12237  ...cfz 12887  ..^cfzo 13028  chash 13685  Word cword 13856   ++ cconcat 13917   substr csubstr 13997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-fzo 13029  df-hash 13686  df-word 13857  df-concat 13918  df-substr 13998
This theorem is referenced by:  pfxccat3  14091  swrdccatin2d  14101
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