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Theorem swrdccatin2 14617
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 7364 . . . . . . . . . 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 7364 . . . . . . . . . . 11 (𝐿 = (♯‘𝐴) → (𝐿 + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
64, 5oveq12d 7375 . . . . . . . . . 10 (𝐿 = (♯‘𝐴) → (𝐿...(𝐿 + (♯‘𝐵))) = ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))))
76eleq2d 2823 . . . . . . . . 9 (𝐿 = (♯‘𝐴) → (𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))) ↔ 𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵)))))
83, 7anbi12d 631 . . . . . . . 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 14421 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
11 elnn0uz 12808 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (ℤ‘0))
12 fzss1 13480 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1311, 12sylbi 216 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...𝑁) ⊆ (0...𝑁))
1413sseld 3943 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑀 ∈ ((♯‘𝐴)...𝑁) → 𝑀 ∈ (0...𝑁)))
15 fzss1 13480 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ (ℤ‘0) → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1611, 15sylbi 216 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ⊆ (0...((♯‘𝐴) + (♯‘𝐵))))
1716sseld 3943 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
1814, 17anim12d 609 . . . . . . . . 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, 20biimtrid 241 . . . . . 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 14612 . . . . 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 13474 . . . . . . 7 (𝑀 ∈ (𝐿...𝑁) → ((𝑁𝐿) − (𝑀𝐿)) = (𝑁𝑀))
2625oveq2d 7373 . . . . . 6 (𝑀 ∈ (𝐿...𝑁) → (0..^((𝑁𝐿) − (𝑀𝐿))) = (0..^(𝑁𝑀)))
2726fneq2d 6596 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))) ↔ ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀))))
2827ad2antrl 726 . . . 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 256 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → ((𝐴 ++ 𝐵) substr ⟨𝑀, 𝑁⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
30 simplr 767 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝐵 ∈ Word 𝑉)
31 elfzmlbm 13551 . . . . 5 (𝑀 ∈ (𝐿...𝑁) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
3231ad2antrl 726 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑀𝐿) ∈ (0...(𝑁𝐿)))
33 lencl 14421 . . . . . . . 8 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
3433nn0zd 12525 . . . . . . 7 (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
3534adantl 482 . . . . . 6 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘𝐵) ∈ ℤ)
36 elfzmlbp 13552 . . . . . 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 714 . . . 4 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
39 swrdvalfn 14539 . . . 4 ((𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
4030, 32, 38, 39syl3anc 1371 . . 3 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 substr ⟨(𝑀𝐿), (𝑁𝐿)⟩) Fn (0..^((𝑁𝐿) − (𝑀𝐿))))
41 simpll 765 . . . . . . 7 ((((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) ∧ 𝑘 ∈ (0..^((𝑁𝐿) − (𝑀𝐿)))) → (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉))
42 elfzelz 13441 . . . . . . . . . 10 (𝑀 ∈ (𝐿...𝑁) → 𝑀 ∈ ℤ)
43 zaddcl 12543 . . . . . . . . . . 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 726 . . . . . . . 8 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℤ → (𝑘 + 𝑀) ∈ ℤ))
47 elfzoelz 13572 . . . . . . . 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 1089 . . . . . . 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 14470 . . . . . 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 13431 . . . . . . . . 9 (𝑀 ∈ (𝐿...𝑁) ↔ ((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)))
54 zre 12503 . . . . . . . . . . . . . . . . 17 (𝐿 ∈ ℤ → 𝐿 ∈ ℝ)
55 zre 12503 . . . . . . . . . . . . . . . . 17 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
5654, 55anim12i 613 . . . . . . . . . . . . . . . 16 ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ))
57 elnn0z 12512 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ0 ↔ (𝑘 ∈ ℤ ∧ 0 ≤ 𝑘))
58 zre 12503 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
59 0re 11157 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ∈ ℝ
6059jctl 524 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
6160ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 ∈ ℝ ∧ 𝐿 ∈ ℝ))
62 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
6362adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
64 le2add 11637 . . . . . . . . . . . . . . . . . . . . . . . 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 11141 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐿 ∈ ℝ → 𝐿 ∈ ℂ)
6766addid2d 11356 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐿 ∈ ℝ → (0 + 𝐿) = 𝐿)
6867ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (0 + 𝐿) = 𝐿)
6968breq1d 5115 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 + 𝐿) ≤ (𝑘 + 𝑀) ↔ 𝐿 ≤ (𝑘 + 𝑀)))
7065, 69sylibd 238 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → ((0 ≤ 𝑘𝐿𝑀) → 𝐿 ≤ (𝑘 + 𝑀)))
71 simprl 769 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → 𝐿 ∈ ℝ)
72 readdcl 11134 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 + 𝑀) ∈ ℝ)
7372adantrl 714 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝑘 + 𝑀) ∈ ℝ)
7471, 73lenltd 11301 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ ℝ ∧ (𝐿 ∈ ℝ ∧ 𝑀 ∈ ℝ)) → (𝐿 ≤ (𝑘 + 𝑀) ↔ ¬ (𝑘 + 𝑀) < 𝐿))
7570, 74sylibd 238 . . . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . . 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 5113 . . . . . . . . . . . . . . . 16 ((𝑘 + 𝑀) < 𝐿 ↔ (𝑘 + 𝑀) < (♯‘𝐴))
8382notbii 319 . . . . . . . . . . . . . . 15 (¬ (𝑘 + 𝑀) < 𝐿 ↔ ¬ (𝑘 + 𝑀) < (♯‘𝐴))
8481, 83syl6ib 250 . . . . . . . . . . . . . 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 1131 . . . . . . . . . . 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 715 . . . . . . . . 9 (((𝐿 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (𝐿𝑀𝑀𝑁)) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9053, 89sylbi 216 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
9190ad2antrl 726 . . . . . . 7 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑘 ∈ ℕ0 → ¬ (𝑘 + 𝑀) < (♯‘𝐴)))
92 elfzonn0 13617 . . . . . . 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 4497 . . . . 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 12504 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℤ → 𝑘 ∈ ℂ)
9695adantl 482 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℂ)
97 zcn 12504 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
9897ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝑀 ∈ ℂ)
99 zcn 12504 . . . . . . . . . . . . . . . 16 (𝐿 ∈ ℤ → 𝐿 ∈ ℂ)
10099ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → 𝐿 ∈ ℂ)
10196, 98, 100addsubassd 11532 . . . . . . . . . . . . . 14 (((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ ℤ) → ((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)))
102 oveq2 7365 . . . . . . . . . . . . . . 15 (𝐿 = (♯‘𝐴) → ((𝑘 + 𝑀) − 𝐿) = ((𝑘 + 𝑀) − (♯‘𝐴)))
103102eqeq1d 2738 . . . . . . . . . . . . . 14 (𝐿 = (♯‘𝐴) → (((𝑘 + 𝑀) − 𝐿) = (𝑘 + (𝑀𝐿)) ↔ ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
104101, 103imbitrid 243 . . . . . . . . . . . . 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 1131 . . . . . . . . . 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 216 . . . . . . . 8 (𝑀 ∈ (𝐿...𝑁) → (𝑘 ∈ ℤ → ((𝑘 + 𝑀) − (♯‘𝐴)) = (𝑘 + (𝑀𝐿))))
110109ad2antrl 726 . . . . . . 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 6846 . . . . 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 2780 . . . 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 14462 . . . . . . 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 215 . . . . . . . . . 10 ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (ℤ‘0))
1171, 116eqeltrid 2842 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ0𝐿 ∈ (ℤ‘0))
118 fzss1 13480 . . . . . . . . 9 (𝐿 ∈ (ℤ‘0) → (𝐿...𝑁) ⊆ (0...𝑁))
11910, 117, 1183syl 18 . . . . . . . 8 (𝐴 ∈ Word 𝑉 → (𝐿...𝑁) ⊆ (0...𝑁))
120119sselda 3944 . . . . . . 7 ((𝐴 ∈ Word 𝑉𝑀 ∈ (𝐿...𝑁)) → 𝑀 ∈ (0...𝑁))
121120ad2ant2r 745 . . . . . 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 3943 . . . . . . . . . . . 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 14463 . . . . . . . . . . . . . 14 ((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
128127oveq2d 7373 . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝑁 ∈ ((♯‘𝐴)...((♯‘𝐴) + (♯‘𝐵))) ∧ (𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉)) → (𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))) ↔ 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵)))))
131126, 130mpbird 256 . . . . . . . . . 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 216 . . . . . . . 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 714 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → 𝑁 ∈ (0...(♯‘(𝐴 ++ 𝐵))))
136115, 121, 1353jca 1128 . . . . 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 726 . . . . . 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 14536 . . . . 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 683 . . . 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 744 . . . . . 6 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝑁𝐿) ∈ (0...(♯‘𝐵)))
14430, 32, 1433jca 1128 . . . . 5 (((𝐴 ∈ Word 𝑉𝐵 ∈ Word 𝑉) ∧ (𝑀 ∈ (𝐿...𝑁) ∧ 𝑁 ∈ (𝐿...(𝐿 + (♯‘𝐵))))) → (𝐵 ∈ Word 𝑉 ∧ (𝑀𝐿) ∈ (0...(𝑁𝐿)) ∧ (𝑁𝐿) ∈ (0...(♯‘𝐵))))
145 swrdfv 14536 . . . . 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 2786 . . 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 6985 . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wss 3910  ifcif 4486  cop 4592   class class class wbr 5105   Fn wfn 6491  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051   + caddc 11054   < clt 11189  cle 11190  cmin 11385  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402   ++ cconcat 14458   substr csubstr 14528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-substr 14529
This theorem is referenced by:  pfxccat3  14622  swrdccatin2d  14632
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