ccatsymb - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ccatsymb		GIF version

Theorem ccatsymb 11228

Description: The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)

**Assertion**
Ref	Expression
ccatsymb	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))

**Proof of Theorem ccatsymb**
Step	Hyp	Ref	Expression
1		simprll 539	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simpr 110	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 < (♯‘𝐴))
3	2	anim2i 342	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴)))
4		simpr 110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
5		0zd 9535	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
6		lencl 11166	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
7	6	nn0zd 9644	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
8	7	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
9		elfzo 10429	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
10	4, 5, 8, 9	syl3anc 1274	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
11	10	ad2antrl 490	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
12	3, 11	mpbird 167	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ (0..^(♯‘𝐴)))
13		df-3an 1007	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
14	1, 12, 13	sylanbrc 417	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
15		ccatval1 11223	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
16	15	eqcomd 2237	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
17	14, 16	syl 14	. . . . . 6 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
18	17	ancoms 268	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) ∧ 0 ≤ 𝐼) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
19		0z 9534	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
20		zltnle 9569	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
21	19, 20	mpan2 425	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
22	21	adantl 277	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
23		simpl 109	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
24	23	anim1i 340	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
25	24	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
26		animorrl 834	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼))
27		wrdsymb0 11195	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼) → (𝐴‘𝐼) = ∅))
28	25, 26, 27	sylc 62	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ∅)
29		ccatcl 11219	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
30	29	anim1i 340	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31	30	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
32		animorrl 834	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
33		wrdsymb0 11195	. . . . . . . . . . . 12 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
34	31, 32, 33	sylc 62	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
35	28, 34	eqtr4d 2267	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
36	35	ex 115	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
37	22, 36	sylbird 170	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
38	37	com12 30	. . . . . . 7 ⊢ (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
39	38	adantrd 279	. . . . . 6 ⊢ (¬ 0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
40	39	impcom 125	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) ∧ ¬ 0 ≤ 𝐼) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
41		simplr 529	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 ∈ ℤ)
42		zdcle 9600	. . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝐼 ∈ ℤ) → DECID 0 ≤ 𝐼)
43	19, 41, 42	sylancr 414	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → DECID 0 ≤ 𝐼)
44		exmiddc 844	. . . . . 6 ⊢ (DECID 0 ≤ 𝐼 → (0 ≤ 𝐼 ∨ ¬ 0 ≤ 𝐼))
45	43, 44	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (0 ≤ 𝐼 ∨ ¬ 0 ≤ 𝐼))
46	18, 40, 45	mpjaodan 806	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
47		simprll 539	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
48		id 19	. . . . . . . . . 10 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
49	6	nn0red 9500	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
50		zre 9527	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
51		lenlt 8297	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
52	49, 50, 51	syl2an 289	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
53	52	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
54	53	biimpar 297	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (♯‘𝐴) ≤ 𝐼)
55	48, 54	anim12ci 339	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
56		lencl 11166	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
57	56	nn0zd 9644	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
58		zaddcl 9563	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
59	7, 57, 58	syl2an 289	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
60	59	adantr 276	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
61		elfzo 10429	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
62	4, 8, 60, 61	syl3anc 1274	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
63	62	ad2antrl 490	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
64	55, 63	mpbird 167	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
65		df-3an 1007	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
66	47, 64, 65	sylanbrc 417	. . . . . . 7 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
67		ccatval2 11224	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (♯‘𝐴))))
68	67	eqcomd 2237	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
69	66, 68	syl 14	. . . . . 6 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
70	69	ancoms 268	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
71	56	nn0red 9500	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
72		readdcl 8201	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
73	49, 71, 72	syl2an 289	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
74		lenlt 8297	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
75	73, 50, 74	syl2an 289	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
76		simplr 529	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
77		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
78	7	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
79	77, 78	zsubcld 9651	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
80	79	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
81	76, 80	jca 306	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
82	81	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
83	49	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℝ)
84	71	ad2antlr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐵) ∈ ℝ)
85	50	adantl 277	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
86	83, 84, 85	leaddsub2d 8769	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
87	86	biimpa 296	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴)))
88	87	olcd 742	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
89		wrdsymb0 11195	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ) → (((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅))
90	82, 88, 89	sylc 62	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅)
91	30	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
92		ccatlen 11221	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
93	92	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
94		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼)
95	93, 94	eqbrtrd 4115	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼)
96	95	olcd 742	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
97	91, 96, 33	sylc 62	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
98	90, 97	eqtr4d 2267	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
99	98	ex 115	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
100	75, 99	sylbird 170	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
101	100	com12 30	. . . . . . 7 ⊢ (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
102	101	adantrd 279	. . . . . 6 ⊢ (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
103	102	impcom 125	. . . . 5 ⊢ (((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) ∧ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
104		simplr 529	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → 𝐼 ∈ ℤ)
105	60	adantr 276	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
106		zdclt 9601	. . . . . . 7 ⊢ ((𝐼 ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → DECID 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
107	104, 105, 106	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → DECID 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
108		exmiddc 844	. . . . . 6 ⊢ (DECID 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∨ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
109	107, 108	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∨ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
110	70, 103, 109	mpjaodan 806	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
111		zdclt 9601	. . . . 5 ⊢ ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝐼 < (♯‘𝐴))
112	4, 8, 111	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → DECID 𝐼 < (♯‘𝐴))
113	46, 110, 112	ifeqdadc 3642	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
114	113	eqcomd 2237	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
115	114	3impa 1221	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ∅c0 3496 ifcif 3607 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℝcr 8074 0cc0 8075 + caddc 8078 < clt 8256 ≤ cle 8257 − cmin 8392 ℤcz 9523 ..^cfzo 10422 ♯chash 11083 Word cword 11162 ++ cconcat 11216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-addass 8177 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-1o 6625 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-inn 9186 df-n0 9445 df-z 9524 df-uz 9800 df-fz 10289 df-fzo 10423 df-ihash 11084 df-word 11163 df-concat 11217

This theorem is referenced by: swrdccatin2 11359

W3C validator