ccatsymb - Metamath Proof Explorer

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

Theorem ccatsymb 13219

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		id 22	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2	1	3adant3 1074	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
3	2	ad2antrl 760	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
4		simpr 476	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → 𝐼 < (#‘𝐴))
5	4	anim2i 591	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (#‘𝐴)))
6		simp3 1056	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
7		0zd 11266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
8		lencl 13179	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ₀)
9	8	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℤ)
10	9	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (#‘𝐴) ∈ ℤ)
11	6, 7, 10	3jca 1235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
12	11	ad2antrl 760	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
13		elfzo 12341	. . . . . . . . 9 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(#‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (#‘𝐴))))
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐼 ∈ (0..^(#‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (#‘𝐴))))
15	5, 14	mpbird 246	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → 𝐼 ∈ (0..^(#‘𝐴)))
16		df-3an 1033	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(#‘𝐴))))
17	3, 15, 16	sylanbrc 695	. . . . . 6 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))))
18		ccatval1 13214	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
19	18	eqcomd 2616	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
20	17, 19	syl 17	. . . . 5 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
21	20	ex 449	. . . 4 ⊢ (0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
22		zre 11258	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23		0red 9920	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 0 ∈ ℝ)
24	22, 23	jca 553	. . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
25	24	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
26		ltnle 9996	. . . . . . . 8 ⊢ ((𝐼 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
27	25, 26	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
28		id 22	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
29	28	3adant2 1073	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
30	29	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31		simpr 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → 𝐼 < 0)
32	31	orcd 406	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼))
33		wrdsymb0 13194	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼) → (𝐴‘𝐼) = ∅))
34	30, 32, 33	sylc 63	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ∅)
35		ccatcl 13212	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
36	35	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
37	36, 6	jca 553	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
38	37	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
39	31	orcd 406	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
40		wrdsymb0 13194	. . . . . . . . . 10 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
41	38, 39, 40	sylc 63	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
42	34, 41	eqtr4d 2647	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
43	42	ex 449	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
44	27, 43	sylbird 249	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
45	44	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
46	45	com12 32	. . . 4 ⊢ (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
47	21, 46	pm2.61i 175	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
48	2	ad2antrl 760	. . . . . . . 8 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
49	8	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℝ)
50		lenlt 9995	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
51	49, 22, 50	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
52	51	3adant2 1073	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
53	52	biimpar 501	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (#‘𝐴) ≤ 𝐼)
54	53	anim2i 591	. . . . . . . . . 10 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ (#‘𝐴) ≤ 𝐼))
55	54	ancomd 466	. . . . . . . . 9 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → ((#‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((#‘𝐴) + (#‘𝐵))))
56		lencl 13179	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ₀)
57	56	nn0zd 11356	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
58		zaddcl 11294	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℤ ∧ (#‘𝐵) ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
59	9, 57, 58	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
60	59	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
61	6, 10, 60	3jca 1235	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
62	61	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
63		elfzo 12341	. . . . . . . . . 10 ⊢ ((𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))) ↔ ((#‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((#‘𝐴) + (#‘𝐵)))))
64	62, 63	syl 17	. . . . . . . . 9 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))) ↔ ((#‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((#‘𝐴) + (#‘𝐵)))))
65	55, 64	mpbird 246	. . . . . . . 8 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))))
66		df-3an 1033	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))))
67	48, 65, 66	sylanbrc 695	. . . . . . 7 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))))
68		ccatval2 13215	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
69	67, 68	syl 17	. . . . . 6 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
70	69	ex 449	. . . . 5 ⊢ (𝐼 < ((#‘𝐴) + (#‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
71	56	nn0red 11229	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℝ)
72		readdcl 9898	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
73	49, 71, 72	syl2an 493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
74	73	3adant3 1074	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
75	22	3ad2ant3 1077	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
76	74, 75	lenltd 10062	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((#‘𝐴) + (#‘𝐵))))
77	37	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
78		ccatlen 13213	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
79	78	3adant3 1074	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
80	79	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
81		simpr 476	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼)
82	80, 81	eqbrtrd 4605	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) ≤ 𝐼)
83	82	olcd 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
84	77, 83, 40	sylc 63	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
85		simp2 1055	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
86		zsubcl 11296	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
87	9, 86	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ ℤ ∧ 𝐴 ∈ Word 𝑉) → (𝐼 − (#‘𝐴)) ∈ ℤ)
88	87	ancoms 468	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
89	88	3adant2 1073	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
90	85, 89	jca 553	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
91	90	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
92		leaddsub2 10384	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
93	49, 71, 22, 92	syl3an 1360	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
94	93	biimpa 500	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘𝐵) ≤ (𝐼 − (#‘𝐴)))
95	94	olcd 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
96		wrdsymb0 13194	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ) → (((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅))
97	91, 95, 96	sylc 63	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅)
98	84, 97	eqtr4d 2647	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
99	98	ex 449	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
100	76, 99	sylbird 249	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
101	100	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
102	101	com12 32	. . . . 5 ⊢ (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
103	70, 102	pm2.61i 175	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
104	103	eqcomd 2616	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (𝐵‘(𝐼 − (#‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
105	47, 104	ifeqda 4071	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → if(𝐼 < (#‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (#‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
106	105	eqcomd 2616	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (#‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (#‘𝐴)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∅c0 3874 ifcif 4036 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℤcz 11254 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156

This theorem is referenced by: swrdccatin2 13338

W3C validator