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Theorem ccatsymb 13366

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 1081	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
3	2	ad2antrl 764	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
4		simpr 477	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → 𝐼 < (#‘𝐴))
5	4	anim2i 593	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (#‘𝐴)))
6		simp3 1063	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
7		0zd 11389	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
8		lencl 13324	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℕ₀)
9	8	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℤ)
10	9	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (#‘𝐴) ∈ ℤ)
11	6, 7, 10	3jca 1242	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
12	11	ad2antrl 764	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ))
13		elfzo 12472	. . . . . . . . 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 247	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → 𝐼 ∈ (0..^(#‘𝐴)))
16		df-3an 1039	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(#‘𝐴))))
17	3, 15, 16	sylanbrc 698	. . . . . 6 ⊢ ((0 ≤ 𝐼 ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))))
18		ccatval1 13361	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
19	18	eqcomd 2628	. . . . . 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 450	. . . 4 ⊢ (0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
22		zre 11381	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
23		0red 10041	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 0 ∈ ℝ)
24	22, 23	jca 554	. . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
25	24	3ad2ant3 1084	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℝ ∧ 0 ∈ ℝ))
26		ltnle 10117	. . . . . . . 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 1080	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
30	29	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → 𝐼 < 0)
32	31	orcd 407	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼))
33		wrdsymb0 13339	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘𝐴) ≤ 𝐼) → (𝐴‘𝐼) = ∅))
34	30, 32, 33	sylc 65	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ∅)
35		ccatcl 13359	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
36	35	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
37	36, 6	jca 554	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
38	37	adantr 481	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
39	31	orcd 407	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
40		wrdsymb0 13339	. . . . . . . . . 10 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
41	38, 39, 40	sylc 65	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
42	34, 41	eqtr4d 2659	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
43	42	ex 450	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
44	27, 43	sylbird 250	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
45	44	adantr 481	. . . . 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 176	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (#‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
48	2	ad2antrl 764	. . . . . . . 8 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
49	8	nn0red 11352	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Word 𝑉 → (#‘𝐴) ∈ ℝ)
50		lenlt 10116	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
51	49, 22, 50	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
52	51	3adant2 1080	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (#‘𝐴)))
53	52	biimpar 502	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (#‘𝐴) ≤ 𝐼)
54	53	anim2i 593	. . . . . . . . . 10 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ (#‘𝐴) ≤ 𝐼))
55	54	ancomd 467	. . . . . . . . 9 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → ((#‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((#‘𝐴) + (#‘𝐵))))
56		lencl 13324	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℕ₀)
57	56	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℤ)
58		zaddcl 11417	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℤ ∧ (#‘𝐵) ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
59	9, 57, 58	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
60	59	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℤ)
61	6, 10, 60	3jca 1242	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
62	61	ad2antrl 764	. . . . . . . . . 10 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → (𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ ∧ ((#‘𝐴) + (#‘𝐵)) ∈ ℤ))
63		elfzo 12472	. . . . . . . . . 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 247	. . . . . . . 8 ⊢ ((𝐼 < ((#‘𝐴) + (#‘𝐵)) ∧ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴))) → 𝐼 ∈ ((#‘𝐴)..^((#‘𝐴) + (#‘𝐵))))
66		df-3an 1039	. . . . . . . 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 698	. . . . . . 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 13362	. . . . . . 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 450	. . . . 5 ⊢ (𝐼 < ((#‘𝐴) + (#‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
71	56	nn0red 11352	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word 𝑉 → (#‘𝐵) ∈ ℝ)
72		readdcl 10019	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
73	49, 71, 72	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
74	73	3adant3 1081	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((#‘𝐴) + (#‘𝐵)) ∈ ℝ)
75	22	3ad2ant3 1084	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
76	74, 75	lenltd 10183	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((#‘𝐴) + (#‘𝐵))))
77	37	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
78		ccatlen 13360	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
79	78	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
80	79	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) = ((#‘𝐴) + (#‘𝐵)))
81		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼)
82	80, 81	eqbrtrd 4675	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘(𝐴 ++ 𝐵)) ≤ 𝐼)
83	82	olcd 408	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (#‘(𝐴 ++ 𝐵)) ≤ 𝐼))
84	77, 83, 40	sylc 65	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
85		simp2 1062	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
86		zsubcl 11419	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ ℤ ∧ (#‘𝐴) ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
87	9, 86	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ ℤ ∧ 𝐴 ∈ Word 𝑉) → (𝐼 − (#‘𝐴)) ∈ ℤ)
88	87	ancoms 469	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
89	88	3adant2 1080	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (#‘𝐴)) ∈ ℤ)
90	85, 89	jca 554	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
91	90	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ))
92		leaddsub2 10505	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝐵) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
93	49, 71, 22, 92	syl3an 1368	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 ↔ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
94	93	biimpa 501	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (#‘𝐵) ≤ (𝐼 − (#‘𝐴)))
95	94	olcd 408	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))))
96		wrdsymb0 13339	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (#‘𝐴)) ∈ ℤ) → (((𝐼 − (#‘𝐴)) < 0 ∨ (#‘𝐵) ≤ (𝐼 − (#‘𝐴))) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅))
97	91, 95, 96	sylc 65	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (#‘𝐴))) = ∅)
98	84, 97	eqtr4d 2659	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ((#‘𝐴) + (#‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
99	98	ex 450	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (((#‘𝐴) + (#‘𝐵)) ≤ 𝐼 → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
100	76, 99	sylbird 250	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((#‘𝐴) + (#‘𝐵)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴)))))
101	100	adantr 481	. . . . . 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 176	. . . 4 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (#‘𝐴))))
104	103	eqcomd 2628	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (#‘𝐴)) → (𝐵‘(𝐼 − (#‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
105	47, 104	ifeqda 4121	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → if(𝐼 < (#‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (#‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
106	105	eqcomd 2628	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∅c0 3915 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℤcz 11377 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301

This theorem is referenced by: swrdccatin2 13487

W3C validator