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Theorem ccatsymb 14616

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 790	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simpr 489	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 < (♯‘𝐴))
3	2	anim2i 628	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴)))
4		simpr 489	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
5		0zd 12599	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
6		lencl 14566	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
7	6	nn0zd 12612	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
8	7	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
9		elfzo 13685	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
10	4, 5, 8, 9	syl3anc 1396	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
11	10	ad2antrl 740	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
12	3, 11	mpbird 260	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ (0..^(♯‘𝐴)))
13		df-3an 1103	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
14	1, 12, 13	sylanbrc 594	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
15		ccatval1 14610	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
16	15	eqcomd 2775	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
17	14, 16	syl 18	. . . . . 6 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
18	17	ex 417	. . . . 5 ⊢ (0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
19		zre 12591	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
20		0red 11207	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	ltnled 11353	. . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
22	21	adantl 486	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
23		simpl 487	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
24	23	anim1i 626	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
25	24	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
26		animorrl 996	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼))
27		wrdsymb0 14582	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼) → (𝐴‘𝐼) = ∅))
28	25, 26, 27	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ∅)
29		ccatcl 14607	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
30	29	anim1i 626	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31	30	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
32		animorrl 996	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
33		wrdsymb0 14582	. . . . . . . . . . 11 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
34	31, 32, 33	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
35	28, 34	eqtr4d 2807	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
36	35	ex 417	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
37	22, 36	sylbird 263	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
38	37	com12 33	. . . . . 6 ⊢ (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
39	38	adantrd 496	. . . . 5 ⊢ (¬ 0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
40	18, 39	pm2.61i 184	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
41		simprll 790	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
42		id 23	. . . . . . . . . 10 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
43	6	nn0red 12562	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
44		lenlt 11284	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
45	43, 19, 44	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
46	45	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
47	46	biimpar 482	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (♯‘𝐴) ≤ 𝐼)
48	42, 47	anim12ci 625	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
49		lencl 14566	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
50	49	nn0zd 12612	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
51		zaddcl 12630	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
52	7, 50, 51	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
53	52	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
54		elfzo 13685	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
55	4, 8, 53, 54	syl3anc 1396	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
56	55	ad2antrl 740	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
57	48, 56	mpbird 260	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
58		df-3an 1103	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
59	41, 57, 58	sylanbrc 594	. . . . . . 7 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))))
60		ccatval2 14611	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (♯‘𝐴))))
61	60	eqcomd 2775	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
62	59, 61	syl 18	. . . . . 6 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
63	62	ex 417	. . . . 5 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
64	49	nn0red 12562	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
65		readdcl 11179	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
66	43, 64, 65	syl2an 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
67		lenlt 11284	. . . . . . . . 9 ⊢ ((((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
68	66, 19, 67	syl2an 607	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
69		simplr 780	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
70		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
71	7	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
72	70, 71	zsubcld 12701	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
73	72	adantlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
74	69, 73	jca 520	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
75	74	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
76	43	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℝ)
77	64	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐵) ∈ ℝ)
78	19	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
79	76, 77, 78	leaddsub2d 11812	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
80	79	biimpa 481	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴)))
81	80	olcd 887	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
82		wrdsymb0 14582	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ) → (((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅))
83	75, 81, 82	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅)
84	30	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
85		ccatlen 14608	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
86	85	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
87		simpr 489	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼)
88	86, 87	eqbrtrd 5134	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼)
89	88	olcd 887	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
90	84, 89, 33	sylc 66	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
91	83, 90	eqtr4d 2807	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
92	91	ex 417	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
93	68, 92	sylbird 263	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
94	93	com12 33	. . . . . 6 ⊢ (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
95	94	adantrd 496	. . . . 5 ⊢ (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
96	63, 95	pm2.61i 184	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
97	40, 96	ifeqda 4526	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
98	97	eqcomd 2775	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
99	98	3impa 1125	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∅c0 4294 ifcif 4489 class class class wbr 5110 ‘cfv 6534 (class class class)co 7408 ℝcr 11095 0cc0 11096 + caddc 11099 < clt 11239 ≤ cle 11240 − cmin 11437 ℤcz 12587 ..^cfzo 13678 ♯chash 14362 Word cword 14546 ++ cconcat 14603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6300 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6490 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-n0 12501 df-z 12588 df-uz 12859 df-fz 13532 df-fzo 13679 df-hash 14363 df-word 14547 df-concat 14604

This theorem is referenced by: swrdccatin2 14762 frlmvscadiccat 43165

W3C validator