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Theorem ccatsymb 14562

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 777	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simpr 483	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 < (♯‘𝐴))
3	2	anim2i 615	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴)))
4		simpr 483	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
5		0zd 12598	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
6		lencl 14513	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
7	6	nn0zd 12612	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
8	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
9		elfzo 13664	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
10	4, 5, 8, 9	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
11	10	ad2antrl 726	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
12	3, 11	mpbird 256	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ (0..^(♯‘𝐴)))
13		df-3an 1086	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
14	1, 12, 13	sylanbrc 581	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
15		ccatval1 14557	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
16	15	eqcomd 2731	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
17	14, 16	syl 17	. . . . . 6 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
18	17	ex 411	. . . . 5 ⊢ (0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
19		zre 12590	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
20		0red 11245	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	ltnled 11389	. . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
22	21	adantl 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
23		simpl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
24	23	anim1i 613	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
25	24	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
26		animorrl 978	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼))
27		wrdsymb0 14529	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼) → (𝐴‘𝐼) = ∅))
28	25, 26, 27	sylc 65	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ∅)
29		ccatcl 14554	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
30	29	anim1i 613	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31	30	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
32		animorrl 978	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
33		wrdsymb0 14529	. . . . . . . . . . 11 ⊢ (((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅))
34	31, 32, 33	sylc 65	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
35	28, 34	eqtr4d 2768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
36	35	ex 411	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
37	22, 36	sylbird 259	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 0 ≤ 𝐼 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
38	37	com12 32	. . . . . 6 ⊢ (¬ 0 ≤ 𝐼 → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
39	38	adantrd 490	. . . . 5 ⊢ (¬ 0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
40	18, 39	pm2.61i 182	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
41		simprll 777	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
42		id 22	. . . . . . . . . 10 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))
43	6	nn0red 12561	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
44		lenlt 11320	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
45	43, 19, 44	syl2an 594	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
46	45	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
47	46	biimpar 476	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (♯‘𝐴) ≤ 𝐼)
48	42, 47	anim12ci 612	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
49		lencl 14513	. . . . . . . . . . . . . 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 594	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
53	52	adantr 479	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
54		elfzo 13664	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
55	4, 8, 53, 54	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
56	55	ad2antrl 726	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
58		df-3an 1086	. . . . . . . 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 581	. . . . . . 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 14558	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (♯‘𝐴))))
61	60	eqcomd 2731	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
62	59, 61	syl 17	. . . . . 6 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
63	62	ex 411	. . . . 5 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
64	49	nn0red 12561	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
65		readdcl 11219	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
66	43, 64, 65	syl2an 594	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
67		lenlt 11320	. . . . . . . . 9 ⊢ ((((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
68	66, 19, 67	syl2an 594	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
69		simplr 767	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
70		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
71	7	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
72	70, 71	zsubcld 12699	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
73	72	adantlr 713	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
74	69, 73	jca 510	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
75	74	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
76	43	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℝ)
77	64	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐵) ∈ ℝ)
78	19	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
79	76, 77, 78	leaddsub2d 11844	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
80	79	biimpa 475	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴)))
81	80	olcd 872	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
82		wrdsymb0 14529	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ) → (((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅))
83	75, 81, 82	sylc 65	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ∅)
84	30	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
85		ccatlen 14555	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
86	85	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
87		simpr 483	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼)
88	86, 87	eqbrtrd 5163	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼)
89	88	olcd 872	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
90	84, 89, 33	sylc 65	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵)‘𝐼) = ∅)
91	83, 90	eqtr4d 2768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
92	91	ex 411	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
93	68, 92	sylbird 259	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
94	93	com12 32	. . . . . 6 ⊢ (¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
95	94	adantrd 490	. . . . 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 182	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
97	40, 96	ifeqda 4558	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
98	97	eqcomd 2731	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
99	98	3impa 1107	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 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∅c0 4316 ifcif 4522 class class class wbr 5141 ‘cfv 6541 (class class class)co 7414 ℝcr 11135 0cc0 11136 + caddc 11139 < clt 11276 ≤ cle 11277 − cmin 11472 ℤcz 12586 ..^cfzo 13657 ♯chash 14319 Word cword 14494 ++ cconcat 14550

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-hash 14320 df-word 14495 df-concat 14551

This theorem is referenced by: swrdccatin2 14709 frlmvscadiccat 41774

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