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Theorem ccatsymb 14620

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 779	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → 𝐼 < (♯‘𝐴))
3	2	anim2i 617	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴)))
4		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
5		0zd 12625	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 0 ∈ ℤ)
6		lencl 14571	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ₀)
7	6	nn0zd 12639	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℤ)
8	7	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
9		elfzo 13701	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ 0 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
10	4, 5, 8, 9	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
11	10	ad2antrl 728	. . . . . . . . 9 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐼 ∈ (0..^(♯‘𝐴)) ↔ (0 ≤ 𝐼 ∧ 𝐼 < (♯‘𝐴))))
12	3, 11	mpbird 257	. . . . . . . 8 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ (0..^(♯‘𝐴)))
13		df-3an 1089	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) ↔ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
14	1, 12, 13	sylanbrc 583	. . . . . . 7 ⊢ ((0 ≤ 𝐼 ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴))) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))))
15		ccatval1 14615	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐴‘𝐼))
16	15	eqcomd 2743	. . . . . . 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 412	. . . . 5 ⊢ (0 ≤ 𝐼 → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < (♯‘𝐴)) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
19		zre 12617	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
20		0red 11264	. . . . . . . . . 10 ⊢ (𝐼 ∈ ℤ → 0 ∈ ℝ)
21	19, 20	ltnled 11408	. . . . . . . . 9 ⊢ (𝐼 ∈ ℤ → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
22	21	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 ↔ ¬ 0 ≤ 𝐼))
23		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → 𝐴 ∈ Word 𝑉)
24	23	anim1i 615	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
25	24	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
26		animorrl 983	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘𝐴) ≤ 𝐼))
27		wrdsymb0 14587	. . . . . . . . . . 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 14612	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝐴 ++ 𝐵) ∈ Word 𝑉)
30	29	anim1i 615	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
31	30	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
32		animorrl 983	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐼 < 0 ∨ (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼))
33		wrdsymb0 14587	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ 𝐼 < 0) → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼))
36	35	ex 412	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 < 0 → (𝐴‘𝐼) = ((𝐴 ++ 𝐵)‘𝐼)))
37	22, 36	sylbird 260	. . . . . . 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 491	. . . . 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 779	. . . . . . . 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 12588	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
44		lenlt 11339	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝐼 ∈ ℝ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
45	43, 19, 44	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
46	45	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) ≤ 𝐼 ↔ ¬ 𝐼 < (♯‘𝐴)))
47	46	biimpar 477	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (♯‘𝐴) ≤ 𝐼)
48	42, 47	anim12ci 614	. . . . . . . . 9 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
49		lencl 14571	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ₀)
50	49	nn0zd 12639	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℤ)
51		zaddcl 12657	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
52	7, 50, 51	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ)
54		elfzo 13701	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
55	4, 8, 53, 54	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))) ↔ ((♯‘𝐴) ≤ 𝐼 ∧ 𝐼 < ((♯‘𝐴) + (♯‘𝐵)))))
56	55	ad2antrl 728	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝐼 < ((♯‘𝐴) + (♯‘𝐵)) ∧ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴))) → 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵))))
58		df-3an 1089	. . . . . . . 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 583	. . . . . . 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 14616	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝐼 ∈ ((♯‘𝐴)..^((♯‘𝐴) + (♯‘𝐵)))) → ((𝐴 ++ 𝐵)‘𝐼) = (𝐵‘(𝐼 − (♯‘𝐴))))
61	60	eqcomd 2743	. . . . . . 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 412	. . . . 5 ⊢ (𝐼 < ((♯‘𝐴) + (♯‘𝐵)) → ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ¬ 𝐼 < (♯‘𝐴)) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
64	49	nn0red 12588	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℝ)
65		readdcl 11238	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℝ) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
66	43, 64, 65	syl2an 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ)
67		lenlt 11339	. . . . . . . . 9 ⊢ ((((♯‘𝐴) + (♯‘𝐵)) ∈ ℝ ∧ 𝐼 ∈ ℝ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
68	66, 19, 67	syl2an 596	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ ¬ 𝐼 < ((♯‘𝐴) + (♯‘𝐵))))
69		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐵 ∈ Word 𝑉)
70		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℤ)
71	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℤ)
72	70, 71	zsubcld 12727	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
73	72	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐼 − (♯‘𝐴)) ∈ ℤ)
74	69, 73	jca 511	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
75	74	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵 ∈ Word 𝑉 ∧ (𝐼 − (♯‘𝐴)) ∈ ℤ))
76	43	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐴) ∈ ℝ)
77	64	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (♯‘𝐵) ∈ ℝ)
78	19	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → 𝐼 ∈ ℝ)
79	76, 77, 78	leaddsub2d 11865	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 ↔ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
80	79	biimpa 476	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴)))
81	80	olcd 875	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐼 − (♯‘𝐴)) < 0 ∨ (♯‘𝐵) ≤ (𝐼 − (♯‘𝐴))))
82		wrdsymb0 14587	. . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((𝐴 ++ 𝐵) ∈ Word 𝑉 ∧ 𝐼 ∈ ℤ))
85		ccatlen 14613	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
86	85	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
87		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼)
88	86, 87	eqbrtrd 5165	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (♯‘(𝐴 ++ 𝐵)) ≤ 𝐼)
89	88	olcd 875	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) ∧ ((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼) → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼))
92	91	ex 412	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → (((♯‘𝐴) + (♯‘𝐵)) ≤ 𝐼 → (𝐵‘(𝐼 − (♯‘𝐴))) = ((𝐴 ++ 𝐵)‘𝐼)))
93	68, 92	sylbird 260	. . . . . . 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 491	. . . . 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 4562	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))) = ((𝐴 ++ 𝐵)‘𝐼))
98	97	eqcomd 2743	. 2 ⊢ (((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) ∧ 𝐼 ∈ ℤ) → ((𝐴 ++ 𝐵)‘𝐼) = if(𝐼 < (♯‘𝐴), (𝐴‘𝐼), (𝐵‘(𝐼 − (♯‘𝐴)))))
99	98	3impa 1110	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ifcif 4525 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 ℤcz 12613 ..^cfzo 13694 ♯chash 14369 Word cword 14552 ++ cconcat 14608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609

This theorem is referenced by: swrdccatin2 14767 frlmvscadiccat 42516

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