Theorem dvsinax 38801

Description: Derivative exercise: the derivative with respect to y of sin(Ay), given a constant 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
dvsinax	⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))

Distinct variable group:   𝑦,𝐴

Proof of Theorem dvsinax

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sinf 14693	. . . . . 6 ⊢ sin:ℂ⟶ℂ
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → sin:ℂ⟶ℂ)
3		mulcl 9899	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝐴 · 𝑦) ∈ ℂ)
4		eqid 2610	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) = (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))
5	3, 4	fmptd 6292	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)):ℂ⟶ℂ)
6		fcompt 6306	. . . . 5 ⊢ ((sin:ℂ⟶ℂ ∧ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)):ℂ⟶ℂ) → (sin ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑤 ∈ ℂ ↦ (sin‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤))))
7	2, 5, 6	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ ℂ → (sin ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑤 ∈ ℂ ↦ (sin‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤))))
8		eqidd 2611	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) = (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
9		oveq2 6557	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝐴 · 𝑦) = (𝐴 · 𝑤))
10	9	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) ∧ 𝑦 = 𝑤) → (𝐴 · 𝑦) = (𝐴 · 𝑤))
11		simpr 476	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → 𝑤 ∈ ℂ)
12		mulcl 9899	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝐴 · 𝑤) ∈ ℂ)
13	8, 10, 11, 12	fvmptd 6197	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤) = (𝐴 · 𝑤))
14	13	fveq2d 6107	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (sin‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤)) = (sin‘(𝐴 · 𝑤)))
15	14	mpteq2dva 4672	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ (sin‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤))) = (𝑤 ∈ ℂ ↦ (sin‘(𝐴 · 𝑤))))
16		oveq2 6557	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝐴 · 𝑤) = (𝐴 · 𝑦))
17	16	fveq2d 6107	. . . . . 6 ⊢ (𝑤 = 𝑦 → (sin‘(𝐴 · 𝑤)) = (sin‘(𝐴 · 𝑦)))
18	17	cbvmptv 4678	. . . . 5 ⊢ (𝑤 ∈ ℂ ↦ (sin‘(𝐴 · 𝑤))) = (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))
19	18	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ (sin‘(𝐴 · 𝑤))) = (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))))
20	7, 15, 19	3eqtrrd 2649	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦))) = (sin ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))))
21	20	oveq2d 6565	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (ℂ D (sin ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))))
22		cnelprrecn 9908	. . . 4 ⊢ ℂ ∈ {ℝ, ℂ}
23	22	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → ℂ ∈ {ℝ, ℂ})
24		dvsin 23549	. . . . . 6 ⊢ (ℂ D sin) = cos
25	24	dmeqi 5247	. . . . 5 ⊢ dom (ℂ D sin) = dom cos
26		cosf 14694	. . . . . 6 ⊢ cos:ℂ⟶ℂ
27	26	fdmi 5965	. . . . 5 ⊢ dom cos = ℂ
28	25, 27	eqtri 2632	. . . 4 ⊢ dom (ℂ D sin) = ℂ
29	28	a1i 11	. . 3 ⊢ (𝐴 ∈ ℂ → dom (ℂ D sin) = ℂ)
30		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
31	30	cbvmptv 4678	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ ↦ 𝑦) = (𝑤 ∈ ℂ ↦ 𝑤)
32	31	oveq2i 6560	. . . . . . . . 9 ⊢ ((ℂ × {𝐴}) ∘𝑓 · (𝑦 ∈ ℂ ↦ 𝑦)) = ((ℂ × {𝐴}) ∘𝑓 · (𝑤 ∈ ℂ ↦ 𝑤))
33	32	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℂ × {𝐴}) ∘𝑓 · (𝑦 ∈ ℂ ↦ 𝑦)) = ((ℂ × {𝐴}) ∘𝑓 · (𝑤 ∈ ℂ ↦ 𝑤)))
34		cnex 9896	. . . . . . . . . . 11 ⊢ ℂ ∈ V
35	34	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ℂ ∈ V)
36		snex 4835	. . . . . . . . . . 11 ⊢ {𝐴} ∈ V
37	36	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → {𝐴} ∈ V)
38		xpexg 6858	. . . . . . . . . 10 ⊢ ((ℂ ∈ V ∧ {𝐴} ∈ V) → (ℂ × {𝐴}) ∈ V)
39	35, 37, 38	syl2anc 691	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}) ∈ V)
40	34	mptex 6390	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℂ ↦ 𝑤) ∈ V
41	40	a1i 11	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ 𝑤) ∈ V)
42		offval3 7053	. . . . . . . . 9 ⊢ (((ℂ × {𝐴}) ∈ V ∧ (𝑤 ∈ ℂ ↦ 𝑤) ∈ V) → ((ℂ × {𝐴}) ∘𝑓 · (𝑤 ∈ ℂ ↦ 𝑤)) = (𝑦 ∈ (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))))
43	39, 41, 42	syl2anc 691	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((ℂ × {𝐴}) ∘𝑓 · (𝑤 ∈ ℂ ↦ 𝑤)) = (𝑦 ∈ (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))))
44		fconst6g 6007	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℂ → (ℂ × {𝐴}):ℂ⟶ℂ)
45		fdm 5964	. . . . . . . . . . . . 13 ⊢ ((ℂ × {𝐴}):ℂ⟶ℂ → dom (ℂ × {𝐴}) = ℂ)
46	44, 45	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → dom (ℂ × {𝐴}) = ℂ)
47		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ ↦ 𝑤) = (𝑤 ∈ ℂ ↦ 𝑤)
48		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ℂ → 𝑤 ∈ ℂ)
49	47, 48	fmpti 6291	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ℂ ↦ 𝑤):ℂ⟶ℂ
50	49	fdmi 5965	. . . . . . . . . . . . 13 ⊢ dom (𝑤 ∈ ℂ ↦ 𝑤) = ℂ
51	50	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → dom (𝑤 ∈ ℂ ↦ 𝑤) = ℂ)
52	46, 51	ineq12d 3777	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) = (ℂ ∩ ℂ))
53		inidm 3784	. . . . . . . . . . . 12 ⊢ (ℂ ∩ ℂ) = ℂ
54	53	a1i 11	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℂ ∩ ℂ) = ℂ)
55	52, 54	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) = ℂ)
56	55	mpteq1d 4666	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))) = (𝑦 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))))
57		fvconst2g 6372	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((ℂ × {𝐴})‘𝑦) = 𝐴)
58		eqidd 2611	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (𝑤 ∈ ℂ ↦ 𝑤) = (𝑤 ∈ ℂ ↦ 𝑤))
59		simpr 476	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℂ ∧ 𝑤 = 𝑦) → 𝑤 = 𝑦)
60		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → 𝑦 ∈ ℂ)
61	58, 59, 60, 60	fvmptd 6197	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦) = 𝑦)
62	61	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦) = 𝑦)
63	57, 62	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦)) = (𝐴 · 𝑦))
64	63	mpteq2dva 4672	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
65	56, 64	eqtrd 2644	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ (dom (ℂ × {𝐴}) ∩ dom (𝑤 ∈ ℂ ↦ 𝑤)) ↦ (((ℂ × {𝐴})‘𝑦) · ((𝑤 ∈ ℂ ↦ 𝑤)‘𝑦))) = (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
66	33, 43, 65	3eqtrrd 2649	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) = ((ℂ × {𝐴}) ∘𝑓 · (𝑦 ∈ ℂ ↦ 𝑦)))
67	66	oveq2d 6565	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (ℂ D ((ℂ × {𝐴}) ∘𝑓 · (𝑦 ∈ ℂ ↦ 𝑦))))
68		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ ℂ ↦ 𝑦) = (𝑦 ∈ ℂ ↦ 𝑦)
69	68, 60	fmpti 6291	. . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↦ 𝑦):ℂ⟶ℂ
70	69	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝑦):ℂ⟶ℂ)
71		id 22	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ)
72	22	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ℂ ∈ {ℝ, ℂ})
73	72	dvmptid 23526	. . . . . . . . . . 11 ⊢ (⊤ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
74	73	trud 1484	. . . . . . . . . 10 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1)
75	74	dmeqi 5247	. . . . . . . . 9 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = dom (𝑦 ∈ ℂ ↦ 1)
76		ax-1cn 9873	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
77	76	rgenw 2908	. . . . . . . . . . 11 ⊢ ∀𝑦 ∈ ℂ 1 ∈ ℂ
78		eqid 2610	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ ↦ 1) = (𝑦 ∈ ℂ ↦ 1)
79	78	fmpt 6289	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ ℂ 1 ∈ ℂ ↔ (𝑦 ∈ ℂ ↦ 1):ℂ⟶ℂ)
80	77, 79	mpbi 219	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ ↦ 1):ℂ⟶ℂ
81	80	fdmi 5965	. . . . . . . . 9 ⊢ dom (𝑦 ∈ ℂ ↦ 1) = ℂ
82	75, 81	eqtri 2632	. . . . . . . 8 ⊢ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = ℂ
83	82	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = ℂ)
84	23, 70, 71, 83	dvcmulf 23514	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D ((ℂ × {𝐴}) ∘𝑓 · (𝑦 ∈ ℂ ↦ 𝑦))) = ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))))
85	67, 84	eqtrd 2644	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))))
86	85	dmeqd 5248	. . . 4 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = dom ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))))
87		ovex 6577	. . . . . . 7 ⊢ (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) ∈ V
88	87	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) ∈ V)
89		offval3 7053	. . . . . 6 ⊢ (((ℂ × {𝐴}) ∈ V ∧ (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) ∈ V) → ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) = (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))))
90	39, 88, 89	syl2anc 691	. . . . 5 ⊢ (𝐴 ∈ ℂ → ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) = (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))))
91	90	dmeqd 5248	. . . 4 ⊢ (𝐴 ∈ ℂ → dom ((ℂ × {𝐴}) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) = dom (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))))
92	46, 83	ineq12d 3777	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) = (ℂ ∩ ℂ))
93	92, 54	eqtrd 2644	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) = ℂ)
94	93	mpteq1d 4666	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))) = (𝑤 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))))
95	94	dmeqd 5248	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))) = dom (𝑤 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))))
96		eqid 2610	. . . . . 6 ⊢ (𝑤 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))) = (𝑤 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤)))
97		fvconst2g 6372	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((ℂ × {𝐴})‘𝑤) = 𝐴)
98	74	fveq1i 6104	. . . . . . . . . . 11 ⊢ ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤) = ((𝑦 ∈ ℂ ↦ 1)‘𝑤)
99	98	a1i 11	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℂ → ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤) = ((𝑦 ∈ ℂ ↦ 1)‘𝑤))
100		eqidd 2611	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℂ → (𝑦 ∈ ℂ ↦ 1) = (𝑦 ∈ ℂ ↦ 1))
101		eqidd 2611	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℂ ∧ 𝑦 = 𝑤) → 1 = 1)
102	76	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ℂ → 1 ∈ ℂ)
103	100, 101, 48, 102	fvmptd 6197	. . . . . . . . . 10 ⊢ (𝑤 ∈ ℂ → ((𝑦 ∈ ℂ ↦ 1)‘𝑤) = 1)
104	99, 103	eqtrd 2644	. . . . . . . . 9 ⊢ (𝑤 ∈ ℂ → ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤) = 1)
105	104	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤) = 1)
106	97, 105	oveq12d 6567	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤)) = (𝐴 · 1))
107		mulcl 9899	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 · 1) ∈ ℂ)
108	76, 107	mpan2 703	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) ∈ ℂ)
109	108	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (𝐴 · 1) ∈ ℂ)
110	106, 109	eqeltrd 2688	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤)) ∈ ℂ)
111	96, 110	dmmptd 5937	. . . . 5 ⊢ (𝐴 ∈ ℂ → dom (𝑤 ∈ ℂ ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))) = ℂ)
112	95, 111	eqtrd 2644	. . . 4 ⊢ (𝐴 ∈ ℂ → dom (𝑤 ∈ (dom (ℂ × {𝐴}) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ 𝑦))) ↦ (((ℂ × {𝐴})‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ 𝑦))‘𝑤))) = ℂ)
113	86, 91, 112	3eqtrd 2648	. . 3 ⊢ (𝐴 ∈ ℂ → dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = ℂ)
114	23, 23, 2, 5, 29, 113	dvcof 23517	. 2 ⊢ (𝐴 ∈ ℂ → (ℂ D (sin ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))))
115	24	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D sin) = cos)
116		coscn 24003	. . . . . . 7 ⊢ cos ∈ (ℂ–cn→ℂ)
117	116	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℂ → cos ∈ (ℂ–cn→ℂ))
118	115, 117	eqeltrd 2688	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D sin) ∈ (ℂ–cn→ℂ))
119	34	mptex 6390	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ V
120	119	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ V)
121		coexg 7010	. . . . 5 ⊢ (((ℂ D sin) ∈ (ℂ–cn→ℂ) ∧ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∈ V) → ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∈ V)
122	118, 120, 121	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ ℂ → ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∈ V)
123		simpl 472	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝐴 ∈ ℂ)
124		0cnd 9912	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
125	23, 71	dvmptc 23527	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ 𝐴)) = (𝑦 ∈ ℂ ↦ 0))
126		simpr 476	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
127	76	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
128	74	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
129	23, 123, 124, 125, 126, 127, 128	dvmptmul 23530	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ ((0 · 𝑦) + (1 · 𝐴))))
130	126	mul02d 10113	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 · 𝑦) = 0)
131	123	mulid2d 9937	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (1 · 𝐴) = 𝐴)
132	130, 131	oveq12d 6567	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 · 𝑦) + (1 · 𝐴)) = (0 + 𝐴))
133	123	addid2d 10116	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0 + 𝐴) = 𝐴)
134	132, 133	eqtrd 2644	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((0 · 𝑦) + (1 · 𝐴)) = 𝐴)
135	134	mpteq2dva 4672	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ ((0 · 𝑦) + (1 · 𝐴))) = (𝑦 ∈ ℂ ↦ 𝐴))
136	129, 135	eqtrd 2644	. . . . 5 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ 𝐴))
137	34	mptex 6390	. . . . . 6 ⊢ (𝑦 ∈ ℂ ↦ 𝐴) ∈ V
138	137	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ 𝐴) ∈ V)
139	136, 138	eqeltrd 2688	. . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∈ V)
140		offval3 7053	. . . 4 ⊢ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∈ V ∧ (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∈ V) → (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = (𝑤 ∈ (dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))))
141	122, 139, 140	syl2anc 691	. . 3 ⊢ (𝐴 ∈ ℂ → (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = (𝑤 ∈ (dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))))
142		frn 5966	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)):ℂ⟶ℂ → ran (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ⊆ ℂ)
143	5, 142	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ran (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ⊆ ℂ)
144	143, 29	sseqtr4d 3605	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ran (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ⊆ dom (ℂ D sin))
145		dmcosseq 5308	. . . . . . . 8 ⊢ (ran (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ⊆ dom (ℂ D sin) → dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
146	144, 145	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
147		ovex 6577	. . . . . . . . 9 ⊢ (𝐴 · 𝑦) ∈ V
148	147, 4	dmmpti 5936	. . . . . . . 8 ⊢ dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) = ℂ
149	148	a1i 11	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) = ℂ)
150	146, 149	eqtrd 2644	. . . . . 6 ⊢ (𝐴 ∈ ℂ → dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = ℂ)
151	150, 113	ineq12d 3777	. . . . 5 ⊢ (𝐴 ∈ ℂ → (dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = (ℂ ∩ ℂ))
152	151, 54	eqtrd 2644	. . . 4 ⊢ (𝐴 ∈ ℂ → (dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = ℂ)
153	152	mpteq1d 4666	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ (dom ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∩ dom (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))) = (𝑤 ∈ ℂ ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))))
154	12	coscld 14700	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (cos‘(𝐴 · 𝑤)) ∈ ℂ)
155		simpl 472	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → 𝐴 ∈ ℂ)
156	154, 155	mulcomd 9940	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((cos‘(𝐴 · 𝑤)) · 𝐴) = (𝐴 · (cos‘(𝐴 · 𝑤))))
157	156	mpteq2dva 4672	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ ((cos‘(𝐴 · 𝑤)) · 𝐴)) = (𝑤 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑤)))))
158	24	coeq1i 5203	. . . . . . . . 9 ⊢ ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (cos ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
159	158	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (cos ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))))
160	159	fveq1d 6105	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) = ((cos ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))
161		ffun 5961	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)):ℂ⟶ℂ → Fun (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
162	5, 161	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → Fun (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
163	162	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → Fun (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
164	11, 148	syl6eleqr 2699	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → 𝑤 ∈ dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))
165		fvco 6184	. . . . . . . 8 ⊢ ((Fun (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)) ∧ 𝑤 ∈ dom (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) → ((cos ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) = (cos‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤)))
166	163, 164, 165	syl2anc 691	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((cos ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) = (cos‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤)))
167	13	fveq2d 6107	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (cos‘((𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))‘𝑤)) = (cos‘(𝐴 · 𝑤)))
168	160, 166, 167	3eqtrd 2648	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) = (cos‘(𝐴 · 𝑤)))
169	136	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) = (𝑦 ∈ ℂ ↦ 𝐴))
170		eqidd 2611	. . . . . . 7 ⊢ (((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐴)
171	169, 170, 11, 155	fvmptd 6197	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) = 𝐴)
172	168, 171	oveq12d 6567	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑤 ∈ ℂ) → ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤)) = ((cos‘(𝐴 · 𝑤)) · 𝐴))
173	172	mpteq2dva 4672	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))) = (𝑤 ∈ ℂ ↦ ((cos‘(𝐴 · 𝑤)) · 𝐴)))
174	9	fveq2d 6107	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (cos‘(𝐴 · 𝑦)) = (cos‘(𝐴 · 𝑤)))
175	174	oveq2d 6565	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝐴 · (cos‘(𝐴 · 𝑦))) = (𝐴 · (cos‘(𝐴 · 𝑤))))
176	175	cbvmptv 4678	. . . . 5 ⊢ (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) = (𝑤 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑤))))
177	176	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))) = (𝑤 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑤)))))
178	157, 173, 177	3eqtr4d 2654	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑤 ∈ ℂ ↦ ((((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤) · ((ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))‘𝑤))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
179	141, 153, 178	3eqtrd 2648	. 2 ⊢ (𝐴 ∈ ℂ → (((ℂ D sin) ∘ (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦))) ∘𝑓 · (ℂ D (𝑦 ∈ ℂ ↦ (𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))
180	21, 114, 179	3eqtrd 2648	1 ⊢ (𝐴 ∈ ℂ → (ℂ D (𝑦 ∈ ℂ ↦ (sin‘(𝐴 · 𝑦)))) = (𝑦 ∈ ℂ ↦ (𝐴 · (cos‘(𝐴 · 𝑦)))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {csn 4125  {cpr 4127   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  sincsin 14633  cosccos 14634  –cn→ccncf 22487   D cdv 23433

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-inf2 8421  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  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-rmo 2904  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-iin 4458  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-se 4998  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-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437

This theorem is referenced by:  dvasinbx  38810  itgcoscmulx  38861  dirkeritg  38995  dirkercncflem2  38997