Theorem stoweidlem22 44728

Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem22.8	⊢ Ⅎ𝑡𝜑
stoweidlem22.9	⊢ Ⅎ𝑡𝐹
stoweidlem22.10	⊢ Ⅎ𝑡𝐺
stoweidlem22.1	⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡)))
stoweidlem22.2	⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
stoweidlem22.3	⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
stoweidlem22.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem22.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem22.6	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem22.7	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)

Assertion

Ref	Expression
stoweidlem22	⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝐹,𝑔   𝑓,𝐺,𝑔   𝑓,𝐼,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑔,𝐿   𝑥,𝑡,𝐴   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐹(𝑥,𝑡)   𝐺(𝑥,𝑡)   𝐻(𝑥,𝑡,𝑓,𝑔)   𝐼(𝑥,𝑡)   𝐿(𝑥,𝑡,𝑓)

Proof of Theorem stoweidlem22

Step	Hyp	Ref	Expression
1		stoweidlem22.8	. . . 4 ⊢ Ⅎ𝑡𝜑
2		stoweidlem22.9	. . . . 5 ⊢ Ⅎ𝑡𝐹
3	2	nfel1 2919	. . . 4 ⊢ Ⅎ𝑡 𝐹 ∈ 𝐴
4		stoweidlem22.10	. . . . 5 ⊢ Ⅎ𝑡𝐺
5	4	nfel1 2919	. . . 4 ⊢ Ⅎ𝑡 𝐺 ∈ 𝐴
6	1, 3, 5	nf3an 1904	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)
7		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
8		simpl1 1191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
9		stoweidlem22.2	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
10		neg1rr 12326	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
11		stoweidlem22.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
12	11	stoweidlem4 44710	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
13	10, 12	mpan2 689	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
14	9, 13	eqeltrid 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
15		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐼 → (𝑓 ∈ 𝐴 ↔ 𝐼 ∈ 𝐴))
16	15	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐼 ∈ 𝐴)))
17		feq1 6698	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → (𝑓:𝑇⟶ℝ ↔ 𝐼:𝑇⟶ℝ))
18	16, 17	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐼 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)))
19		stoweidlem22.4	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
20	18, 19	vtoclg 3556	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝐴 → ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ))
21	20	anabsi7 669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)
22	8, 14, 21	syl2anc2 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐼:𝑇⟶ℝ)
23	22, 7	ffvelcdmd 7087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) ∈ ℝ)
24		simpl3 1193	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐺 ∈ 𝐴)
25		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
26	25	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
27		feq1 6698	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
28	26, 27	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)))
29	28, 19	vtoclg 3556	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
30	29	anabsi7 669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)
31	30	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝐺:𝑇⟶ℝ)
32		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
33	31, 32	ffvelcdmd 7087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
34	8, 24, 7, 33	syl3anc 1371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
35	23, 34	remulcld 11243	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ)
36		stoweidlem22.3	. . . . . . . 8 ⊢ 𝐿 = (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
37	36	fvmpt2 7009	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
38	7, 35, 37	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
39	9	fvmpt2 7009	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ -1 ∈ ℝ) → (𝐼‘𝑡) = -1)
40	10, 39	mpan2 689	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 → (𝐼‘𝑡) = -1)
41	40	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) = -1)
42	41	oveq1d 7423	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) = (-1 · (𝐺‘𝑡)))
43	34	recnd 11241	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
44	43	mulm1d 11665	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (-1 · (𝐺‘𝑡)) = -(𝐺‘𝑡))
45	38, 42, 44	3eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = -(𝐺‘𝑡))
46	45	oveq2d 7424	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐿‘𝑡)) = ((𝐹‘𝑡) + -(𝐺‘𝑡)))
47		simpl2 1192	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹 ∈ 𝐴)
48		eleq1 2821	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
49	48	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
50		feq1 6698	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
51	49, 50	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)))
52	51, 19	vtoclg 3556	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
53	52	anabsi7 669	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)
54	8, 47, 53	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹:𝑇⟶ℝ)
55	54, 7	ffvelcdmd 7087	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
56	55	recnd 11241	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
57	56, 43	negsubd 11576	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -(𝐺‘𝑡)) = ((𝐹‘𝑡) − (𝐺‘𝑡)))
58	46, 57	eqtr2d 2773	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝐺‘𝑡)) = ((𝐹‘𝑡) + (𝐿‘𝑡)))
59	6, 58	mpteq2da 5246	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))))
60	14	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐼 ∈ 𝐴)
61		nfmpt1 5256	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -1)
62	9, 61	nfcxfr 2901	. . . . . . 7 ⊢ Ⅎ𝑡𝐼
63	62	nfeq2 2920	. . . . . 6 ⊢ Ⅎ𝑡 𝑓 = 𝐼
64	4	nfeq2 2920	. . . . . 6 ⊢ Ⅎ𝑡 𝑔 = 𝐺
65		stoweidlem22.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
66	63, 64, 65	stoweidlem6 44712	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
67	60, 66	syld3an2 1411	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
68	36, 67	eqeltrid 2837	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐿 ∈ 𝐴)
69		stoweidlem22.5	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
70		nfmpt1 5256	. . . . 5 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
71	36, 70	nfcxfr 2901	. . . 4 ⊢ Ⅎ𝑡𝐿
72	69, 2, 71	stoweidlem8 44714	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐿 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
73	68, 72	syld3an3 1409	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
74	59, 73	eqeltrd 2833	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  Ⅎwnf 1785   ∈ wcel 2106  Ⅎwnfc 2883   ↦ cmpt 5231  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  ℝcr 11108  1c1 11110   + caddc 11112   · cmul 11114   − cmin 11443  -cneg 11444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-sub 11445  df-neg 11446

This theorem is referenced by:  stoweidlem33  44739