Theorem stoweidlem22 46662

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 2947	. . . 4 ⊢ Ⅎ𝑡 𝐹 ∈ 𝐴
4		stoweidlem22.10	. . . . 5 ⊢ Ⅎ𝑡𝐺
5	4	nfel1 2947	. . . 4 ⊢ Ⅎ𝑡 𝐺 ∈ 𝐴
6	1, 3, 5	nf3an 1928	. . 3 ⊢ Ⅎ𝑡(𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴)
7		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
8		simpl1 1208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
9		stoweidlem22.2	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
10		neg1rr 12204	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
11		stoweidlem22.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
12	11	stoweidlem4 46644	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
13	10, 12	mpan2 703	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
14	9, 13	eqeltrid 2873	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
15		eleq1 2857	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐼 → (𝑓 ∈ 𝐴 ↔ 𝐼 ∈ 𝐴))
16	15	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐼 ∈ 𝐴)))
17		feq1 6684	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → (𝑓:𝑇⟶ℝ ↔ 𝐼:𝑇⟶ℝ))
18	16, 17	imbi12d 347	. . . . . . . . . . . 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 3531	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝐴 → ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ))
21	20	anabsi7 683	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ)
22	8, 14, 21	syl2anc2 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐼:𝑇⟶ℝ)
23	22, 7	ffvelcdmd 7081	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) ∈ ℝ)
24		simpl3 1210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐺 ∈ 𝐴)
25		eleq1 2857	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
26	25	anbi2d 641	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
27		feq1 6684	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
28	26, 27	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)))
29	28, 19	vtoclg 3531	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
30	29	anabsi7 683	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)
31	30	3adant3 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝐺:𝑇⟶ℝ)
32		simp3 1154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
33	31, 32	ffvelcdmd 7081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
34	8, 24, 7, 33	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
35	23, 34	remulcld 11239	. . . . . . 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 7002	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐼‘𝑡) · (𝐺‘𝑡)) ∈ ℝ) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
38	7, 35, 37	syl2anc 595	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = ((𝐼‘𝑡) · (𝐺‘𝑡)))
39	9	fvmpt2 7002	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ -1 ∈ ℝ) → (𝐼‘𝑡) = -1)
40	10, 39	mpan2 703	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 → (𝐼‘𝑡) = -1)
41	40	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) = -1)
42	41	oveq1d 7426	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) = (-1 · (𝐺‘𝑡)))
43	34	recnd 11237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
44	43	mulm1d 11666	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (-1 · (𝐺‘𝑡)) = -(𝐺‘𝑡))
45	38, 42, 44	3eqtrd 2808	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = -(𝐺‘𝑡))
46	45	oveq2d 7427	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐿‘𝑡)) = ((𝐹‘𝑡) + -(𝐺‘𝑡)))
47		simpl2 1209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹 ∈ 𝐴)
48		eleq1 2857	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
49	48	anbi2d 641	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
50		feq1 6684	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
51	49, 50	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)))
52	51, 19	vtoclg 3531	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
53	52	anabsi7 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)
54	8, 47, 53	syl2anc 595	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹:𝑇⟶ℝ)
55	54, 7	ffvelcdmd 7081	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
56	55	recnd 11237	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
57	56, 43	negsubd 11575	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -(𝐺‘𝑡)) = ((𝐹‘𝑡) − (𝐺‘𝑡)))
58	46, 57	eqtr2d 2805	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝐺‘𝑡)) = ((𝐹‘𝑡) + (𝐿‘𝑡)))
59	6, 58	mpteq2da 5207	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))))
60	14	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐼 ∈ 𝐴)
61		nfmpt1 5214	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -1)
62	9, 61	nfcxfr 2929	. . . . . . 7 ⊢ Ⅎ𝑡𝐼
63	62	nfeq2 2948	. . . . . 6 ⊢ Ⅎ𝑡 𝑓 = 𝐼
64	4	nfeq2 2948	. . . . . 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 46646	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
67	60, 66	syld3an2 1436	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
68	36, 67	eqeltrid 2873	. . 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 5214	. . . . 5 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
71	36, 70	nfcxfr 2929	. . . 4 ⊢ Ⅎ𝑡𝐿
72	69, 2, 71	stoweidlem8 46648	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐿 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
73	68, 72	syld3an3 1434	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
74	59, 73	eqeltrd 2869	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916   ↦ cmpt 5196  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℝcr 11099  1c1 11101   + caddc 11103   · cmul 11105   − cmin 11441  -cneg 11442

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-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176

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-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-po 5570  df-so 5571  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-ltxr 11248  df-sub 11443  df-neg 11444

This theorem is referenced by:  stoweidlem33  46673