Theorem stoweidlem22 43563

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 2923	. . . 4 ⊢ Ⅎ𝑡 𝐹 ∈ 𝐴
4		stoweidlem22.10	. . . . 5 ⊢ Ⅎ𝑡𝐺
5	4	nfel1 2923	. . . 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 1190	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
9		stoweidlem22.2	. . . . . . . . . . 11 ⊢ 𝐼 = (𝑡 ∈ 𝑇 ↦ -1)
10		neg1rr 12088	. . . . . . . . . . . 12 ⊢ -1 ∈ ℝ
11		stoweidlem22.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
12	11	stoweidlem4 43545	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
13	10, 12	mpan2 688	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ -1) ∈ 𝐴)
14	9, 13	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ 𝐴)
15		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐼 → (𝑓 ∈ 𝐴 ↔ 𝐼 ∈ 𝐴))
16	15	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐼 ∈ 𝐴)))
17		feq1 6581	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐼 → (𝑓:𝑇⟶ℝ ↔ 𝐼:𝑇⟶ℝ))
18	16, 17	imbi12d 345	. . . . . . . . . . . 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 3505	. . . . . . . . . . 11 ⊢ (𝐼 ∈ 𝐴 → ((𝜑 ∧ 𝐼 ∈ 𝐴) → 𝐼:𝑇⟶ℝ))
21	20	anabsi7 668	. . . . . . . . . 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	ffvelrnd 6962	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) ∈ ℝ)
24		simpl3 1192	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐺 ∈ 𝐴)
25		eleq1 2826	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑓 ∈ 𝐴 ↔ 𝐺 ∈ 𝐴))
26	25	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐺 ∈ 𝐴)))
27		feq1 6581	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐺 → (𝑓:𝑇⟶ℝ ↔ 𝐺:𝑇⟶ℝ))
28	26, 27	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐺 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)))
29	28, 19	vtoclg 3505	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ))
30	29	anabsi7 668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴) → 𝐺:𝑇⟶ℝ)
31	30	3adant3 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝐺:𝑇⟶ℝ)
32		simp3 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
33	31, 32	ffvelrnd 6962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝐴 ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
34	8, 24, 7, 33	syl3anc 1370	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℝ)
35	23, 34	remulcld 11005	. . . . . . 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 6886	. . . . . . 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 6886	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑇 ∧ -1 ∈ ℝ) → (𝐼‘𝑡) = -1)
40	10, 39	mpan2 688	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 → (𝐼‘𝑡) = -1)
41	40	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐼‘𝑡) = -1)
42	41	oveq1d 7290	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐼‘𝑡) · (𝐺‘𝑡)) = (-1 · (𝐺‘𝑡)))
43	34	recnd 11003	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐺‘𝑡) ∈ ℂ)
44	43	mulm1d 11427	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (-1 · (𝐺‘𝑡)) = -(𝐺‘𝑡))
45	38, 42, 44	3eqtrd 2782	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐿‘𝑡) = -(𝐺‘𝑡))
46	45	oveq2d 7291	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + (𝐿‘𝑡)) = ((𝐹‘𝑡) + -(𝐺‘𝑡)))
47		simpl2 1191	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝐹 ∈ 𝐴)
48		eleq1 2826	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
49	48	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
50		feq1 6581	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
51	49, 50	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)))
52	51, 19	vtoclg 3505	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
53	52	anabsi7 668	. . . . . . . 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	ffvelrnd 6962	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
56	55	recnd 11003	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
57	56, 43	negsubd 11338	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) + -(𝐺‘𝑡)) = ((𝐹‘𝑡) − (𝐺‘𝑡)))
58	46, 57	eqtr2d 2779	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝐺‘𝑡)) = ((𝐹‘𝑡) + (𝐿‘𝑡)))
59	6, 58	mpteq2da 5172	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))))
60	14	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → 𝐼 ∈ 𝐴)
61		nfmpt1 5182	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ -1)
62	9, 61	nfcxfr 2905	. . . . . . 7 ⊢ Ⅎ𝑡𝐼
63	62	nfeq2 2924	. . . . . 6 ⊢ Ⅎ𝑡 𝑓 = 𝐼
64	4	nfeq2 2924	. . . . . 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 43547	. . . . 5 ⊢ ((𝜑 ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
67	60, 66	syld3an2 1410	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡))) ∈ 𝐴)
68	36, 67	eqeltrid 2843	. . 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 5182	. . . . 5 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐼‘𝑡) · (𝐺‘𝑡)))
71	36, 70	nfcxfr 2905	. . . 4 ⊢ Ⅎ𝑡𝐿
72	69, 2, 71	stoweidlem8 43549	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐿 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
73	68, 72	syld3an3 1408	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) + (𝐿‘𝑡))) ∈ 𝐴)
74	59, 73	eqeltrd 2839	1 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝐺‘𝑡))) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℝcr 10870  1c1 10872   + caddc 10874   · cmul 10876   − cmin 11205  -cneg 11206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-ltxr 11014  df-sub 11207  df-neg 11208

This theorem is referenced by:  stoweidlem33  43574